1 Introduction
In the study of certain questions in mathematics, there are often two extreme cases that come to mind instantly.
For example, the subgroups of a certain group can either be “small” (e.g. finite/virtually abelian/virtually solvable) or “large” (e.g. contain a non-abelian free group/have a subgroup of finite index with a non-abelian free quotient) or anything in between.
Sometimes, when the object in question is particularly well-behaved, only the two extreme cases can happen.
In group theory, the above dichotomy is called the Tits alternative.
This name is derived from the result of Tits who showed that a finitely generated linear group is virtually solvable or contains a non-abelian free group as a subgroup [28].
1.1 Normal subgroups in automorphism groups of right-angled Artin groups
Following the notion of [4], a group 𝐺 is called full-sized if it contains a non-abelian free group
F
2
↪
G
. Here we are interested in the structure of non-full-sized normal subgroups in the automorphism group of a group that is defined by a graph.
Given a finite simplicial graph Γ with vertex set
V
(
Γ
)
and edge set
E
(
Γ
)
, the right-angled Artin group
A
Γ
is defined as follows:
A
Γ
:
=
⟨
V
(
Γ
)
∣
v
w
=
w
v
whenever
{
v
,
w
}
∈
E
(
Γ
)
⟩
.
For instance, if Γ is a clique, then
A
Γ
is the free abelian group
Z
|
V
(
Γ
)
|
, and if Γ has no edges, then
A
Γ
is the free group of rank
|
V
(
Γ
)
|
.
The starting point for our investigations on normal subgroups in
Aut
(
A
Γ
)
is the well-studied group
Aut
(
Z
n
)
≅
GL
n
(
Z
)
.
Given a normal subgroup
N
⊴
GL
n
(
Z
)
, it is known that 𝑁 is finite or full-sized; see [23] and Lemma 2.16.
Since
GL
n
(
Z
)
shares many properties with the automorphism group of a free group and to some extent also with the automorphism group of an arbitrary right-angled Artin group, our goal here is to understand the structure of non-full-sized normal subgroups in
Aut
(
A
Γ
)
.
It was proven by Horbez in [18] that a non-full-sized subgroup in
Aut
(
A
Γ
)
is virtually solvable.
Thus we are particularly interested in the structure of virtually solvable normal subgroups in
Aut
(
A
Γ
)
.
We start by considering normal subgroups in the automorphism group of a centerless right-angled Artin group.
Proposition A
Let
Aut
(
A
Γ
)
be the automorphism group of a centerless right-angled Artin group
A
Γ
and let
N
⊴
Aut
(
A
Γ
)
be a normal subgroup.
If 𝑁 is non-trivial, then 𝑁 is full-sized.
Given a right-angled Artin group
A
Γ
, the center of
A
Γ
is a direct factor of
A
Γ
and is itself a right-angled Artin group given by the induced subgraph Δ whose vertices 𝑣 are precisely those satisfying
st
(
v
)
=
V
(
Γ
)
, where
st
(
v
)
:
=
{
w
∈
V
(
Γ
)
|
{
v
,
w
}
∈
E
(
Γ
)
}
∪
{
v
}
.
Hence many right-angled Artin groups have non-trivial centers.
We denote the vertices in
V
(
Δ
)
by
{
v
1
,
…
,
v
n
}
and those in
V
(
Γ
)
−
V
(
Δ
)
by
{
w
1
,
…
,
w
m
}
.
Let us discuss the center and some other properties of the right-angled Artin group defined by the graph Γ in Figure 1.
The center of
A
Γ
is generated by
v
1
and is a direct factor of
A
Γ
; thus
A
Γ
≅
⟨
v
1
⟩
×
⟨
w
1
,
w
2
,
w
3
,
w
4
⟩
.
For
w
i
∈
V
(
Γ
)
−
V
(
Δ
)
, we define an automorphism
ρ
w
i
v
1
∈
Aut
(
A
Γ
)
as follows:
ρ
w
i
v
1
(
w
i
)
=
w
i
v
1
,
ρ
w
i
v
1
(
w
j
)
=
w
j
,
i
≠
j
and
ρ
w
i
v
1
(
v
1
)
=
v
1
.
One can show that the subgroup
⟨
ρ
w
i
v
1
∣
i
=
1
,
…
,
4
⟩
⊆
Aut
(
A
Γ
)
is normal and it is isomorphic to
Z
4
.
This normal subgroup is not maximal under all non-full-sized normal subgroups in
Aut
(
A
Γ
)
since the subgroup
⟨
ρ
w
i
v
1
,
ι
∣
i
=
1
,
…
,
4
⟩
, where
ι
(
v
1
)
=
v
1
−
1
and
ι
(
w
i
)
=
w
i
for
i
=
1
,
…
,
4
, is also normal.
Furthermore, this normal subgroup is isomorphic to
Z
4
⋊
Z
/
2
Z
and it is indeed maximal under all non-full-sized normal subgroups in
Aut
(
A
Γ
)
.
We show that a maximal non-full-sized normal subgroup in the automorphism group of a general right-angled Artin group is unique.
Theorem B
Let
Aut
(
A
Γ
)
be the automorphism group of a right-angled Artin group
A
Γ
and let
N
⊴
Aut
(
A
Γ
)
be a non-trivial maximal (with respect to inclusion) normal subgroup.
If 𝑁 is not full-sized, then either
V
(
Γ
)
−
V
(
Δ
)
=
∅
and
N
=
{
id
,
ι
}
, where
ι
(
v
i
)
=
v
i
−
1
for all
v
i
∈
V
(
Δ
)
, or
V
(
Δ
)
=
{
v
1
,
…
,
v
n
}
,
V
(
Γ
)
−
V
(
Δ
)
=
{
w
1
,
…
,
w
m
}
for
n
,
m
≥
1
and
N
≅
T
Δ
⋊
{
id
,
ι
}
≅
Z
n
⋅
m
⋊
Z
/
2
Z
,
where
T
Δ
:=
{
f
∈
Aut
(
A
Γ
)
∣
w
i
−
1
f
(
w
i
)
∈
A
Δ
and
f
(
v
j
)
=
v
j
for all
i
=
1
,
…
,
m
and
j
=
1
,
…
,
n
}
.
Furthermore, let
N
⊴
Aut
(
A
Γ
)
be a non-trivial minimal non-full-sized normal subgroup and set
n
:
=
|
V
(
Δ
)
|
and
m
:
=
|
V
(
Γ
)
−
V
(
Δ
)
|
.
If
n
,
m
≥
1
, then
N
≅
Z
l
for some
n
≤
l
≤
n
⋅
m
.
We have the following immediate corollary.
Corollary C
The automorphism group of a right-angled Artin group
Aut
(
A
Γ
)
has non-trivial finite normal subgroups if and only if Γ is a clique.
It is a known fact that a centerless group has a centerless automorphism group.
The center of a right-angled Artin group can be easily read off the defining graph Γ as discussed above.
Thus there are many right-angled Artin groups with non-trivial center.
Nevertheless, we show that the automorphism group of a right-angled Artin group is in most cases centerless.
Corollary D
The automorphism group of a right-angled Artin group
Aut
(
A
Γ
)
is centerless if and only if Γ is not a clique.
The center of the automorphism group of a right-angled Artin group was also determined in [12], but the proof of [12, Proposition 5.1] contains a small gap.
Structure of the proofs of Proposition A and Theorem B
The crucial ingredient in the proof of Proposition A is a result by Baudisch which says that a two generated subgroup in
A
Γ
is abelian or isomorphic to
F
2
(see [2, Theorem 1.2]).
This result is used to prove that a normal subgroup in
A
Γ
is contained in the center or is full-sized.
Using the subgroup consisting of inner automorphisms of
A
Γ
and the characterization of non-full-sized normal subgroups in
A
Γ
, we are able to prove Proposition A.
For the proof of Theorem B, we use Proposition A and the following short exact sequence which is given by the fact that
A
Δ
is a characteristic subgroup:
{
id
}
→
T
Δ
→
Aut
(
A
Γ
)
→
Aut
(
A
Δ
)
×
Aut
(
A
Γ
−
Δ
)
→
{
id
}
.
1.2 Normal subgroups in automorphism groups of graph products
Right-angled Artin groups are closely related to graph products of groups.
Given a finite simplicial graph Γ and a collection of non-trivial groups
{
G
u
∣
u
∈
V
(
Γ
)
}
indexed by the vertex-set
V
(
Γ
)
of Γ, the graph product
G
Γ
is defined as the quotient
(
∗
u
∈
V
(
Γ
)
G
u
)
/
⟨
⟨
[
g
,
h
]
=
1
,
g
∈
G
u
,
h
∈
G
v
,
{
u
,
v
}
∈
E
(
Γ
)
⟩
⟩
,
where
E
(
Γ
)
denotes the edge set of Γ.
If all vertex groups are infinite cyclic, then
G
Γ
is a right-angled Artin group, and if all vertex groups are cyclic of order two, then
G
Γ
is called a right-angled Coxeter group and will be denoted by
W
Γ
.
A natural question is if the results of Proposition A and Theorem B also hold for graph products of cyclic groups.
Let us focus on the result of Proposition A that tells us that a non-trivial normal subgroup in the automorphism group of a centerless right-angled Artin group is always full-sized.
Consider the centerless right-angled Coxeter group
Z
/
2
Z
∗
Z
/
2
Z
and its automorphism group.
The group of inner automorphisms in
Aut
(
Z
/
2
Z
∗
Z
/
2
Z
)
is isomorphic to
Z
/
2
Z
∗
Z
/
2
Z
.
Hence it follows that the group
Aut
(
Z
/
2
Z
∗
Z
/
2
Z
)
has a normal subgroup isomorphic to
Z
/
2
Z
∗
Z
/
2
Z
and this group is obviously not full-sized.
In the centerless case, this is however the only way for non-trivial non-full-sized normal subgroups to exist.
Proposition E
Proposition E (see Propositions 2.15 and 3.5)
Let
G
Γ
be a graph product of non-trivial groups.
We decompose
G
Γ
≅
G
Γ
1
×
⋯
×
G
Γ
n
as a direct product, where the factors are directly indecomposable special parabolic subgroups.
Assume that
G
Γ
i
≇
Z
/
2
Z
∗
Z
/
2
Z
for all
i
=
1
,
…
,
n
.
If
|
V
(
Γ
i
)
|
≥
2
for
i
=
1
,
…
,
n
, then a non-trivial normal subgroup in
Aut
(
G
Γ
)
is full-sized.
In particular, if
G
Γ
is a centerless graph product of abelian groups, then a non-trivial normal subgroup in
Aut
(
G
Γ
)
is full-sized.
If all vertex groups are finite, then a normal subgroup in
Aut
(
G
Γ
)
is finite or full-sized.
In the special case where the graph product is a right-angled Coxeter group, we prove that if one of the factors in the direct decomposition of
W
Γ
is isomorphic to the infinite dihedral group, then
Aut
(
W
Γ
)
has an infinite virtually abelian normal subgroup.
More precisely, we define
J
:
=
{
j
∈
{
1
,
…
,
n
}
∣
W
Γ
j
≅
Z
/
2
Z
∗
Z
/
2
Z
}
and we show that
∏
j
∈
J
W
Γ
j
as a subgroup of
Inn
(
W
Γ
)
is normal in
Aut
(
W
Γ
)
.
Thus the infinite dihedral group is a poison subgroup when it comes to the non-existence of infinite non-full-sized normal subgroups in
Aut
(
W
Γ
)
as the following corollary shows.
Corollary F
Let
W
Γ
be a right-angled Coxeter group.
We decompose
W
Γ
≅
W
Γ
1
×
⋯
×
W
Γ
n
as a direct product, where the factors are directly indecomposable special parabolic subgroups.
There exists an infinite virtually abelian normal subgroup in
Aut
(
W
Γ
)
if and only if there exists an
i
∈
{
1
,
…
,
n
}
such that
W
Γ
i
≅
Z
/
2
Z
∗
Z
/
2
Z
.
Otherwise, every normal subgroup in
Aut
(
W
Γ
)
is either finite or full-sized.
Now, for a graph product of arbitrary cyclic groups, we are interested in conditions on Γ such that
Aut
(
G
Γ
)
has no non-trivial finite normal subgroups.
For this, it is easier to “blow up” the graph product a bit, by replacing the vertex whose associated vertex group is
Z
/
n
Z
, where
n
=
p
1
m
1
⋅
…
⋅
p
k
m
k
, by a complete graph on 𝑘 vertices with vertex groups isomorphic to
Z
/
(
p
i
m
i
Z
)
for
1
≤
i
≤
k
.
This yields isomorphic graph products, but allows us to assume that the vertex groups are infinite cyclic or have prime power order.
We prove the following theorem.
Theorem G
Let
G
Γ
be a graph product of cyclic groups, where the finite vertex groups have prime power orders.
Let Δ be the induced subgraph of Γ generated by the vertices
v
∈
V
(
Γ
)
such that
st
(
v
)
=
V
(
Γ
)
.
The automorphism group
Aut
(
G
Γ
)
has no non-trivial finite normal subgroup if and only if one of the following conditions holds:
V
(
Δ
)
=
{
v
1
}
,
G
v
1
≅
Z
/
2
Z
,
V
(
Γ
)
−
V
(
Δ
)
=
{
w
1
,
…
,
w
m
}
,
m
≥
1
and
ord
(
G
w
j
)
<
∞
and
2
∤
ord
(
G
w
j
)
for all
1
≤
j
≤
m
or
for every
v
∈
V
(
Δ
)
, we have
G
v
≅
Z
and there exists a
w
∈
V
(
Γ
)
−
V
(
Δ
)
such that
G
w
≅
Z
.
Structure of the proofs of Proposition E and Theorem G
First we give a description of non-full-sized normal subgroups of a graph product of arbitrary vertex groups; see Proposition 2.8 and Corollary 2.9.
Using several characteristic subgroups of
G
Γ
and the induced group homomorphisms, we prove Proposition E and use this result to show Theorem G.
1.3 The center of the automorphism group of a graph product
Methods similar to the ones used for the proof of Theorem G allow us to precisely compute the center of the automorphism group of a graph product of cyclic groups in many situations.
Since the center is heavily dependent on the structure of the graph, one has to distinguish between many cases.
First we investigate the center of the automorphism group of a finitely generated abelian group and collect known and new results in Propositions 7.2 and 7.9.
A description of the center of
Aut
(
G
Γ
)
where Γ is not a clique is more complicated; see Proposition 7.13.
It is known that the center of a Coxeter group is a finite abelian 2-group.
We prove that the center of the automorphism group of a right-angled Coxeter group is in most cases trivial, and if the center is non-trivial, then it is also a finite abelian 2-group.
Corollary H
Corollary H (see Corollary 7.15)
Let
W
Γ
be a right-angled Coxeter group.
Let Δ be the induced subgraph of Γ generated by the vertices
v
∈
V
(
Γ
)
such that
st
(
v
)
=
V
(
Γ
)
.
Furthermore, we denote the vertices of Δ by
V
(
Δ
)
=
{
v
1
,
…
,
v
n
}
and
V
(
Γ
)
−
V
(
Δ
)
=
{
w
1
,
…
,
w
m
}
.
The center
Z
(
Aut
(
W
Γ
)
)
is non-trivial if and only if
n
=
1
,
m
≥
1
and there exists a vertex
w
j
∈
V
(
Γ
)
−
V
(
Δ
)
such that
st
(
w
j
)
⊈
st
(
w
i
)
and
st
(
w
i
)
⊈
st
(
w
j
)
for all
i
∈
{
1
,
…
,
m
}
,
i
≠
j
.
Moreover, let Ω be a subset of
V
(
Γ
)
−
V
(
Δ
)
defined as follows:
Ω
:=
{
w
j
|
st
(
w
j
)
⊈
st
(
w
i
)
and
st
(
w
i
)
⊈
st
(
w
j
)
for all
i
∈
{
1
,
…
,
m
}
,
i
≠
j
}
.
Then Ω is preserved under the action of
Isom
(
Γ
−
Δ
)
and
Z
(
Aut
(
W
Γ
)
)
≅
(
Z
/
2
Z
)
l
,
where 𝑙 is equal to the cardinality of the set of orbits under this action.
1.4 Automatic continuity
The results of Theorem G have an application to automatic continuity.
The aim of the concept of automatic continuity is to bring together two areas of mathematics: locally compact Hausdorff groups and discrete groups.
The main question for us in this direction is the following.
Question
Is every algebraic epimorphism
φ
:
L
↠
Aut
(
G
Γ
)
from a locally compact Hausdorff group 𝐿 to the discrete group
Aut
(
G
Γ
)
continuous?
A direct consequence of [20, Theorem B] is that any algebraic epimorphism
φ
:
L
↠
G
Γ
from a locally compact Hausdorff group 𝐿 into a graph product of cyclic groups
G
Γ
is continuous if the diameter of Γ is at least 3.
Using [19, Theorem B], we obtain a similar result where the target group is the automorphism group of a graph product of cyclic groups.
Corollary I
Corollary I (see Corollaries 8.7 and 8.8)
Let
φ
:
L
↠
Aut
(
G
Γ
)
be an algebraic epimorphism from a locally compact Hausdorff group 𝐿 to the automorphism group of a graph product of cyclic groups
G
Γ
.
If
Aut
(
G
Γ
)
has no non-trivial finite normal subgroups, then 𝜑 is continuous.
In particular, if
V
(
Δ
)
=
∅
or
V
(
Δ
)
and
V
(
Γ
)
−
V
(
Δ
)
are both non-empty and all vertex groups in
V
(
Δ
)
are infinite cyclic and there exists an infinite vertex group in
V
(
Γ
)
−
V
(
Δ
)
, then 𝜑 is continuous.
Moreover, any algebraic epimorphism from a locally compact Hausdorff group to
Aut
(
A
Γ
)
is continuous if and only if Γ is not a clique.
1.5 The stable rank of the reduced group
C
∗
-algebra of the automorphism group of a graph product
Associated to a group 𝐺, there is an associated reduced group
C
∗
-algebra
C
r
∗
(
G
)
.
We say that
C
r
∗
(
G
)
has stable rank1 if the set of invertible elements is dense in
C
r
∗
(
G
)
.
Additional information about this in the context of acylindrically hyperbolic groups can be found in [14].
Proposition E and Theorem G have an application to
C
r
∗
(
Aut
(
G
Γ
)
)
-algebras; more precisely, using work of Gerasimova–Osin [14] and Genevois [13], we obtain the following corollary.
Corollary J
Let
G
Γ
be a graph product of finitely generated groups.
If Γ is not a join, contains at least two vertices and
G
Γ
≇
Z
/
2
Z
∗
Z
/
2
Z
, then the stable rank of the reduced group
C
∗
-algebra of
Aut
(
G
Γ
)
is equal to 1.
In particular, if
G
Γ
is a right-angled Artin group and Γ is not a join and contains at least two vertices, then the stable rank of the reduced group
C
∗
-algebra of
Aut
(
G
Γ
)
is equal to 1.
Structure of the paper
In Section 2, we review some of the standard facts on graph products of groups
G
Γ
.
It is natural to try to relate combinatorial properties of the defining graph Γ and algebraic properties of the graph product
G
Γ
.
We discuss these relations focusing on the question when a special subgroup of a graph product can be normal and when a graph product of groups can have a non-trivial non-full-sized normal subgroup.
These results allow us to prove Proposition A and Proposition E (i).
Section 3 contains a brief summary of characteristic subgroups and some examples.
In particular, we investigate the special subgroup
G
Δ
of
G
Γ
, where
V
(
Δ
)
=
{
v
∈
V
(
Γ
)
∣
st
(
v
)
=
V
(
Γ
)
}
,
and give a condition on the vertex groups so that
G
Δ
is characteristic.
We also consider a canonical map induced by the fact that
G
Δ
is characteristic and compute the kernel of this map which leads to the proof of Proposition E (ii).
We start Section 4 with a brief exposition of different types of automorphisms of a graph product whose vertex groups are cyclic.
This section also contains a proof for the fact that torsion subgroups in
Aut
(
G
Γ
)
are finite and
Aut
(
G
Γ
)
contains no subgroup isomorphic to ℚ.
These two properties are important for the application to automatic continuity.
Section 5 contains the proof of Theorem B and the proofs of Corollaries C and D.
Section 6 is dedicated to the proof of Theorem G.
We start Section 7 with some known facts about the center of the automorphism group and move on to the calculation of the center of the automorphism group of a graph product of cyclic groups.
The last section, Section 8, provides applications of Theorem G to automatic continuity and to reduced group
C
∗
-algebras of automorphism groups of graph products.
2 Graph products of groups
In this section, we first recall the basics of graph products of groups.
This includes their definition, parabolic subgroups and results about subgroups of directly indecomposable graph products.
We define what the term full-sized means and then study normal subgroups of graph products and of their automorphism groups with regard to being finite or full-sized.
To finish off the section, we prove Proposition A.
In group theory, it is common to build new groups out of given ones using free or direct product constructions.
Furthermore, we can also use amalgamation to obtain a new group out of given ones.
For example,
Z
/
4
Z
∗
Z
/
6
Z
is an infinite group,
Z
/
4
Z
×
Z
/
6
Z
is a finite abelian group and
Z
/
4
Z
∗
Z
/
2
Z
Z
/
6
Z
≅
SL
2
(
Z
)
.
Graph products generalize free/direct products and special kind of amalgamations.
We start by recalling the definition of a graph product of groups.
Definition 2.1
Given a finite simplicial graph Γ and a collection of groups
G
=
{
G
u
∣
u
∈
V
(
Γ
)
}
indexed by the vertex-set
V
(
Γ
)
of Γ, the graph product
G
Γ
is defined as the quotient
(
∗
u
∈
V
(
Γ
)
G
u
)
/
⟨
⟨
[
g
,
h
]
=
1
,
g
∈
G
u
,
h
∈
G
v
,
{
u
,
v
}
∈
E
(
Γ
)
⟩
⟩
,
where
E
(
Γ
)
denotes the edge set of Γ.
Convention
All vertex groups are always non-trivial.
If a vertex group
G
v
is cyclic for a vertex 𝑣, we also write 𝑣 for a generator of
G
v
by slight abuse of notation.
Graph products of groups were introduced and studied by Green in her PhD thesis [15].
If the vertex groups are all cyclic of the same type, then it is common to use another name for the graph product.
Definition 2.2
Let
G
Γ
be a graph product.
If all vertex groups are infinite cyclic, then
G
Γ
is called a right-angled Artin group which we denote by
A
Γ
.
If all vertex groups are cyclic of order 2, then
G
Γ
is called a right-angled Coxeter group which we denote by
W
Γ
.
Given a simplicial graph Γ and a vertex
v
∈
V
(
Γ
)
, we define the link of 𝑣 as
lk
(
v
)
:
=
{
w
∈
V
(
Γ
)
∣
{
v
,
w
}
∈
E
(
Γ
)
}
and the star of 𝑣 as
st
(
v
)
:
=
{
v
}
∪
lk
(
v
)
.
The valency of 𝑣 is defined as
val
(
v
)
:
=
|
{
w
∈
V
(
Γ
)
∣
{
v
,
w
}
∈
E
(
Γ
)
}
|
.
We denote by
(
V
2
)
the set of all subsets of 𝑉 with cardinality 2.
A subgraph
Ω
⊆
Γ
is called induced if, for every
{
v
,
w
}
∈
(
V
2
)
, we have
{
v
,
w
}
∈
E
(
Ω
)
if and only if
{
v
,
w
}
∈
E
(
Γ
)
.
Given a graph product
G
Γ
and an induced subgraph
Ω
⊆
Γ
, the subgroup
⟨
G
v
∣
v
∈
V
(
Ω
)
⟩
⊆
G
Γ
is canonically isomorphic to
G
Ω
(see [15, Lemma 3.20]) and is called a special parabolic subgroup.
Furthermore, a conjugate of a special parabolic subgroup is called a parabolic subgroup.
By definition, a graph product
G
Γ
is called directly decomposable if
G
Γ
is a direct product of proper special parabolic subgroups of
G
Γ
.
For example, if Γ is a cycle of length
n
≥
3
, then
G
Γ
is directly decomposable if and only if
n
=
3
or
n
=
4
.
Furthermore, a graph Γ is a join if there exists a partition
V
(
Γ
)
=
A
∪
B
, where 𝐴 and 𝐵 are both non-empty, such that, for every
(
a
,
b
)
∈
A
×
B
, we have
{
a
,
b
}
∈
E
(
Γ
)
.
If Γ is a join, then we write
Γ
=
Γ
1
∗
Γ
2
, where
Γ
1
is the induced subgraph by 𝐴 and
Γ
2
is the induced subgraph by 𝐵.
Hence
G
Γ
is directly decomposable if and only if Γ is a join and then
G
Γ
≅
G
Γ
1
×
G
Γ
2
.
Since the main focus of this article is on normal subgroups, we ask the following question.
Let
G
Γ
be a graph product of arbitrary groups and
G
Ω
a special parabolic subgroup.
Under which combinatorial conditions on the graph Γ is
G
Ω
normal in
G
Γ
?
The answer to this question was given by Antolin and Minasyan in [1].
They showed that
G
Ω
is normal in
G
Γ
if and only if
G
Ω
is a direct factor in a direct decomposition of
G
Γ
in special parabolic subgroups (see [1, Proposition 3.13]).
In particular,
G
Ω
⊴
G
Γ
if and only if
V
(
Ω
)
⊆
lk
(
v
)
for all
v
∈
V
(
Γ
)
−
V
(
Ω
)
.
Given a non-trivial subgroup
H
⊆
G
Γ
, one can consider parabolic subgroups
g
G
Ω
g
−
1
that contain 𝐻 as a subgroup.
The natural question regarding these parabolic subgroups is if their intersection is again parabolic.
Proposition 2.3
Proposition 2.3 ([1, Proposition 3.10])
Let
G
Γ
be a graph product.
For every non-trivial subgroup
H
⊆
G
Γ
, there exists a unique minimal (with respect to inclusion) parabolic subgroup
g
G
Ω
g
−
1
such that
H
⊆
g
G
Ω
g
−
1
.
This parabolic subgroup is called the parabolic closure of 𝐻 and is denoted by
Pc
(
H
)
.
Let us consider the right-angled Coxeter group
W
Γ
defined by the graph Γ in Figure 2.
We have
W
Γ
≅
(
⟨
v
1
⟩
∗
⟨
v
3
⟩
)
×
⟨
v
2
⟩
≅
(
Z
/
2
Z
∗
Z
/
2
Z
)
×
Z
/
2
Z
.
Since the infinite cyclic subgroup
⟨
v
1
v
3
⟩
is normal in
⟨
v
1
⟩
∗
⟨
v
3
⟩
, we know that
⟨
v
1
v
3
⟩
is also normal in the entire Coxeter group
W
Γ
.
We note also that
Pc
(
⟨
v
1
v
3
⟩
)
=
⟨
v
1
⟩
∗
⟨
v
3
⟩
and furthermore
Pc
(
⟨
v
1
v
3
⟩
)
is also normal in
W
Γ
.
In general, we have the following lemma.
Lemma 2.4
Let
G
Γ
be a graph product and
N
⊆
G
Γ
a subgroup.
If
N
⊴
G
Γ
is normal, then the parabolic closure
Pc
(
N
)
⊴
G
Γ
is normal and
Pc
(
N
)
is a direct factor in a direct decomposition of
G
Γ
in special parabolic subgroups.
In particular, if
G
Γ
is directly indecomposable, then
Pc
(
N
)
=
G
Γ
.
Proof
First we remark that the normalizer of a subgroup 𝐻 in
G
Γ
is contained in the normalizer of the parabolic closure of 𝐻; see [1, Lemma 3.12].
Thus, given a normal subgroup
N
⊴
G
Γ
, we know that
Pc
(
N
)
⊴
G
Γ
is normal.
Furthermore, [1, Proposition 3.13] tells us that
Pc
(
N
)
is a direct factor of
G
Γ
.
More precisely,
G
Γ
=
Pc
(
N
)
×
G
Ω
, where
G
Ω
is a special parabolic subgroup of
G
Γ
.
∎
In the special case where
G
Γ
is directly indecomposable, we have a description of all subgroups of
G
Γ
.
In particular, of those subgroups without
F
2
as a subgroup.
Proposition 2.5
Proposition 2.5 ([1, Theorem 4.1])
Let
G
Γ
be a graph product of arbitrary groups and let
H
⊆
G
Γ
be a subgroup.
If
G
Γ
is directly indecomposable and
|
V
(
Γ
)
|
≥
2
, then
𝐻 is contained in a proper parabolic subgroup or
H
≅
Z
/
2
Z
∗
Z
/
2
Z
or
𝐻 has a subgroup isomorphic to
F
2
.
The following property of a group is important for us.
Definition 2.6
A group 𝐺 is called full-sized if it contains a non-abelian free group
F
2
↪
G
.
Now the natural question for us is the following.
Question 2.7
Let
G
Γ
be a (directly indecomposable) graph product of arbitrary groups and let
N
⊴
G
Γ
be a non-trivial normal subgroup.
Assume that 𝑁 is not full-sized.
What can we say about the structure of Γ?
To answer the question, we need to recall elementary Bass–Serre theory.
More precisely, we need the fact that, to every amalgamated product
A
∗
C
B
, there is an associated tree
T
A
∗
C
B
on which
A
∗
C
B
acts without a global fixed point.
The vertices of the tree correspond to the cosets
g
A
and
h
B
and there is an edge between
g
A
and
h
B
whenever their intersection is equal to
g
C
.
Clearly, the amalgamated product
A
∗
C
B
acts by left multiplication on
T
A
∗
C
B
.
Most important for us is that the stabilizers of vertices are conjugates of 𝐴 and 𝐵 respectively.
A good reference concerning Bass–Serre theory is [27].
Before we move on to the next proposition, let us discuss an example of a Bass–Serre tree.
The Coxeter group
W
Γ
associated to the graph Γ in Figure2 can also be written as an amalgamated product as follows:
W
Γ
≅
⟨
v
1
,
v
2
⟩
∗
⟨
v
2
⟩
⟨
v
2
,
v
3
⟩
.
We define
A
:
=
⟨
v
1
,
v
2
⟩
,
B
:
=
⟨
v
2
,
v
3
⟩
and
C
:
=
⟨
v
2
⟩
.
Figure 3 shows a part of the infinite tree
T
A
∗
C
B
.
Note that each vertex in
T
A
∗
C
B
has valency two.
Let us consider the action of
A
∗
C
B
by left multiplication on
T
A
∗
C
B
.
For example,
v
2
acts trivially on
T
A
∗
C
B
,
v
1
acts as an elliptic isometry with the fixed point set consisting of the single vertex labeled by 𝐴 and
v
1
v
3
acts as a translation.
Moreover, the isometry group of
T
A
∗
C
B
is isomorphic to the infinite dihedral group.
Note also that the Bass–Serre tree associated to the free product
Z
/
n
Z
∗
Z
/
n
Z
is infinite and each vertex has valency 𝑛.
Proposition 2.8
Let
G
Γ
be a directly indecomposable graph product,
|
V
(
Γ
)
|
≥
2
, and let
N
⊴
G
Γ
be a non-trivial normal subgroup.
If 𝑁 is not full-sized, then
G
Γ
≅
Z
/
2
Z
∗
Z
/
2
Z
.
Proof
First we note that if a group 𝐺 has a normal subgroup
N
≅
Z
/
2
Z
∗
Z
/
2
Z
=
⟨
a
,
b
∣
a
2
=
b
2
=
1
⟩
,
then the subgroup
⟨
a
b
⟩
≅
Z
is normal in 𝐺.
This is because the only elements of infinite order in
Z
/
2
Z
∗
Z
/
2
Z
are of the form
(
a
b
)
k
for some
k
∈
Z
−
{
0
}
and conjugation does not change the order of an element.
Let 𝑁 be a non-trivial non-full-sized normal subgroup in
G
Γ
.
By Proposition 2.5, we know that
N
≅
Z
or
N
≅
Z
/
2
Z
∗
Z
/
2
Z
.
If
N
≅
Z
/
2
Z
∗
Z
/
2
Z
, then by the above remark, we know that
G
Γ
has a normal subgroup
N
′
≅
Z
.
Our goal is to show that
G
Γ
is isomorphic to
Z
/
2
Z
∗
Z
/
2
Z
.
By assumption, Γ is not complete and has at least 2 vertices; therefore,
G
Γ
decomposes as an amalgamated product of special parabolic subgroups
G
Γ
≅
G
Γ
1
∗
G
Γ
3
G
Γ
2
.
More precisely, we choose a vertex
v
∈
V
(
Γ
)
, and then
V
(
Γ
1
)
=
V
(
Γ
)
−
{
v
}
,
V
(
Γ
3
)
=
lk
(
v
)
and
V
(
Γ
2
)
=
st
(
v
)
.
By Bass–Serre theory [27], there is a simplicial tree 𝑇 on which
G
Γ
1
∗
G
Γ
3
G
Γ
2
acts simplicially without a global fixed point.
We restrict this action to the infinite cyclic normal subgroup 𝑁 (resp.
N
′
).
The generator 𝑛 of this normal subgroup must act by a hyperbolic isometry since otherwise 𝑁 has a global fixed point and hence is contained in a proper parabolic subgroup.
In that case, we would obtain
Pc
(
N
)
≠
G
Γ
, which is impossible by Lemma 2.4.
Hence we have a group homomorphism
N
→
Isom
(
Min
(
N
)
)
≅
Z
/
2
Z
∗
Z
/
2
Z
, where
Min
(
N
)
denotes the straight path in 𝑇 that is invariant under 𝑁 and on which 𝑛 induces a translation of non-zero amplitude (see [27, I, 6.4, Proposition 25]).
Since 𝑁 is normal and, for
g
∈
G
Γ
, we have
g
Min
(
N
)
=
Min
(
g
N
g
−
1
)
=
Min
(
N
)
,
we obtain a group homomorphism
φ
:
G
Γ
→
Isom
(
Min
(
N
)
)
≅
Z
/
2
Z
∗
Z
/
2
Z
.
Furthermore, we know that
ker
(
φ
)
has a global fixed point and is therefore contained in a proper parabolic subgroup.
By the previous arguments, it follows that
ker
(
φ
)
=
{
1
}
.
More precisely, since
ker
(
φ
)
is normal,
Pc
(
ker
(
φ
)
)
is also normal.
We know that
G
Γ
is directly indecomposable, so by Lemma 2.4,
Pc
(
ker
(
φ
)
)
is either
G
Γ
or trivial.
We additionally have
ker
(
φ
)
⊆
Pc
(
ker
(
φ
)
)
=
{
1
}
because it is also contained in a proper parabolic subgroup.
Hence 𝜑 is injective.
The map 𝜑 is also surjective since Γ has at least 2 vertices.
Thus
G
Γ
≅
Z
/
2
Z
∗
Z
/
2
Z
.
∎
Using Proposition 2.8, we are able to characterize non-full-sized normal subgroups in directly decomposable graph products.
But first, we need to recall the connection between a combinatorial property of the defining graph Γ and the center of
G
Γ
.
The center of a graph product can be easily read off the defining graph Γ if the vertex groups are all abelian. More precisely, a graph product
G
Γ
decomposes as a direct product of special parabolic subgroups
G
Γ
≅
G
Δ
×
G
Γ
−
Δ
, where
V
(
Δ
)
=
{
v
∈
V
(
Γ
)
∣
st
(
v
)
=
V
(
Γ
)
}
.
The elements in
V
(
Δ
)
are called central vertices.
Furthermore, the center of
G
Γ
is contained in the special parabolic subgroup
G
Δ
, and if the vertex groups in
V
(
Δ
)
are abelian, then
Z
(
G
Γ
)
≅
G
Δ
.
Corollary 2.9
Let
G
Γ
be a graph product.
We decompose
G
Γ
≅
G
Γ
1
×
⋯
×
G
Γ
n
as a direct product, where the factors are directly indecomposable special parabolic subgroups.
Furthermore, we define
J
:
=
{
j
∈
{
1
,
…
,
n
}
|
|
V
(
Γ
j
)
|
≥
2
,
G
Γ
j
≅
Z
/
2
Z
∗
Z
/
2
Z
}
.
Let
N
⊴
G
Γ
be a non-trivial normal subgroup.
If 𝑁 is not full-sized, then
N
⊴
G
Δ
×
∏
j
∈
J
G
Γ
j
,
where
V
(
Δ
)
=
{
v
∈
V
(
Γ
)
∣
st
(
v
)
=
V
(
Γ
)
}
and the image of the projection of 𝑁 to
G
Γ
j
is trivial or infinite for all
j
∈
J
.
In particular, if 𝑁 is finite, then
N
⊴
G
Δ
.
Proof
First we note that if a group 𝐺 is not full-sized, then no quotient of 𝐺 is full-sized.
Let
N
⊴
G
Γ
be a non-full-sized normal subgroup.
Now let us consider the projection
π
i
:
G
Γ
↠
G
Γ
i
.
Since
π
i
is surjective, we know that
π
i
(
N
)
⊴
G
Γ
i
.
Furthermore, by the above remark, the normal subgroup
π
i
(
N
)
is not full-sized.
Therefore, Proposition 2.8 implies that if
|
V
(
Γ
i
)
|
≥
2
and
G
Γ
i
is not isomorphic to
Z
/
2
Z
∗
Z
/
2
Z
, then
π
i
(
N
)
is trivial.
Thus
N
⊴
G
Δ
×
∏
j
∈
J
G
Γ
j
, where
V
(
Δ
)
=
{
v
∈
V
(
Γ
)
∣
st
(
v
)
=
V
(
Γ
)
}
and
J
:
=
{
j
∈
{
1
,
…
,
n
}
|
|
V
(
Γ
j
)
|
≥
2
,
G
Γ
j
≅
Z
/
2
Z
∗
Z
/
2
Z
}
.
Since
π
j
(
N
)
⊴
G
Γ
j
≅
Z
/
2
Z
∗
Z
/
2
Z
, we know that
π
j
(
N
)
≅
Z
or
π
j
(
N
)
≅
Z
/
2
Z
∗
Z
/
2
Z
.
Hence if 𝑁 is finite, then
N
⊴
G
Δ
.
∎
Since the center of a graph product of abelian groups is isomorphic to
G
Δ
, the following result is a direct consequence of Corollary 2.9.
Corollary 2.10
Let
G
Γ
be a graph product of abelian groups.
We decompose
G
Γ
≅
G
Γ
1
×
⋯
×
G
Γ
n
as a direct product, where the factors are directly indecomposable special parabolic subgroups.
If
G
Γ
i
is not isomorphic to
Z
/
2
Z
∗
Z
/
2
Z
for all
i
=
1
,
…
,
n
, then a normal subgroup in
G
Γ
is contained in the center of
G
Γ
or is full-sized.
As an immediate corollary, we obtain the following result concerning normal subgroups in right-angled Artin groups.
This result can also be proven directly as follows using [2, Theorem 1.2].
Lemma 2.11
Lemma 2.11 ([29])
Let
A
Γ
be a right-angled Artin group and 𝑁 a subgroup.
If 𝑁 is normal, then 𝑁 is contained in the center of
A
Γ
or 𝑁 is full-sized.
Proof
It was proven by Baudisch in [2, Theorem 1.2] that every two generated subgroup
⟨
u
,
v
⟩
in
A
Γ
such that 𝑢 and 𝑣 do not commute is isomorphic to
F
2
.
We decompose
A
Γ
as
A
Γ
≅
A
Δ
×
A
Γ
−
Δ
.
Note that the special parabolic subgroup
A
Γ
−
Δ
is centerless.
If 𝑁 is not contained in the center of
A
Γ
, which is equal to
A
Δ
, then the normal subgroup
N
∩
A
Γ
−
Δ
of the special parabolic subgroup
A
Γ
−
Δ
is non-trivial.
Thus there exist
n
∈
N
∩
A
Γ
−
Δ
,
n
≠
1
, and
x
∈
A
Γ
−
Δ
such that
x
n
≠
n
x
.
By Baudisch’s result, we know that
⟨
x
,
n
⟩
≅
F
2
.
Therefore, the elements 𝑛 and
x
n
x
−
1
do not commute either, and again by Baudisch’s result, we have
⟨
n
,
x
n
x
−
1
⟩
≅
F
2
.
Since
N
∩
A
Γ
−
Δ
is normal, it follows that
x
n
x
−
1
∈
N
∩
A
Γ
−
Δ
.
Thus we have
F
2
≅
⟨
n
,
x
n
x
−
1
⟩
⊆
N
∩
A
Γ
−
Δ
⊆
N
.
∎
We note that the structure of finitely generated normal subgroups in right-angled Artin groups was investigated in [5].
2.12 Normal subgroups in automorphism groups
For a group 𝐺, we denote by
Aut
(
G
)
the group of all bijective group homomorphisms from 𝐺 to 𝐺 and by
Inn
(
G
)
the normal subgroup consisting of all inner automorphisms.
More precisely, for
g
∈
G
, we denote the inner automorphism that maps
h
∈
G
to
g
h
g
−
1
by
γ
g
.
Hence the map
φ
:
G
→
Inn
(
G
)
,
φ
(
g
)
=
γ
g
is an epimorphism with
ker
(
φ
)
=
Z
(
G
)
.
Thus if 𝐺 is centerless and
γ
g
1
=
γ
g
2
, then the equality
g
1
=
g
2
holds, and therefore
Inn
(
G
)
≅
G
.
We begin by proving the following result for the automorphism group of a centerless group.
Lemma 2.13
Let 𝒫 be a property of groups that is inherited by subgroups.
Let 𝐺 be a centerless group.
If 𝐺 has no non-trivial normal subgroups with property 𝒫, then
Aut
(
G
)
also has no non-trivial normal subgroups with property 𝒫.
Proof
Let 𝐺 be a centerless group that has no non-trivial normal subgroups with property 𝒫.
Furthermore, let 𝑁 be a normal subgroup in
Aut
(
G
)
with property 𝒫.
We consider the subgroup
N
∩
Inn
(
G
)
which is a normal subgroup in
Inn
(
G
)
≅
G
with property 𝒫.
Thus, by assumption,
N
∩
Inn
(
G
)
is trivial.
For any
g
∈
G
and
n
∈
N
, we have
n
∘
γ
g
∘
n
−
1
=
γ
n
(
g
)
.
We can multiply the above equation by
γ
g
−
1
on the right and obtain
n
∘
γ
g
∘
n
−
1
∘
γ
g
−
1
=
γ
n
(
g
)
∘
γ
g
−
1
.
Since 𝑁 is normal, the left side of the equation is an element in 𝑁 and the right side of the equation is an element in
Inn
(
G
)
.
Hence
γ
n
(
g
)
∘
γ
g
−
1
=
id
.
Thus
n
(
g
)
g
−
1
∈
Z
(
G
)
=
{
1
}
and 𝑁 is trivial.
∎
Remark 2.14
The proof of the above lemma shows the following: given a centerless group 𝐺 and a normal subgroup
N
⊴
Aut
(
G
)
, for every
n
∈
N
and
g
∈
G
, the inner automorphism
γ
n
(
g
)
g
−
1
is contained in 𝑁.
Proposition 2.15
Let
G
Γ
be a graph product of arbitrary groups.
We decompose
G
Γ
≅
G
Γ
1
×
⋯
×
G
Γ
n
as a direct product, where the factors are directly indecomposable special parabolic subgroups.
If
V
(
Δ
)
=
∅
, then
Aut
(
G
Γ
)
has no non-trivial finite normal subgroups.
Assume that
G
Γ
i
≇
Z
/
2
Z
∗
Z
/
2
Z
for all
i
=
1
,
…
,
n
.
If
|
V
(
Γ
i
)
|
≥
2
for
i
=
1
…
,
n
, then a non-trivial normal subgroup in
Aut
(
G
Γ
)
is full-sized.
In particular, if
G
Γ
is a centerless graph product of abelian groups, then a non-trivial normal subgroup in
Aut
(
G
Γ
)
is full-sized.
Proof
If
V
(
Δ
)
=
∅
, then
Z
(
G
Γ
)
=
{
1
}
.
Corollary 2.9 says that
G
Γ
has no non-trivial finite normal subgroups.
Since the property of being finite is inherited by subgroups, Lemma 2.13 implies that
Aut
(
G
Γ
)
has no non-trivial finite normal subgroups.
Let 𝒫 be the property of a group being not full-sized.
Clearly, 𝒫 is inherited by subgroups.
Now Corollary 2.9 tells us that, in case (ii),
G
Γ
has no non-trivial non-full-sized normal subgroups.
Thus, by Lemma 2.13, a non-trivial normal subgroup in
Aut
(
G
Γ
)
is full-sized.
∎
As an immediate corollary, we obtain Proposition A from the introduction.
We end this section by recalling a description of non-full-sized normal subgroups in the automorphism group of a graph product
G
Γ
, where Γ is a complete graph and all vertex groups are infinite cyclic; in that case,
Aut
(
G
Γ
)
≅
Aut
(
Z
n
)
≅
GL
n
(
Z
)
.
Note that a subgroup in
GL
n
(
Z
)
is virtually solvable or is full-sized (see [28]).
Let us first describe normal subgroups in
GL
n
(
Z
)
in the cases where
n
=
1
,
2
.
If
n
=
1
, then
GL
1
(
Z
)
≅
Z
/
2
Z
.
If
n
=
2
, then
PSL
2
(
Z
)
≅
Z
/
2
Z
∗
Z
/
3
Z
and Proposition 2.8 then implies that a non-trivial normal subgroup in
PSL
2
(
Z
)
is full-sized.
Hence a non-trivial non-full-sized normal subgroup in
GL
2
(
Z
)
is equal to the center
Z
(
GL
2
(
Z
)
)
=
{
I
,
−
I
}
, where 𝐼 is the identity matrix.
Recall that a group 𝐺 is called almost simple if every normal subgroup is either finite and is contained in the center or has finite index in 𝐺.
It follows from the Margulis normal subgroups theorem [23] that
SL
n
(
Z
)
is almost simple for
n
≥
3
.
Since
SL
n
(
Z
)
has index 2 in
GL
n
(
Z
)
and every finite index subgroup in
SL
n
(
Z
)
is full-sized (see [22, Exercise 4.E.20 (4)]), we know that a non-full-sized normal subgroup in
GL
n
(
Z
)
is contained in the center of
GL
n
(
Z
)
.
We summarize these results in the following lemma.
Lemma 2.16
Let 𝑁 be a non-trivial normal subgroup in
GL
n
(
Z
)
.
If 𝑁 is not full-sized, then
N
=
Z
(
GL
n
(
Z
)
)
=
{
I
,
−
I
}
.
3 Characteristic subgroups
To construct quotients of the entire automorphism group
Aut
(
G
Γ
)
, it is common to consider characteristic subgroups of
G
Γ
.
This can be helpful since, in some cases, it allows us to reduce a question about a complicated graph product to a question about two smaller graph products and a short exact sequence.
We start this section by recalling the definition of a characteristic subgroup.
Using this, we obtain a very important short exact sequence that will be used throughout the rest of the article.
Furthermore, we prove some basic but useful results about a splitting of the automorphism group of a graph product.
Finally, we show that if a right-angled Coxeter group decomposes as a direct product of an infinite dihedral group and another right-angled Coxeter group, it has a characteristic subgroup isomorphic to a product of copies of the infinite dihedral group.
Definition 3.1
Let 𝐺 be a group and
H
⊆
G
a subgroup.
The group 𝐻 is called characteristic in 𝐺 if, for every
f
∈
Aut
(
G
)
, the equality
f
(
H
)
=
H
holds.
A characteristic subgroup 𝐻 of a group 𝐺 is always normal in 𝐺 and we have two group homomorphisms
π
1
:
Aut
(
G
)
→
Aut
(
H
)
and
π
2
:
Aut
(
G
)
→
Aut
(
G
/
H
)
defined as follows:
π
1
(
f
)
=
f
|
H
and
π
2
(
f
)
(
g
H
)
:
=
f
(
g
)
H
.
For a graph product
G
Γ
, we always have the direct decomposition
G
Δ
×
G
Γ
−
Δ
, where
V
(
Δ
)
=
{
v
∈
V
(
Γ
)
∣
st
(
v
)
=
V
(
Γ
)
}
.
If the vertex groups in Δ are abelian, then
Z
(
G
Γ
)
=
G
Δ
is a characteristic subgroup of
G
Γ
and we obtain two group homomorphisms
π
1
:
Aut
(
G
Γ
)
→
Aut
(
G
Δ
)
and
π
2
:
Aut
(
G
Γ
)
→
Aut
(
G
Γ
−
Δ
)
which are obviously surjective.
But if there exists a vertex group in Δ that is not abelian, then
G
Δ
is in general not characteristic.
For example, consider the graph product of the defining graph Γ in Figure 4.
It is obvious that the subgroup
G
Δ
≅
Z
/
2
Z
∗
Z
/
2
Z
is not characteristic.
This observation leads us to the following question.
Question 3.2
Let
G
Γ
be a graph product.
Under which conditions on the vertex groups in Δ is the subgroup
G
Δ
characteristic in
G
Γ
?
Following Genevois [13], a group 𝐺 is called graphically irreducible if, for every finite simplicial graph Γ and every collection of vertex groups indexed by
V
(
Γ
)
such that
G
≅
G
Γ
, the graph Γ is a clique.
Lemma 3.3
Lemma 3.3 ([13, Proposition 3.11])
Let
G
Γ
be a graph product.
If the vertex groups are graphically irreducible, then the special parabolic subgroup
G
Δ
is characteristic in
G
Γ
.
Hence we obtain two surjective group homomorphisms
π
1
:
Aut
(
G
Γ
)
→
Aut
(
G
Δ
)
and
π
2
:
Aut
(
G
Γ
)
→
Aut
(
G
Γ
−
Δ
)
.
We define
π
:
Aut
(
G
Γ
)
→
Aut
(
G
Δ
)
×
Aut
(
G
Γ
−
Δ
)
as follows:
π
(
f
)
=
(
π
1
(
f
)
,
π
2
(
f
)
)
.
The group homomorphism 𝜋 is surjective by construction.
Furthermore, the following proposition gives a concrete description of the kernel of 𝜋.
Proposition 3.4
Let
G
Γ
be a graph product of graphically irreducible vertex groups.
The kernel of
π
:
Aut
(
G
Γ
)
→
Aut
(
G
Δ
)
×
Aut
(
G
Γ
−
Δ
)
is equal to
T
Δ
:
=
{
f
∈
Aut
(
G
Γ
)
∣
for all
g
∈
G
Γ
,
g
−
1
f
(
g
)
∈
G
Δ
and
f
|
G
Δ
=
id
}
.
Moreover, the short exact sequence
{
1
}
→
T
Δ
→
Aut
(
G
Γ
)
→
Aut
(
G
Δ
)
×
Aut
(
G
Γ
−
Δ
)
→
{
1
}
splits and we have
Aut
(
G
Γ
)
≅
T
Δ
⋊
(
Aut
(
G
Γ
)
×
Aut
(
G
Γ
−
Δ
)
)
.
Proof
We follow the ideas of the proof of [21, Lemma 7.7].
Let
f
∈
ker
(
π
)
=
ker
(
π
1
)
∩
ker
(
π
2
)
.
Since
f
∈
ker
(
π
1
)
, we have
f
(
g
)
=
g
for all
g
∈
G
Δ
.
Furthermore, since
f
∈
ker
(
π
2
)
, we have, for all
h
∈
G
Γ
,
π
2
(
f
)
(
h
G
Δ
)
=
f
(
h
)
G
Δ
=
h
G
Δ
.
Therefore,
h
−
1
f
(
h
)
∈
G
Δ
and hence
f
∈
T
Δ
.
It follows by definition of
T
Δ
that
T
Δ
⊆
ker
(
π
)
.
Hence we obtain a short exact sequence
{
1
}
→
T
Δ
→
Aut
(
G
Γ
)
→
Aut
(
G
Δ
)
×
Aut
(
G
Γ
−
Δ
)
→
{
1
}
that splits since, for the canonical inclusion
i
:
Aut
(
G
Δ
)
×
Aut
(
G
Γ
−
Δ
)
→
Aut
(
G
Γ
)
,
we have
π
∘
i
=
id
.
Thus
Aut
(
G
Γ
)
≅
T
Δ
⋊
(
Aut
(
G
Γ
)
×
Aut
(
G
Γ
−
Δ
)
)
.
∎
Now we have collected all the tools to prove Proposition E (ii).
Proposition 3.5
Let
G
Γ
be a graph product.
We decompose
G
Γ
≅
G
Γ
1
×
⋯
×
G
Γ
n
as a direct product, where the factors are directly indecomposable special parabolic subgroups.
Assume that
G
Γ
i
≇
Z
/
2
Z
∗
Z
/
2
Z
for all
i
=
1
,
…
,
n
.
If all vertex groups are finite, then a normal subgroup in
Aut
(
G
Γ
)
is finite or full-sized.
Proof
First we note that if
V
(
Δ
)
is empty, then by Proposition 2.15 (i), a non-trivial normal subgroup in
Aut
(
G
Γ
)
is full-sized.
Let
N
⊴
Aut
(
G
Γ
)
be an infinite normal subgroup.
By assumption, all vertex groups are finite; thus, in particular, all vertex groups are graphically irreducible.
By Proposition 3.4, we have the exact sequence
{
1
}
→
T
Δ
→
Aut
(
G
Γ
)
→
π
=
(
π
1
×
π
2
)
Aut
(
G
Δ
)
×
Aut
(
G
Γ
−
Δ
)
→
{
1
}
.
We claim that 𝑁 is full-sized.
Indeed, if
π
2
(
N
)
⊴
Aut
(
G
Γ
−
Δ
)
is non-empty, we know that
π
2
(
N
)
is full-sized, and therefore 𝑁 is also full-sized.
Now we turn to the case where
π
2
(
N
)
is trivial; then we have the following exact sequence:
{
1
}
→
N
∩
T
Δ
→
N
→
π
1
(
N
)
→
{
1
}
.
The group
Aut
(
G
Δ
)
is finite because, by assumption, the vertex groups of Δ are finite.
Thus
π
1
(
N
)
⊆
Aut
(
G
Δ
)
is a finite group.
Furthermore, the group
N
∩
T
Δ
is also finite because
T
Δ
is a finite group.
Hence 𝑁 is a finite normal subgroup.
∎
The next result will be useful in the proof of Theorem G.
Namely, in the case where the graph product
G
Γ
decomposes as a direct product of characteristic subgroups, we completely understand the structure of
Aut
(
G
Γ
)
by the automorphism groups of direct factors of
G
Γ
.
Lemma 3.6
Let 𝐺 be a group that decomposes as direct product
G
=
G
1
×
G
2
.
If both factors
G
1
and
G
2
are characteristic subgroups of 𝐺, then
Aut
(
G
)
≅
Aut
(
G
1
)
×
Aut
(
G
2
)
.
Proof
It is obvious that
Aut
(
G
1
)
×
Aut
(
G
2
)
can naturally be embedded in
Aut
(
G
)
by the natural map
(
f
1
,
f
2
)
↦
f
, where
f
(
v
)
=
f
i
(
v
)
when
v
∈
G
i
for
i
=
1
,
2
.
We claim this map is surjective.
Given a general
f
∈
Aut
(
G
)
, we know that
f
(
G
i
)
=
G
i
since both factors are characteristic.
Hence we obtain
f
i
∈
Aut
(
G
i
)
by setting
f
i
:
=
f
|
G
i
.
It is easy to see that
(
f
1
,
f
2
)
is mapped to 𝑓 under the natural embedding, which is what we wanted to show.
∎
Let us discuss one example of the above lemma in the context of graph products and their automorphism groups.
Lemma 3.7
Let
G
Γ
be a graph product of cyclic groups.
If all vertex groups in
V
(
Δ
)
are infinite cyclic and all vertex groups in
V
(
Γ
)
−
V
(
Δ
)
are finite, then
G
Γ
−
Δ
is characteristic and thus
Aut
(
G
Γ
)
≅
Aut
(
G
Δ
)
×
Aut
(
G
Γ
−
Δ
)
.
Proof
The subgroup
G
Δ
is characteristic since it is the center of
G
Γ
.
Let
f
∈
Aut
(
G
Γ
)
and
w
i
∈
V
(
Γ
)
−
V
(
Δ
)
.
To show that
G
Γ
−
Δ
is characteristic, it is sufficient to prove that
f
(
w
i
)
∈
G
Γ
−
Δ
.
It was proven in [15, Theorem 3.26] that a finite order element is contained in a parabolic subgroup whose defining graph is a clique.
Thus
f
(
w
i
)
∈
g
G
Ω
g
−
1
, where Ω is a clique subgraph of
Γ
−
Δ
and
g
∈
G
Γ
.
So there exists an element
h
∈
G
Ω
such that
f
(
w
i
)
=
g
h
g
−
1
.
Since
G
Δ
commutes with
G
Ω
, we can assume that
g
∈
G
Γ
−
Δ
.
Hence
g
h
g
−
1
∈
G
Γ
−
Δ
.
By Lemma 3.6, it follows that
Aut
(
G
Γ
)
≅
Aut
(
G
Δ
)
×
Aut
(
G
Γ
−
Δ
)
.
∎
We end this section with an investigation of characteristic subgroups in right-angled Coxeter groups.
Given a right-angled Coxeter group
W
Γ
, it was proven by Leder in [21, Proposition 7.17] that, for
k
∈
N
, the normal closure of the special subgroup
W
Ω
, where
V
(
Ω
)
=
{
v
∈
V
(
Γ
)
∣
val
(
v
)
≥
k
}
, is a characteristic subgroup.
Let us consider the right-angled Coxeter group defined by the graph Γ given in Figure 5.
The vertices
w
1
and
w
2
have valency 6 and all the other vertices have smaller valencies.
So the normal closure of the subgroup
⟨
w
1
,
w
2
⟩
is characteristic in
W
Γ
.
Furthermore, the special parabolic subgroup
⟨
w
1
,
w
2
⟩
≅
Z
/
2
Z
∗
Z
/
2
Z
is a direct factor of
W
Γ
; thus the normal closure of
⟨
w
1
,
w
2
⟩
is equal to
⟨
w
1
,
w
2
⟩
, and this special parabolic subgroup is characteristic in
W
Γ
.
Lemma 3.8
Let
W
Γ
be a centerless right-angled Coxeter group.
We decompose
W
Γ
as a direct product of directly indecomposable special parabolic subgroups
W
Γ
=
W
Γ
1
×
⋯
×
W
Γ
n
.
If at least one of the factors is isomorphic to
Z
/
2
Z
∗
Z
/
2
Z
, then
∏
j
∈
J
W
Γ
j
, where
J
:
=
{
j
∈
{
1
,
…
,
n
}
∣
W
j
≅
Z
/
2
Z
∗
Z
/
2
Z
}
is a characteristic subgroup.
In particular,
∏
j
∈
J
W
Γ
j
as a subgroup of
Inn
(
W
Γ
)
is an infinite virtually abelian normal subgroup in
Aut
(
W
Γ
)
.
Proof
We write
V
(
Γ
)
=
{
w
1
,
…
,
w
n
}
.
First we need to show the following combinatorial lemma from graph theory.
Let Ω denote a finite simplicial graph with 𝑛 vertices.
Assume that every vertex has valency at least
n
−
2
.
Then we have
Ω
=
A
1
∗
A
2
∗
⋯
∗
A
k
, where every
A
j
is an induced subgraph of Ω and has at most 2 vertices for
j
=
1
,
…
,
k
.
This is true since, starting with a vertex
v
0
∈
V
(
Ω
)
,
v
0
is connected to at least
n
−
2
other vertices, of which there are
n
−
1
.
If
v
0
is connected to all of them, set
A
1
=
{
v
0
}
and continue with
Ω
−
⟨
v
0
⟩
.
If not, there is a vertex
v
1
∈
V
(
Ω
)
, which is not connected to
v
0
by an edge.
However,
v
1
has valency
n
−
2
; hence it is connected to all other vertices apart from
v
0
.
In other words,
Ω
=
⟨
v
0
,
v
1
⟩
∗
(
Ω
−
⟨
{
v
0
,
v
1
}
⟩
)
.
Since Ω is finite, this proves the combinatorial lemma in finitely many steps.
Note that this proof actually shows that the existence of two vertices
x
,
y
which are not connected by an edge and which have valency
n
−
2
always implies that Γ decomposes as a join
Γ
=
(
{
x
,
y
}
,
∅
)
∗
(
Γ
−
(
{
x
,
y
}
,
∅
)
)
.
To show the result about right-angled Coxeter groups, simply notice that, given
W
Γ
j
≅
Z
/
2
Z
∗
Z
/
2
Z
with canonical generators
w
i
and
w
j
, the vertices
w
i
,
w
j
in the defining graph for the right-angled Coxeter group have valency
|
V
(
Γ
)
|
−
2
.
Furthermore, there are no vertices with valency
|
V
(
Γ
)
|
−
1
since
W
Γ
is centerless.
Hence
∏
j
∈
J
W
Γ
j
is nothing but
⟨
V
(
D
)
⟩
, where
D
=
{
v
∈
V
(
Γ
)
∣
val
(
v
)
≥
|
V
(
Γ
)
|
−
2
}
.
Thus [21, Proposition 7.17] implies that
∏
j
∈
J
W
Γ
j
is a characteristic subgroup in
W
Γ
.
Furthermore, since
W
Γ
is centerless, we have
Inn
(
W
Γ
)
≅
W
Γ
, and therefore, we can consider the special parabolic subgroup
∏
j
∈
J
W
Γ
j
as a subgroup of
Inn
(
W
Γ
)
.
Let
f
∈
Aut
(
W
Γ
)
and
γ
w
∈
Inn
(
W
Γ
)
, where
w
∈
∏
j
∈
J
W
Γ
j
.
We have
f
∘
γ
w
∘
f
−
1
=
γ
f
(
w
)
.
Since
∏
j
∈
J
W
Γ
j
is a characteristic subgroup, we know that
f
(
w
)
∈
∏
j
∈
J
W
Γ
j
.
Hence it follows that
∏
j
∈
J
W
Γ
j
is a normal subgroup in
Aut
(
W
Γ
)
and it is indeed virtually abelian since each direct factor is isomorphic to the infinite dihedral group
Z
/
2
Z
∗
Z
/
2
Z
which is virtually abelian.
∎
4 The automorphism group of a graph product of cyclic groups
In this section, we specialize to the case of finitely generated abelian groups.
First we remark that this is essentially the same as assuming the vertex groups are cyclic.
Then we discuss certain types of automorphisms that typically occur in this situation and recall an important result by Corredor–Gutierrez which shows these automorphisms generate the entire automorphism group.
Finally, we show that torsion subgroups in the automorphism group of a graph product of cyclic groups are finite and that no subgroup is isomorphic to ℚ.
Let
G
Γ
be a graph product of finitely generated abelian groups.
We remark that
G
Γ
is isomorphic to a graph product of cyclic groups.
More precisely, given a finitely generated abelian vertex group
G
v
, we know that
G
v
is isomorphic to the direct sum
Z
l
⊕
Z
/
n
1
Z
⊕
⋯
⊕
Z
/
n
k
Z
.
Thus replacing the vertex 𝑣 by a complete graph on
k
+
l
vertices with vertex groups isomorphic to
Z
/
n
i
Z
or ℤ respectively yields an isomorphic graph product.
Moreover, this graph product is isomorphic to a graph product of cyclic groups, where the finite groups have prime power order.
In order to achieve that, we simply replace the vertex whose vertex group is
Z
/
n
Z
, where
n
=
p
1
k
1
⋅
…
⋅
p
r
k
r
, by a complete graph on 𝑟 vertices with vertex groups
Z
/
p
i
k
i
Z
, where
p
i
are prime numbers for
i
=
1
,
…
,
r
.
It was proven by Corredor and Gutierrez in [7] that the automorphism group of a graph product of cyclic groups is generated by a finite collection of different types of automorphisms: labeled graph automorphisms, local automorphisms, partial conjugations and dominated transvections.
In the following, we explain these different types of automorphisms that will prove to be useful especially when calculating the center.
Let
G
Γ
be a graph product of cyclic groups and
Aut
(
G
Γ
)
the automorphism group of
G
Γ
.
We start by introducing labeled graph automorphisms.
The group of labeled graph automorphisms is defined as follows:
Aut
(
Γ
)
:
=
{
σ
∈
Sym
(
Γ
)
∣
G
v
≅
G
σ
(
v
)
for all
v
∈
V
(
Γ
)
}
.
It is not hard to show that each element in
Aut
(
Γ
)
canonically induces an automorphism in
Aut
(
G
Γ
)
; therefore, we call also this induced automorphism a labeled graph automorphism.
Now we investigate automorphisms in
Aut
(
G
Γ
)
that come from the automorphism group of the vertex groups.
Given a non-identity automorphism in
Aut
(
G
v
)
, where
v
∈
V
(
Γ
)
, this automorphism induces an automorphism of the entire graph product
G
Γ
by sending the generator 𝑤 of
G
w
,
v
≠
w
, to itself.
We call this kind of automorphism a local automorphism.
Let
v
∈
V
(
Γ
)
and let
C
⊆
Γ
be a connected component of the graph induced by the vertex set
V
(
Γ
)
−
st
(
v
)
.
We define a map
ρ
v
C
:
V
(
Γ
)
→
G
Γ
as follows:
ρ
c
C
(
z
)
=
v
z
v
−
1
for all
z
∈
V
(
C
)
and
ρ
x
C
(
w
)
=
w
for
w
∈
V
(
Γ
)
−
V
(
C
)
.
This map induces an automorphism of
G
Γ
which we call a partial conjugation.
The last type of automorphism that we introduce is dominated transvections.
Let
v
,
w
∈
V
(
Γ
)
be two different vertices.
We define a map
ρ
v
w
j
:
V
(
Γ
)
→
G
Γ
as follows:
ρ
v
w
j
(
v
)
=
v
w
j
and
ρ
v
w
j
is the identity on all other vertices in
V
(
Γ
)
.
We should be careful here since this map does not always induce a well-defined automorphism of
G
Γ
.
But if
ord
(
v
)
=
∞
and
lk
(
v
)
⊆
st
(
w
)
, then
ρ
v
w
induces a well-defined automorphism of
G
Γ
.
Furthermore, if
ord
(
v
)
=
p
i
and
ord
(
w
)
=
p
j
and
st
(
v
)
⊆
st
(
w
)
, then we have two cases: if
j
≤
i
, then
ρ
v
w
is a well-defined automorphism of
G
Γ
, and if
j
>
i
, then
ρ
v
w
p
j
−
i
is a well-defined automorphism (see [7, Proposition 5.5]).
In all these cases, we call the induced automorphism of
G
Γ
a dominated transvection.
Theorem 4.1
Theorem 4.1 ([7])
Let
G
Γ
be a graph product of cyclic groups.
Then
Aut
(
G
Γ
)
is generated by labeled graph automorphisms, local automorphisms, partial conjugations and dominated transvections.
In particular, the group
Aut
(
G
Γ
)
is finitely generated.
Since we are now working with cyclic vertex groups, the center of
G
Γ
is
G
Δ
.
Therefore, we will denote the subgroup
T
Δ
of
Aut
(
G
Γ
)
by
T
Z
, where the 𝑍 should emphasize that, for
f
∈
T
Z
and
g
∈
G
Γ
, the element
g
−
1
f
(
g
)
is now always a central element.
Corollary 4.2
Let
G
Γ
be a graph product of cyclic groups.
Let
Δ
fin
⊆
Δ
be the induced subgraph generated by
{
v
∈
V
(
Δ
)
∣
ord
(
G
v
)
<
∞
}
.
We define a subgroup
T
Z
fin
⊆
T
Z
as follows:
T
Z
fin
:
=
{
f
∈
T
Z
∣
for all
w
i
∈
V
(
Γ
)
−
V
(
Δ
)
,
w
i
−
1
f
(
w
i
)
∈
G
Δ
fin
}
.
The subgroup
T
Z
fin
⊆
Aut
(
G
Γ
)
is normal and it is non-trivial if and only if
V
(
Δ
fin
)
≠
∅
and there exists a vertex
w
∈
V
(
Γ
)
−
V
(
Δ
)
such that
ord
(
G
w
)
=
∞
or there exist a vertex
v
∈
V
(
Δ
fin
)
and
w
∈
V
(
Γ
)
−
V
(
Δ
)
such that
gcd
(
|
G
v
|
,
|
G
w
|
)
=
p
.
Proof
By definition, an automorphism
f
∈
T
Z
acts trivially on
G
Δ
.
Hence the order of 𝑓 is finite if and only if, for every vertex
w
∈
V
(
Γ
)
−
V
(
Δ
)
, the element
z
w
defined by
z
w
:
=
w
−
1
f
(
w
)
has finite order.
Hence
T
Z
fin
is nothing but the torsion part of the finitely generated abelian normal subgroup
T
Z
of
Aut
(
G
Γ
)
and therefore normal itself.
Since it is generated by the dominated transvections
ρ
w
v
k
for vertices
w
∈
V
(
Γ
)
−
V
(
Δ
)
and
v
∈
V
(
Δ
)
,
it is non-trivial if and only if at least one of these is well-defined.
It is easy to see that this is the case if and only if the conditions in the corollary are satisfied.
∎
Concerning automatic continuity, we will need the facts that torsion subgroups in
Aut
(
G
Γ
)
are finite and
Aut
(
G
Γ
)
does not have ℚ as a subgroup.
The first property follows from the fact that
Out
(
G
Γ
)
:
=
Aut
(
G
Γ
)
/
Inn
(
G
Γ
)
satisfies a stronger version of the Tits alternative, namely every subgroup in
Out
(
G
Γ
)
is either full-sized or virtually polycyclic (see [26, Corollary 4.4]).
Hence a torsion subgroup in
Out
(
G
Γ
)
is virtually polycyclic.
Recall that, by definition, a group 𝐺 is polycyclic if there exists a descending chain of subgroups
{
1
}
=
G
n
⊆
G
n
−
1
⊆
⋯
⊆
G
1
=
G
such that
G
i
+
1
is normal in
G
i
and the quotient
G
i
/
G
i
+
1
is cyclic for
i
=
1
,
…
,
n
−
1
.
Corollary 4.3
Let
Aut
(
G
Γ
)
be the automorphism group of a graph product of cyclic groups and let
H
⊆
Aut
(
G
Γ
)
be a subgroup.
If 𝐻 is a torsion group, then 𝐻 is finite.
Proof
First we show that a torsion polycyclic group is finite.
Let 𝑇 be a torsion polycyclic group.
Since 𝑇 is polycyclic, there exists a chain of subgroups
{
1
}
=
T
n
⊆
T
n
−
1
⊆
⋯
⊆
T
1
=
T
such that
T
i
+
1
is normal in
T
i
and
T
i
/
T
i
+
1
is cyclic.
Now we prove that
T
n
−
1
is finite.
We know that
T
n
−
1
/
T
n
is cyclic, so there exists
x
n
−
1
∈
T
such that
⟨
x
n
−
1
⟩
=
T
n
−
1
.
By assumption, 𝑇 is a torsion group; thus
x
n
−
1
has finite order, and therefore
T
n
−
1
is finite.
The same strategy shows that
T
n
−
2
and then
T
j
for
j
=
1
,
…
,
n
are finite groups.
Hence
T
1
=
T
is finite.
Moreover, a torsion virtually polycyclic group is also finite.
More precisely, let 𝑇 be a torsion virtually polycyclic group.
Then there exists a normal subgroup
N
⊴
T
such that
T
/
N
is finite and 𝑁 is a torsion polycyclic group.
We have
{
1
}
→
N
→
T
→
T
/
N
→
{
1
}
.
Since 𝑁 is a torsion polycyclic group, this group is finite.
Thus 𝑁 and
T
/
N
are finite groups and therefore 𝑇 is also finite.
Now we show that a torsion subgroup 𝐻 in
Aut
(
G
Γ
)
is finite.
Let
π
:
Aut
(
G
Γ
)
→
Out
(
G
Γ
)
be the canonical projection.
Since
π
(
H
)
is not full-sized, we know that
π
(
H
)
is a torsion virtually polycyclic group which is finite by the above arguments.
So we have
π
(
H
)
≅
H
/
H
∩
Inn
(
G
Γ
)
.
The group
H
∩
Inn
(
G
Γ
)
is a torsion subgroup in
Inn
(
G
Γ
)
≅
G
Γ
−
Δ
.
Lemma 3.6 in [20] implies that this group is contained in a parabolic subgroup whose defining graph is a clique.
Hence
H
∩
Inn
(
G
Γ
)
is finite and therefore 𝐻 has to be finite too.
∎
The next lemma shows that the group of rational numbers cannot be embedded in
Aut
(
G
Γ
)
.
Lemma 4.4
Let
G
Γ
be a graph product of finitely generated residually finite groups.
The group
Aut
(
G
Γ
)
has no subgroup isomorphic to ℚ.
Proof
By [15, Corollary 5.4], the group
G
Γ
is residually finite.
Since all vertex groups are finitely generated, so is
G
Γ
.
Furthermore, it was proven in [3] that the automorphism group of a finitely generated residually finite group is residually finite.
Hence
Aut
(
G
Γ
)
is residually finite.
Lemma 5.1 in [19] implies that a residually finite group does not have ℚ as a subgroup.
Hence
Aut
(
G
Γ
)
has no subgroup isomorphic to ℚ.
∎
5 Proof of Theorem B
We start with a characterization of non-full-sized normal subgroups in
Aut
(
A
Γ
)
where both vertex sets
V
(
Δ
)
=
{
v
1
…
,
v
n
}
and
V
(
Γ
)
−
V
(
Δ
)
=
{
w
1
,
…
,
w
m
}
are non-empty.
In that case,
A
Γ
has a non-trivial center and is not abelian.
In particular, all groups, except the left and the right one, in the following exact sequence are non-trivial:
{
1
}
→
T
Z
→
Aut
(
A
Γ
)
→
π
=
(
π
1
×
π
2
)
Aut
(
A
Δ
)
×
Aut
(
A
Γ
−
Δ
)
→
{
1
}
.
Note also that
T
Z
is generated by dominated transvections
ρ
w
i
v
j
for
i
=
1
,
…
,
m
and
j
=
1
,
…
,
n
whose order is infinite and any pair of these dominated transvections generate a free abelian group of rank 2; hence
T
Z
≅
Z
n
⋅
m
.
For the following proposition, we recall the definition of the automorphism 𝜄 of a right-angled Artin group
A
Γ
.
We define
ι
(
v
i
)
=
v
i
−
1
and
ι
(
w
j
)
=
w
j
for
1
≤
i
≤
n
and
1
≤
j
≤
m
and extend this to a homomorphism.
Proposition 5.1
Let
A
Γ
be a right-angled Artin group.
Let
V
(
Δ
)
=
{
v
1
,
…
,
v
n
}
and
V
(
Γ
)
−
V
(
Δ
)
=
{
w
1
,
…
,
w
m
}
be both non-empty.
Let
N
⊴
Aut
(
A
Γ
)
be a non-trivial normal subgroup.
If 𝑁 is not full-sized, then
N
⊆
T
Z
⋊
{
id
,
ι
}
≅
Z
n
⋅
m
⋊
Z
/
2
Z
.
Moreover, if
N
⊆
T
Z
, then
N
≅
Z
l
, where
n
≤
l
≤
n
⋅
m
.
Proof
Let
N
⊴
Aut
(
A
Γ
)
be a non-full-sized normal subgroup.
Since 𝑁 is not full-sized,
π
1
(
N
)
⊴
Aut
(
A
Δ
)
≅
GL
n
(
Z
)
is not full-sized as well.
Lemma 2.11 implies that
π
1
(
N
)
=
{
id
}
or
π
1
(
N
)
=
{
id
,
ι
}
.
Furthermore,
π
2
(
N
)
is also a normal subgroup of
Aut
(
A
Γ
−
Δ
)
and is not full-sized.
Hence, by Proposition 2.15, the normal subgroup
π
2
(
N
)
is trivial.
Now we consider the short exact sequence
{
1
}
→
T
Z
→
Aut
(
A
Γ
)
→
π
=
(
π
1
×
π
2
)
Aut
(
A
Δ
)
×
Aut
(
A
Γ
−
Δ
)
→
{
1
}
and restrict the map 𝜋 to 𝑁.
We obtain two possibilities.
1
→
N
∩
T
Z
→
N
↠
{
id
}
×
{
id
}
→
{
1
}
.
In that case, we have
N
=
N
∩
T
Z
.
1
→
N
∩
T
Z
→
N
↠
{
id
,
ι
}
×
{
id
}
→
{
1
}
.
In that case, we have
N
⊆
⟨
T
Z
,
ι
⟩
≅
T
Z
⋊
{
id
,
ι
}
≅
Z
n
⋅
m
⋊
Z
/
2
Z
.
It remains to prove that, in the first case,
N
∩
T
Z
is isomorphic to the free abelian group of rank 𝑙, where
n
≤
l
≤
n
⋅
m
.
Since we need to explicitly work with dominated transvections, let us first consider an easy example given by the dominated transvection
ρ
w
1
v
1
.
Conjugating
ρ
w
1
v
1
with the labeled graph automorphism permuting the vertices of Δ as 𝜎 in
Sym
(
n
)
yields the dominated transvection
ρ
w
1
v
σ
(
1
)
.
Similarly, if there exists a labeled graph automorphism permuting (some of) the vertices in
Γ
−
Δ
stemming from a
τ
∈
Sym
(
m
)
, we obtain the dominated transvection
ρ
w
τ
(
1
)
v
1
by conjugation.
So, in this case, the normal closure of
ρ
w
1
v
1
is isomorphic to
Z
l
for some
n
≤
l
≤
n
⋅
m
, where the upper bound comes from the observation that
T
Z
≅
Z
n
⋅
m
.
Thus
N
∩
T
Z
is always isomorphic to some
Z
k
for some
k
≤
n
⋅
m
, so all that remains to show is that
k
≥
n
=
|
V
(
Δ
)
|
.
We know that, given
w
i
∈
V
(
Γ
)
−
V
(
Δ
)
, the automorphism
f
∈
N
must satisfy
f
(
w
i
)
=
w
i
z
i
for some
z
i
∈
A
Δ
.
This means
f
(
w
i
)
=
w
i
v
1
k
1
⋅
…
⋅
v
n
k
n
(
k
j
∈
Z
) and we obtain a homomorphism
h
i
:
N
→
Z
n
by collecting the exponents, i.e.
h
i
(
f
)
:
=
(
k
1
,
…
,
k
n
)
.
Note that this is a homomorphism since all elements of
T
Z
are products of dominated transvections.
Suppose that 𝑓 is not the identity homomorphism.
Then at least one of the maps
h
i
is non-trivial.
Let
σ
∈
Sym
(
n
)
denote a permutation and 𝛼 the corresponding labeled graph automorphism permuting the vertices of
A
Δ
.
The completeness of Δ ensures there always exists such a labeled graph automorphism.
Then
h
i
(
α
∘
f
∘
α
−
1
)
=
(
k
σ
(
1
)
,
…
,
k
σ
(
n
)
)
.
If
{
(
k
σ
(
1
)
,
…
,
k
σ
(
n
)
)
∣
σ
∈
Sym
(
n
)
}
is a generating system for
Z
n
, then we are done.
If not, that means that
k
1
=
k
2
=
⋯
=
k
n
≠
0
since
h
i
(
f
)
≠
0
and
h
i
(
α
∘
f
∘
α
−
1
)
=
(
k
σ
(
1
)
,
…
,
k
σ
(
n
)
)
.
Now we conjugate 𝑓 by the dominated transvection
ρ
v
1
v
2
.
Note that there are always at least two vertices in Δ because
⟨
k
1
⟩
is isomorphic to ℤ for every
k
1
≠
0
.
Evaluating the conjugation of 𝑓 by
ρ
v
1
v
2
on
w
1
gives us
ρ
v
1
v
2
∘
f
∘
ρ
v
1
v
2
−
1
(
w
i
)
=
ρ
v
1
v
2
∘
f
(
w
i
)
=
w
i
z
i
v
2
k
1
.
Thus we obtain
h
i
(
ρ
v
1
v
2
∘
f
∘
ρ
v
1
v
2
−
1
)
=
(
k
1
,
k
2
+
k
1
,
…
,
k
n
)
.
But now not all entries are equal; thus the group generated by all permutations is isomorphic to
Z
n
, which completes the proof.
∎
Now we have all the ingredients to prove Theorem B.
Theorem B
Let
Aut
(
A
Γ
)
be the automorphism group of a right-angled Artin group
A
Γ
and let
N
⊴
Aut
(
A
Γ
)
be a non-trivial maximal (with respect to inclusion) normal subgroup.
If 𝑁 is not full-sized, then either
V
(
Γ
)
−
V
(
Δ
)
=
∅
and
N
=
{
id
,
ι
}
, where
ι
(
v
i
)
=
v
i
−
1
for all
v
i
∈
V
(
Δ
)
, or
V
(
Δ
)
=
{
v
1
,
…
,
v
n
}
,
V
(
Γ
)
−
V
(
Δ
)
=
{
w
1
,
…
,
w
m
}
for
n
,
m
≥
1
and
N
≅
T
Δ
⋊
{
id
,
ι
}
≅
Z
n
⋅
m
⋊
Z
/
2
Z
,
where
T
Δ
:=
{
f
∈
Aut
(
A
Γ
)
∣
w
i
−
1
f
(
w
i
)
∈
A
Δ
and
f
(
v
j
)
=
v
j
for all
i
=
1
,
…
,
m
and
j
=
1
,
…
,
n
}
.
Furthermore, let
N
⊴
Aut
(
A
Γ
)
be a non-trivial minimal non-full-sized normal subgroup and set
n
:
=
|
V
(
Δ
)
|
and
m
:
=
|
V
(
Γ
)
−
V
(
Δ
)
|
.
If
n
,
m
≥
1
, then
N
≅
Z
l
for some
n
≤
l
≤
n
⋅
m
.
Proof
Let
N
⊴
Aut
(
A
Γ
)
be a non-full-sized normal subgroup.
As algebraic properties of
Aut
(
A
Γ
)
depend on the combinatorial structure of the defining graph Γ, we have to consider several cases.
First we investigate the case where
V
(
Γ
)
−
V
(
Δ
)
=
∅
.
In that case, we have
Aut
(
A
Γ
)
≅
GL
n
(
Z
)
and Lemma 2.16 implies that 𝑁 is trivial or isomorphic to the center of
GL
n
(
Z
)
which is equal to
{
I
,
−
I
}
.
The next case deals with the situation where
V
(
Δ
)
is empty.
Proposition 2.15 implies directly that 𝑁 is trivial.
Now we consider the last case: assume that
V
(
Δ
)
≠
∅
and
V
(
Γ
)
−
V
(
Δ
)
≠
∅
and let 𝑁 be maximal.
By Proposition 5.1, we have
N
⊆
T
Z
⋊
{
id
,
ι
}
.
Using the generating set of
Aut
(
A
Γ
)
from Theorem 4.1, it is straightforward to prove that
T
Z
⋊
{
id
,
ι
}
is normal in
Aut
(
A
Γ
)
.
If 𝑁 is a minimal, non-trivial normal subgroup in
Aut
(
A
Γ
)
, then Proposition 5.1 implies that
N
⊆
T
Z
and
N
≅
Z
l
, where
n
≤
l
≤
n
⋅
m
.
More precisely, first note that
{
id
,
ι
}
is not normal in
Aut
(
A
Γ
)
since
ρ
w
1
v
1
∘
ι
∘
ρ
w
1
v
1
−
1
∉
{
id
,
ι
}
; thus
N
∩
T
Z
is a non-trivial normal subgroup.
Hence if
N
⊈
T
Z
, then
N
∩
T
Z
is a smaller non-trivial normal subgroup of
Aut
(
A
Γ
)
.
This finishes the proof.
∎
In particular, Theorem B implies Corollary C from the introduction.
6 Proof of Theorem G
Before we move on to the proof of Theorem G, let us recall a result about the structure of the automorphism group of a finitely generated abelian group.
If Γ is a complete graph, then
G
Γ
is a finitely generated abelian group.
Thus there exist
n
∈
N
and a finite abelian group 𝑇 such that
G
Γ
≅
Z
n
×
T
.
Since 𝑇 is a characteristic subgroup, we always have two group homomorphisms
ρ
1
:
Aut
(
Z
n
×
T
)
→
Aut
(
Z
n
)
≅
GL
n
(
Z
)
and
ρ
2
:
Aut
(
Z
n
×
T
)
→
Aut
(
T
)
.
Note that both maps are surjective.
Furthermore, the map
ρ
=
(
ρ
1
×
ρ
2
)
is also surjective by construction.
Hence the sequence
{
1
}
→
ker
(
ρ
)
→
Aut
(
Z
n
×
T
)
→
ρ
GL
n
(
Z
)
×
Aut
(
T
)
→
{
1
}
is exact.
Additionally, it was proven in [26, Lemma 2.16] that
ker
(
ρ
)
≅
T
n
.
Thus if
Γ
=
Δ
and it has at least one infinite cyclic vertex group and at least one finite vertex group, then
ker
(
ρ
)
is a non-trivial finite normal subgroup in
Aut
(
G
Γ
)
.
Using methods similar to the proof of Theorem B, we prove Theorem G.
Theorem G
Let
G
Γ
be a graph product of cyclic groups, where the finite vertex groups have prime power orders.
Let Δ be the induced subgraph of Γ generated by the vertices
v
∈
V
(
Γ
)
such that
st
(
v
)
=
V
(
Γ
)
.
The automorphism group
Aut
(
G
Γ
)
has no non-trivial finite normal subgroup if and only if one of the following conditions holds:
V
(
Δ
)
=
{
v
1
}
,
G
v
1
≅
Z
/
2
Z
,
V
(
Γ
)
−
V
(
Δ
)
=
{
w
1
,
…
,
w
m
}
,
m
≥
1
and
ord
(
G
w
j
)
<
∞
and
2
∤
ord
(
G
w
j
)
for all
1
≤
j
≤
m
or
for every
v
∈
V
(
Δ
)
, we have
G
v
≅
Z
and there exists a
w
∈
V
(
Γ
)
−
V
(
Δ
)
such that
G
w
≅
Z
.
Proof
It follows directly from Proposition 2.15 that if
V
(
Δ
)
=
∅
, then any finite normal subgroup in
Aut
(
G
Γ
)
is trivial.
Now assume that
V
(
Γ
)
−
V
(
Δ
)
is empty.
The goal is to prove that
Aut
(
G
Γ
)
has no non-trivial finite normal subgroups if and only if
V
(
Γ
)
=
{
v
1
}
and
ord
(
G
v
1
)
=
2
.
The “if” statement is clear since
Aut
(
Z
/
2
Z
)
=
{
id
}
.
For the “only if” statement, we differentiate between two cases.
First assume that all vertex groups are infinite.
Then
Aut
(
G
Γ
)
≅
GL
n
(
Z
)
and hence it has a non-trivial finite normal subgroup, the center, which is isomorphic to
Z
/
2
Z
.
In the other case, there either exists at least one infinite vertex group and at least one finite vertex group, in which case
ker
(
ρ
)
is a non-trivial finite normal subgroup, or all vertex groups are finite.
If all vertex groups are finite, then
Aut
(
G
Γ
)
is itself finite and this group is non-trivial if and only if
Γ
≠
{
v
1
}
, where
ord
(
v
1
)
=
2
.
Now assume that
V
(
Δ
)
=
{
v
1
,
…
,
v
n
}
is non-empty and
V
(
Γ
)
−
V
(
Δ
)
=
{
w
1
,
…
,
w
m
}
is non-empty.
We have the splitting short exact sequence
{
1
}
→
T
Z
→
Aut
(
G
Γ
)
→
π
=
(
π
1
×
π
2
)
Aut
(
G
Δ
)
×
Aut
(
G
Γ
−
Δ
)
→
{
1
}
.
Assume that all vertex groups in
V
(
Δ
)
are infinite cyclic and that there is an infinite vertex group
G
w
in
Γ
−
Δ
.
Let
N
⊴
Aut
(
G
Γ
)
denote a finite normal subgroup.
We know that
T
Z
is a finitely generated abelian group.
If
G
Δ
is torsion free, it follows immediately that
T
Z
is torsion free; hence
T
Z
∩
N
=
{
1
}
.
Furthermore, from results above, we know that
π
1
(
N
)
⊆
{
id
,
ι
}
and
π
2
(
N
)
=
{
id
}
.
Hence 𝑁 can only be isomorphic to
Z
/
2
Z
and generated by an element 𝑓 which inverts all vertices in Δ and maps every vertex
w
∈
V
(
Γ
−
Δ
)
to
w
z
w
for some
z
w
∈
G
Δ
.
But such an 𝑓 does not commute with the dominated transvection
ρ
w
v
since
ρ
w
v
∘
f
(
w
)
=
w
z
w
v
and
f
∘
ρ
w
v
(
w
)
=
w
z
w
v
−
1
.
Hence 𝑓 has to be trivial.
Note that if all vertex groups in
V
(
Γ
)
−
V
(
Δ
)
are finite, then Lemma 3.6 implies that
Aut
(
G
Γ
)
≅
Aut
(
G
Δ
)
×
Aut
(
G
Γ
−
Δ
)
and hence the subgroup
{
id
,
ι
}
is normal in
Aut
(
G
Γ
)
.
We have proved that if all vertex groups in
V
(
Δ
)
are infinite cyclic, then
Aut
(
G
Γ
)
has no non-trivial finite normal subgroups if and only if there exists an infinite vertex group
G
w
j
.
If there exists
v
i
∈
V
(
Δ
)
such that
G
v
i
is finite, then we have to differentiate two cases and prove the following statements.
If
n
=
1
and
ord
(
G
v
1
)
=
2
, then
Aut
(
G
Γ
)
has no non-trivial finite normal subgroups if and only if
ord
(
G
w
j
)
<
∞
and
2
∤
ord
(
G
w
j
)
for
j
=
1
,
…
,
m
.
If
n
≥
2
or
n
=
1
and
ord
(
G
v
1
)
≠
2
, then
Aut
(
G
Γ
)
does have non-trivial finite normal subgroups.
For the “only if” statement in (a), simply note that
T
Z
fin
is a finite normal subgroup of
Aut
(
G
Γ
)
and it is non-trivial if one of the conditions is satisfied; see Corollary 4.2. If those conditions are not satisfied, it is easy to check that
G
Δ
and
G
Γ
−
Δ
are characteristic subgroups, and hence, by Lemma 3.6, we have
Aut
(
G
Γ
)
=
Aut
(
G
Δ
)
×
Aut
(
G
Γ
−
Δ
)
.
But
Aut
(
G
Δ
)
=
{
1
}
and
Aut
(
G
Γ
−
Δ
)
has no non-trivial finite normal subgroups, which finishes off (a).
Finally, consider case (b).
If there exists at least one dominated transvection
ρ
w
v
for a vertex
w
∈
V
(
Γ
−
Δ
)
and a vertex
v
∈
V
(
Γ
)
with
ord
(
v
)
<
∞
, then
T
Z
fin
is a non-trivial, finite normal subgroup of
Aut
(
G
Γ
)
.
If no such dominated transvection exists, then
Aut
(
G
Γ
)
=
Aut
(
G
Δ
)
×
Aut
(
G
Γ
−
Δ
)
.
We know that a non-trivial finite normal subgroup in
Aut
(
G
Δ
)
exists which in turn needs to be normal in
Aut
(
G
Γ
)
due to the direct product structure, which proves (b).
∎
7 The center of the automorphism group of a graph product
In this section, we study the center of the automorphism group of a graph product of cyclic groups.
First we discuss two known but useful results and then show that the center is always finite.
We obtain conditions when the automorphism group is centerless and in particular prove Corollary D.
Afterwards, we explicitly calculate the center of the automorphism group if the defining graph is complete.
Then we study the case where the defining graph is a join of one vertex with the rest and also calculate the center.
This is then applied to right-angled Coxeter groups and an explicit example is discussed where the center of the automorphism group is isomorphic to
(
Z
/
2
Z
)
12
.
We start this section by recalling a result regarding the center of the automorphism group that is well known.
Since we could not find a good reference, we include the proof of it here.
Lemma 7.1
Let 𝐺 be a group and let
Aut
(
G
)
be the automorphism group.
If
f
∈
Z
(
Aut
(
G
)
)
, then for
g
∈
G
, there exists
z
g
∈
Z
(
G
)
such that
f
(
g
)
=
g
z
g
.
In particular, if 𝐺 is centerless, then the center of
Aut
(
G
)
is trivial.
Proof
For every
g
∈
G
, we denote by
γ
g
∈
Inn
(
G
)
the conjugation by 𝑔.
Let us consider the equation
γ
g
∘
f
∘
γ
g
−
1
=
f
, which holds because 𝑓 commutes with all automorphisms of 𝐺.
Thus, for each
x
∈
G
, we obtain
γ
g
∘
f
∘
γ
g
−
1
(
x
)
=
f
(
x
)
which is equivalent to
g
f
(
g
)
−
1
f
(
x
)
f
(
g
)
g
−
1
=
f
(
x
)
.
This shows that
g
f
(
g
)
−
1
commutes with
f
(
x
)
, and since 𝑥 is an arbitrary element in 𝐺, it follows that
g
f
(
g
)
−
1
∈
Z
(
f
(
G
)
)
=
Z
(
G
)
.
Hence there exists
z
g
∈
Z
(
G
)
such that
f
(
g
)
=
g
z
g
.
Furthermore, if
Z
(
G
)
=
{
1
}
, then
z
g
=
1
for every
g
∈
G
, and therefore 𝑓 is the identity.
Thus if 𝐺 is centerless, the center of
Aut
(
G
)
is trivial.
∎
The center of the automorphism group of a free abelian group and of a finite abelian group have been explicitly calculated; for completeness, we state the results here.
Proposition 7.2
Let
G
Γ
be a graph product of cyclic groups.
Assume that Γ is a clique.
If all vertex groups are infinite cyclic, then
Z
(
Aut
(
G
Γ
)
≅
Z
(
GL
n
(
Z
)
)
=
{
I
,
−
I
}
.
If all vertex groups are finite, then
Z
(
Aut
(
G
Γ
)
)
≅
∏
p
∈
J
(
Z
/
p
l
p
Z
)
∗
,
where
J
:
=
{
p
∈
P
∣
there exists a vertex
v
∈
V
(
Γ
)
with
p
∣
ord
(
v
)
}
and
p
l
p
is the maximal order of a vertex group in Γ with 𝑝 power order.
The result of Proposition 7.2 (i) is well known and that of Proposition 7.2 (ii) was shown in [16, Proposition 3.2].
Before we move on to the concrete calculation of the center of the automorphism group of a graph product of cyclic groups, we show that the center is always finite.
Lemma 7.3
Let
G
Γ
be a graph product of cyclic groups.
The center of
Aut
(
G
Γ
)
is finite.
Proof
First we note that if there is no vertex
v
∈
V
(
Γ
)
such that
st
(
v
)
=
V
(
Γ
)
, then
G
Γ
is centerless, and therefore, by Lemma 7.1, the group
Aut
(
G
Γ
)
is also centerless.
Now let us assume that Γ is a complete graph.
Then
G
Γ
is a finitely generated abelian group, and therefore there exist
n
∈
N
and a finite abelian group 𝑇 such that
G
Γ
≅
Z
n
×
T
.
We consider the exact sequence
{
1
}
→
ker
(
ρ
)
→
Aut
(
Z
n
×
T
)
→
GL
n
(
Z
)
×
Aut
(
T
)
→
{
1
}
,
where
ρ
=
(
ρ
1
×
ρ
2
)
and
ker
(
ρ
)
≅
T
n
.
Consider the restriction of this exact sequence to
Z
(
Aut
(
Z
n
×
T
)
)
.
Note that
ρ
1
(
Z
(
Aut
(
Z
n
×
T
)
)
)
⊆
Z
(
GL
n
(
Z
)
)
=
{
I
,
−
I
}
.
We obtain
ker
(
ρ
)
∩
Z
(
Aut
(
Z
n
×
T
)
)
→
Z
(
Aut
(
Z
n
×
T
)
)
→
ρ
1
(
Z
(
Aut
(
Z
n
×
T
)
)
)
×
ρ
2
(
Z
(
Aut
(
Z
n
×
T
)
)
)
.
The group
ker
(
ρ
)
∩
Z
(
Aut
(
Z
n
×
T
)
)
⊆
T
n
is finite and the group
ρ
1
(
Z
(
Aut
(
Z
n
×
T
)
)
)
×
ρ
2
(
Z
(
Aut
(
Z
n
×
T
)
)
)
⊆
{
I
,
−
I
}
×
Aut
(
T
)
is finite too; hence
Z
(
Aut
(
Z
n
×
T
)
)
is finite.
In the last case, we assume that
V
(
Δ
)
=
{
v
1
,
…
,
v
n
}
and
V
(
Γ
)
−
V
(
Δ
)
=
{
w
1
,
…
,
w
m
}
for
n
,
m
≥
1
.
Let
f
∈
Z
(
Aut
(
G
Γ
)
)
.
Our goal is to show that 𝑓 has finite order.
Since
G
Δ
is characteristic, we get a group homomorphism
π
1
:
Aut
(
G
Γ
)
→
Aut
(
G
Δ
)
that is surjective.
Note that
π
1
(
Z
(
Aut
(
G
Γ
)
)
)
⊆
Z
(
Aut
(
G
Δ
)
)
.
Hence we know that
f
|
G
Δ
has finite order since
f
|
G
Δ
∈
Z
(
Aut
(
G
Δ
)
)
.
By Lemma 7.1, it follows that, for
w
j
, there exists
z
j
∈
G
Δ
such that
f
(
w
j
)
=
w
j
z
j
.
Thus 𝑓 has finite order if and only if all
z
j
are of finite order.
Assume for contradiction that one of the
z
i
’s has infinite order.
Then we decompose
z
i
=
x
1
x
2
…
x
k
y
1
…
y
l
,
where
x
o
,
y
j
∈
V
(
Δ
)
and
ord
(
x
o
)
=
∞
for all
o
∈
{
1
,
…
,
k
}
and
ord
(
y
j
)
<
∞
for all
j
∈
{
1
,
…
,
l
}
.
Let
h
∈
Aut
(
G
Γ
)
be the automorphism defined as
h
(
x
1
)
=
x
1
−
1
and identity elsewhere.
Then 𝑓 does not commute with ℎ.
More precisely, we have
f
∘
h
(
w
i
)
=
f
(
w
i
)
=
w
i
x
1
x
2
…
x
k
y
1
…
y
l
and
h
∘
f
(
w
i
)
=
h
(
w
i
x
1
x
2
…
x
k
y
1
…
y
l
)
=
w
i
x
1
−
1
x
2
…
x
k
y
1
…
y
l
.
Hence 𝑓 has finite order, and therefore
Z
(
Aut
(
G
Γ
)
)
is an abelian torsion group.
By Lemma 4.3, we know that any torsion subgroup in
Aut
(
G
Γ
)
is finite; thus
Z
(
Aut
(
G
Γ
)
)
is finite.
∎
The above lemma together with Theorem G implies the following corollary.
Corollary 7.4
Let
G
Γ
be a graph product of cyclic groups.
If
V
(
Δ
)
=
{
v
1
}
,
ord
(
v
1
)
=
2
and
V
(
Γ
)
−
V
(
Δ
)
is non-empty and all vertex groups in this set are finite and
2
∤
ord
(
w
j
)
for
j
=
1
,
…
,
m
or
V
(
Δ
)
is non-empty and
V
(
Γ
)
−
V
(
Δ
)
is non-empty and all vertex groups in Δ are infinite cyclic and there exists an infinite vertex group in
V
(
Γ
)
−
V
(
Δ
)
,
then
Aut
(
G
Γ
)
is centerless.
In particular, we have proven Corollary D from the introduction.
7.5 The center of the automorphism group of an infinite finitely generated abelian group with torsion
Let us now focus on the calculation of the center of the automorphism group of an infinite finitely generated abelian group with torsion elements
G
≅
Z
n
×
T
, where
n
≥
1
and 𝑇 is a non-trivial finite abelian group.
We denote by
{
x
1
,
…
,
x
n
}
a generating set of the free abelian part of 𝐺 and by
{
y
1
,
…
,
y
m
}
a generating set of the torsion part of 𝐺, where each
y
j
has a prime power order.
Note that we can write 𝐺 as a graph product
G
Δ
, where
V
(
Δ
)
=
{
x
1
,
…
,
x
n
,
y
1
,
…
,
y
m
}
.
Lemma 7.6
Let
G
=
Z
n
×
T
be a finitely generated infinite abelian group, where 𝑇 is a finite non-trivial abelian group.
Let
ι
∈
Aut
(
G
)
be the automorphism induced by
x
i
↦
x
i
−
1
and
y
j
↦
y
j
for all
i
=
1
,
…
,
n
,
j
=
1
,
…
,
m
.
We have
ι
∈
Z
(
Aut
(
G
)
)
if and only if 𝑇 is a product of
Z
/
2
Z
’s.
Proof
We write 𝐺 as a graph product of a complete graph Δ, where
x
1
,
…
,
x
n
are the infinite order vertices and
y
1
,
…
,
y
m
the finite order ones; hence
G
≅
G
Δ
.
For the “only if” part, we assume there exists a vertex
y
j
∈
V
(
Δ
)
with
ord
(
y
j
)
≠
2
.
Then the dominated transvection
ρ
:
=
ρ
x
1
y
j
∈
Aut
(
G
Δ
)
does not commute with 𝜄 since
ι
∘
ρ
(
x
1
)
=
x
1
−
1
y
j
and
ρ
∘
ι
(
x
1
)
=
x
1
−
1
y
j
−
1
and the equality
y
j
=
y
j
−
1
holds if and only if
ord
(
y
j
)
=
2
.
For the “if” part, the above calculation shows that 𝜄 commutes with all dominated transvections of the form
ρ
x
i
y
l
for
i
=
1
,
…
,
n
and
l
=
1
,
…
,
m
.
It is an easy exercise to show that 𝜄 commutes with local automorphisms and graph automorphisms as well.
∎
Given a finitely generated abelian group
G
≅
Z
n
×
T
, we always have a surjective group homomorphism
ρ
:
=
(
ρ
1
×
ρ
2
)
:
Aut
(
G
)
→
Aut
(
Z
n
)
×
Aut
(
T
)
induced by the fact that 𝑇 is a characteristic subgroup.
The next lemma describes when the restriction of 𝜌 to the center of
Aut
(
G
)
is injective.
Lemma 7.7
Let
G
≅
Z
n
×
T
be a finitely generated infinite abelian group, where 𝑇 is a finite group.
Let
ρ
:
Aut
(
G
)
→
Aut
(
Z
n
)
×
Aut
(
T
)
be as above.
The restriction
ρ
|
Z
(
Aut
(
G
)
)
is injective if and only if there exist no non-trivial central transvections, which is equivalent to the following condition.
Write
T
=
⟨
y
1
⟩
×
⋯
×
⟨
y
m
⟩
and consider the maximum order of
y
i
, that is equal to a power of 2.
Call this order 𝑘.
Then there exist non-trivial central transvections if and only if
n
=
1
and
|
{
y
i
∣
ord
(
y
i
)
=
k
}
|
=
1
.
If the kernel of the restriction is non-trivial, then it is isomorphic to
Z
/
2
Z
.
Proof
The kernel of 𝜌 is isomorphic to
T
n
as discussed in the proof of Lemma 7.3.
The fact that
ρ
|
Z
(
Aut
(
G
)
)
is injective if and only if there exist no non-trivial central transvections follows from the short exact sequence used in the proof of Lemma 7.3.
For the “only if” part, first assume that, in the torsion part, no vertex order is a power of 2.
Then no elements of order 2 exist, and hence, by the calculation in the proof of the previous lemma, no dominated transvection
ρ
x
i
y
j
for
i
=
1
,
…
,
n
and
j
=
1
,
…
,
m
commutes with 𝜄.
Now suppose there are multiple vertices with order 𝑘.
Without loss of generality, we can assume that
ord
(
y
1
)
=
k
=
ord
(
y
2
)
.
Towards a contradiction, assume there exists a non-trivial element 𝑓 in the kernel of
ρ
|
Z
(
Aut
(
G
)
)
.
We know that
f
(
x
i
)
=
x
i
z
i
and that the order of
z
i
is 1 or 2 for every 𝑖.
Without loss of generality,
z
1
≠
1
.
Write
z
1
=
v
1
l
1
…
v
o
l
o
, where the order of all
v
j
is a power of 2 and
l
1
,
…
,
l
o
∈
Z
−
{
0
}
.
We first show that
ord
(
v
i
)
≠
k
.
Assume for the contrary that
ord
(
v
1
)
=
k
.
Then all vertices with order 𝑘 need to appear as factors in
z
1
since they can be permuted by a labeled graph automorphism and 𝑓 needs to commute with these.
So we can assume that
ord
(
v
2
)
=
k
as well.
Hence the dominated transvection
ρ
v
1
v
2
is well-defined.
But 𝑓 does not commute with
ρ
v
1
v
2
since
ρ
v
1
v
2
∘
f
(
x
1
)
=
x
1
v
1
l
1
v
3
l
3
v
4
l
4
…
v
o
l
o
,
f
∘
ρ
v
1
v
2
(
x
1
)
=
x
1
v
1
l
1
v
2
l
2
…
v
o
l
o
.
Now we show that the other orders cannot appear either.
Suppose that the order of
v
1
is
2
a
and that
ord
(
y
1
)
=
k
=
2
b
with
b
>
a
.
Then the dominated transvection
ρ
v
1
y
1
2
b
−
a
is well-defined and does not commute with 𝑓 since
f
∘
ρ
v
1
y
1
2
b
−
a
(
x
1
)
=
x
1
z
1
and
ρ
v
1
y
1
2
b
−
a
∘
f
(
x
1
)
=
x
1
z
1
y
1
2
b
−
1
and
y
1
has order
2
b
, so
y
1
2
b
−
1
is non-trivial.
Hence
z
1
=
1
, a contradiction.
The final case is that there is exactly one vertex of order 𝑘 but
n
>
1
.
Then, by the previous case, the only possible central element would be of the form
f
(
x
i
)
=
x
i
y
k
/
2
, where 𝑦 is the unique vertex of order 𝑘.
Now there exists the dominated transvection
ρ
x
1
x
2
and this does not commute with 𝑓 since
f
∘
ρ
x
1
x
2
(
x
1
)
=
x
1
x
2
but
ρ
x
1
x
2
∘
f
(
x
1
)
=
x
1
x
2
y
k
/
2
,
finishing the proof of the “only if” direction.
For the “if” part, assume that the order of
y
1
is 𝑘.
Then the dominated transvection
ρ
x
1
y
1
k
/
2
is central and hence in the kernel of
ρ
|
Z
(
Aut
(
G
)
)
.
This can be seen by doing essentially the same calculations as in the previous lemma.
The calculations of the other direction show that this dominated transvection is the only possible central element of the kernel.
∎
Before calculating the center of the automorphism group of a graph product
G
Δ
in full generality, let us discuss the definition of one very important automorphism.
If there exists at least one vertex whose order is a power of 2, then let 𝑘 denote the maximum order of the
y
j
such that
ord
(
y
j
)
is a power of 2.
Write
{
z
1
,
…
,
z
l
}
:
=
{
y
j
∣
ord
(
y
j
)
=
k
}
.
Then define
α
∈
Aut
(
G
Δ
)
by
α
(
x
i
)
=
x
i
z
1
…
z
l
and
α
(
y
j
)
=
y
j
.
First we investigate the center of
Aut
(
G
Δ
)
, where
ord
(
y
i
)
=
2
for
i
=
1
,
…
,
m
.
Corollary 7.8
Let
G
Δ
≅
Z
n
×
(
Z
/
2
Z
)
m
, where
n
,
m
≥
1
.
The center of
Aut
(
G
Δ
)
is isomorphic to
⟨
ι
,
α
⟩
≅
Z
/
2
Z
×
Z
/
2
Z
if and only if
n
=
1
and
m
=
1
.
In all other cases, the center of
Aut
(
G
Δ
)
is isomorphic to
{
1
,
ι
}
.
Proof
For the “if” statement, simply note that
Aut
(
G
Δ
)
≅
⟨
ι
,
α
⟩
≅
Z
/
2
Z
×
Z
/
2
Z
by Theorem 4.1, since the only generators are the local automorphism 𝜄 and one dominated transvection mapping the generator of ℤ to the product of itself with the generator of
Z
/
2
Z
.
In the other cases, we know by Lemma 7.7 that
ρ
|
Z
(
Aut
(
G
Δ
)
)
is injective.
Thus we have
ρ
|
Aut
(
G
Δ
)
:
Z
(
Aut
(
G
Δ
)
)
↪
Z
(
Aut
(
Z
n
)
)
×
Z
(
Aut
(
Z
/
2
Z
)
m
)
=
{
1
,
ι
}
×
{
1
}
.
Hence the center of
Aut
(
G
Δ
)
has at most 2 elements.
Moreover, it is easy to check that 𝜄 is a central element using the generating set from Theorem 4.1.∎
Now we move on to the general case.
Proposition 7.9
Let
G
Δ
be a graph product of cyclic groups, where Δ is a clique.
We write
V
(
Δ
)
=
{
x
1
,
…
,
x
n
,
y
1
,
…
,
y
m
}
, where the order of
x
i
for
i
=
1
,
…
,
n
is infinite and each order of
y
j
for
j
=
1
,
…
,
m
is a prime power.
If
n
,
m
≥
1
, then the center of
Aut
(
G
Γ
)
is a subgroup of
⟨
ι
,
α
⟩
≅
Z
/
2
Z
×
Z
/
2
Z
.
Furthermore, let 𝑘 denote the maximal order of a vertex group which is also a power of 2.
Then the center is
isomorphic to
{
1
,
α
}
if there exists exactly one vertex of infinite order and precisely one vertex of order 𝑘 and
k
≠
2
or precisely one vertex of order 2,
k
=
2
and at least one vertex of finite order not equal to 2;
isomorphic to
⟨
ι
,
α
⟩
if there is precisely one vertex order of finite order and this order is 2 and precisely one vertex of infinite order;
isomorphic to
{
1
,
ι
}
if all finite vertex groups have order 2 and there are more than 2 vertices;
Proof
We start by invoking Lemma 7.7 which tells us that
ρ
|
Z
(
Aut
(
G
Δ
)
)
is injective in cases (iii) and (iv) and otherwise not injective (this follows by carefully checking all possibilities).
In the cases where
ρ
|
Z
(
Aut
(
G
Δ
)
)
is injective, we know that the center is a subgroup of
{
1
,
ι
}
×
Z
(
Aut
(
⟨
y
1
,
…
,
y
m
⟩
)
.
However, no automorphism of
Z
(
Aut
(
⟨
y
1
,
…
,
y
m
⟩
)
)
can appear in
Z
(
Aut
(
G
Δ
)
)
since these appear as local automorphisms by Proposition 7.2 (ii) and
⟨
y
1
,
…
,
y
m
⟩
is characteristic.
These do not commute with the product of the dominated transvections
ρ
x
1
y
1
ρ
x
1
y
2
…
ρ
x
1
y
m
.
More precisely, let
ρ
1
:
Aut
(
G
Δ
)
→
Aut
(
Z
n
)
and
ρ
2
:
Aut
(
G
Δ
)
→
Aut
(
⟨
y
1
,
…
,
y
m
⟩
)
denote the respective projections.
Since the center of
Aut
(
G
Δ
)
is mapped to the center of the direct product, we know a central element
f
∈
Aut
(
G
Δ
)
has to map an infinite order vertex 𝑥 to
x
t
or
x
−
1
t
for some torsion element
t
∈
⟨
y
1
,
…
,
y
m
⟩
and
f
(
y
)
=
y
k
y
for every vertex in
{
y
1
,
…
,
y
m
}
.
If
k
y
is not equal to 1, then this 𝑓 does not commute with the dominated transvection
ρ
x
y
.
Therefore, we conclude that a central element 𝑓 acts as the identity on
⟨
y
1
,
…
,
y
m
⟩
.
Hence, in cases (iii) and (iv), the only possible non-trivial central element is 𝜄, so we use Lemma 7.6 to finish these two cases.
Corollary 7.8 implies case (ii).
The remaining case is (i).
It is clear by Lemma 7.7 that we have the following inclusion:
Z
(
Aut
(
G
Δ
)
)
⊆
⟨
α
,
ι
,
Aut
(
⟨
y
1
,
…
,
y
m
⟩
)
⟩
.
By the above argument, we can see that a central element once again acts trivially on
⟨
y
1
,
…
,
y
m
⟩
; hence
Z
(
Aut
(
G
Δ
)
)
⊆
⟨
ι
,
α
⟩
, and since 𝜄 is not central by Lemma 7.6 and 𝛼 is by the same argument as in the proof of Lemma 7.7, we obtain the desired statement.
∎
Let us discuss some examples.
We denote the graphs in Figure 6 from left to right by
Δ
1
,
Δ
2
,
Δ
3
and
Δ
4
.
Proposition 7.9 (iii) implies that the center of
Aut
(
G
Δ
1
)
is generated by 𝜄; thus
Z
(
Aut
(
G
Δ
1
)
)
≅
Z
/
2
Z
.
Furthermore, by Proposition 7.9 (iv), we know that
Z
(
Aut
(
G
Δ
2
)
)
is trivial.
The center of
Aut
(
G
Δ
3
)
is equal to
{
id
,
α
}
by Proposition 7.9 (i).
We have
Z
(
Aut
(
G
Δ
4
)
)
=
{
id
}
by Proposition 7.9 (iv).
7.10 The center of the automorphism group of a non-abelian graph product of cyclic groups
Now that we completely understand the abelian case, let us move towards the more general case of a non-abelian graph product.
Given a graph product
G
Γ
, we always have the epimorphism
π
=
(
π
1
×
π
2
)
:
Aut
(
G
Γ
)
↠
Aut
(
G
Δ
)
×
Aut
(
G
Γ
−
Δ
)
(see Section 3).
Since
π
1
and
π
2
are surjective maps and
Z
(
Aut
(
G
Γ
−
Δ
)
)
is trivial, we have
π
1
(
Z
(
Aut
(
G
Γ
)
)
)
⊆
Z
(
Aut
(
G
Δ
)
)
)
and
π
2
(
Z
(
Aut
(
G
Γ
)
)
)
=
{
id
}
.
Proposition 7.11
Let
Aut
(
G
Γ
)
be the automorphism group of the graph product
G
Γ
, where
V
(
Δ
)
and
V
(
Γ
)
−
V
(
Δ
)
are both non-empty.
Let 𝑘 denote the maximal order of a vertex in Δ which is also a power of 2.
The restriction
π
|
Z
(
Aut
(
G
Γ
)
)
:
Z
(
Aut
(
G
Γ
)
)
→
Aut
(
G
Δ
)
is injective if and only if
for
v
∈
V
(
Δ
)
, we have
ord
(
v
)
=
∞
or
ord
(
v
)
=
p
n
for some 𝑛,
p
≠
2
, or
there are multiple vertices of order 𝑘 in Δ, or
there is precisely one vertex 𝑣 of order 𝑘 in Δ and either there is no vertex in
Γ
−
Δ
such that
ρ
w
v
ord
(
v
)
/
2
is well-defined or, for all
w
i
∈
V
(
Γ
−
Δ
)
such that
ρ
w
i
v
is well-defined, which lie in the same orbit of the action of the labeled graph automorphisms on Γ, also
ρ
w
i
w
j
or
ρ
w
j
w
i
is well-defined.
Proof
The kernel of 𝜋 is given by
T
Z
; recall that these are automorphisms 𝑓 such that
f
(
w
)
=
w
z
w
for every vertex
w
∈
V
(
Γ
−
Δ
)
and
f
(
v
)
=
v
for every vertex
v
∈
V
(
Δ
)
.
Let
ι
:
G
Γ
→
G
Γ
denote the automorphism induced by
v
↦
v
−
1
for all
v
∈
V
(
Δ
)
and
w
↦
w
for all
V
(
Γ
−
Δ
)
.
We can now calculate
ι
∘
f
(
w
)
=
w
z
w
−
1
and
f
∘
ι
(
w
)
=
w
z
w
; hence
ord
(
z
w
)
=
2
or
z
w
=
1
if
f
∈
Z
(
Aut
(
G
Γ
)
)
.
We first prove the “if” statement.
In case (i), there are no elements of order 2 in
G
Δ
; thus
z
w
=
1
, and hence we obtain the injectivity of
π
|
Z
(
Aut
(
G
Γ
)
)
.
In the setting of case (ii), suppose that there exists
w
∈
V
(
Γ
−
Δ
)
such that
z
w
≠
1
.
Then we can write
z
w
=
v
1
l
1
v
2
l
2
…
v
o
l
o
, where the order of every
v
i
is a power of 2 and
l
1
,
…
,
l
o
∈
Z
−
{
0
}
.
Moreover, if there are multiple vertices of the same order in
V
(
Δ
)
, then either all of these appear as a factor of
z
w
or none do, since there exists an automorphism of the labeled graph permuting those vertices.
Additionally, only a vertex whose order is the maximal power of 2 appearing in Δ can show up as a factor here since otherwise a dominated transvection
ρ
v
i
x
exists,
x
∈
V
(
Δ
)
, with
ord
(
x
)
>
ord
(
v
i
)
and then
ρ
v
i
x
∘
f
(
w
)
=
w
z
w
x
ord
(
v
i
/
2
)
and
f
∘
ρ
v
i
x
(
w
)
=
w
z
w
.
However, there cannot be multiple factors of the same order either since the dominated transvection
ρ
v
1
v
2
does not commute with 𝑓 either, by essentially the same argument.
Therefore, we are done with part (ii).
Finally, for part (iii), notice that if no dominated transvection
ρ
w
v
ord
(
v
)
/
2
exists, then
f
(
w
)
=
w
v
ord
(
v
)
/
2
will not be well-defined.
For the other part of the statement, first note that two vertices 𝑤 and
w
′
in the same orbit need to satisfy
z
w
=
z
w
′
, i.e.
f
(
w
′
)
=
w
′
z
w
=
w
′
z
w
′
.
The existence of a dominated transvection
ρ
w
w
′
now implies that 𝑓 is not central if
z
w
≠
1
, since
f
∘
ρ
w
w
′
(
w
)
=
w
w
′
z
w
z
w
′
=
w
w
′
and
ρ
w
w
′
∘
f
(
w
)
=
w
w
′
z
w
.
This proves the “if” statement.
For the other direction, we essentially mimic the construction of the automorphism 𝛼 from the clique case.
Let
v
∈
V
(
Δ
)
denote the unique vertex in Δ whose order is the maximal order of 2 appearing, and let
w
∈
V
(
Γ
−
Δ
)
denote a vertex such that
ρ
w
v
ord
(
v
)
/
2
is well-defined and neither
ρ
w
w
′
nor
ρ
w
′
w
is well-defined for all vertices
w
′
in the orbit of 𝑤 under the action of the labeled graph automorphisms.
Define
β
:
G
Γ
→
G
Γ
as the automorphism induced by
w
′
↦
w
′
v
ord
(
v
)
/
2
for all vertices
w
′
in the orbit of 𝑤 under the action of the labeled graph automorphisms and
x
↦
x
for all other vertices.
One can check that this is indeed a central element using the generating set from Theorem 4.1.
∎
Let
W
Γ
be a right-angled Coxeter group.
It is known that
Aut
(
W
Δ
)
≅
GL
n
(
Z
/
2
Z
)
,
where
n
=
|
V
(
Δ
)
|
.
Hence
Z
(
Aut
(
W
Δ
)
)
is trivial.
Combining this fact with Proposition 7.11 (ii) implies the following result.
Corollary 7.12
Let
W
Γ
be a right-angled Coxeter group.
If
|
V
(
Δ
)
|
≥
2
, then
Z
(
Aut
(
W
Γ
)
)
is trivial.
Now we move on to the case where Δ has precisely one vertex but there are no further restrictions.
Proposition 7.13
Let
G
Γ
be a graph product, where
Δ
=
{
v
}
and
V
(
Γ
)
−
V
(
Δ
)
≠
∅
.
Let
V
(
Γ
)
−
V
(
Δ
)
=
{
w
1
,
…
,
w
m
}
.
If
ord
(
v
)
=
∞
, then
Z
(
Aut
(
G
Γ
)
)
is non-trivial if and only if
ord
(
w
j
)
<
∞
for all
j
=
1
,
…
,
m
.
In the non-trivial case, we have
Z
(
Aut
(
G
Γ
)
)
=
{
id
,
ι
}
.
Assume that
ord
(
v
)
=
p
k
and
p
≠
2
.
If
ord
(
w
j
)
=
p
j
k
j
and
p
j
≠
p
for all
j
=
1
,
…
,
m
, then
Z
(
Aut
(
G
Γ
)
)
≅
(
Z
/
p
k
Z
)
∗
.
If there exists
w
i
with
ord
(
w
i
)
=
∞
or
p
k
∣
ord
(
w
i
)
, then
Z
(
Aut
(
G
Γ
)
)
is trivial.
If there exist vertices
w
i
1
,
…
,
w
i
m
such that
ord
(
w
i
r
)
=
p
k
i
r
,
k
i
r
<
k
, then
Z
(
Aut
(
G
Γ
)
)
≅
Z
/
p
l
Z
for
l
=
min
{
p
k
−
k
i
r
∣
r
=
1
,
…
,
m
}
.
Assume that
ord
(
v
)
=
2
k
.
Assume that
k
=
1
.
Let
N
i
denote the number of orbits of the action of the graph isometry group on the vertices of order
2
i
.
Let
k
i
denote the number of orbits of vertices of order
2
i
such that
st
(
w
)
⊈
st
(
w
′
)
for all vertices
w
,
w
′
in the orbit. Finally, let
m
i
denote the number of orbits which additionally satisfy
st
(
x
)
⊈
st
(
y
)
for all vertices 𝑥 in orbits of vertices of order less than
2
i
and all vertices 𝑦 in orbits of vertices of order
2
i
.
Then
Z
(
Aut
(
G
Γ
)
≅
∏
i
∈
N
>
0
(
Z
/
2
Z
)
m
i
.
If
ord
(
v
)
=
2
k
for some
k
>
1
, then the center is the direct product of the centers obtained by (ii) (a)–(c) and (iii) (a).
Proof
With regard to (i), if
ord
(
v
)
=
∞
and
ord
(
w
j
)
<
∞
for all
j
=
1
,
…
,
m
, then the special subgroup
G
Γ
−
Δ
is characteristic by Lemma 3.7.
Thus both factors of
G
Γ
≅
G
Δ
×
G
Γ
−
Δ
are characteristic, and by Lemma 3.6, it follows that
Aut
(
G
Γ
)
≅
Aut
(
G
Δ
)
×
Aut
(
G
Γ
−
Δ
)
.
For the center of
Aut
(
G
Γ
)
, we have
Z
(
Aut
(
G
Γ
)
)
=
Z
(
Aut
(
G
Δ
)
)
×
Z
(
Aut
(
G
Γ
−
Δ
)
)
=
Z
(
Aut
(
G
Δ
)
)
=
{
id
,
ι
}
.
If
ord
(
v
)
=
∞
and there exists a vertex
w
i
with
ord
(
w
i
)
=
∞
, then we first show that a central element
f
∈
Z
(
Aut
(
G
Γ
)
)
maps 𝑣 to 𝑣.
By Lemma 7.1, we know that there exists
z
i
∈
G
Δ
such that
f
(
w
i
)
=
w
i
z
i
.
Furthermore, we also know that
f
(
v
)
=
v
ϵ
, where
ϵ
=
1
or −1.
Since the order of
w
i
is infinite, there exists the dominated transvection
ρ
w
i
v
.
Thus
f
∘
ρ
w
i
v
(
w
i
)
=
f
(
w
i
v
)
=
w
i
z
i
v
ϵ
and
ρ
w
i
v
∘
f
(
w
i
)
=
ρ
w
i
v
(
w
i
z
i
)
=
w
i
v
z
i
.
Thus
ϵ
=
1
and
f
(
v
)
=
v
.
Furthermore, for a vertex
w
j
, we know by Lemma 7.1 that there exists
v
k
∈
G
Δ
=
⟨
v
⟩
such that
f
(
w
j
)
=
w
j
v
k
.
We have
f
∘
ι
(
w
i
)
=
f
(
w
i
)
=
w
i
v
k
and
ι
∘
f
(
w
i
)
=
ι
(
w
i
v
k
)
=
w
i
v
−
k
Thus
v
k
=
v
−
k
and therefore
k
=
0
which shows that
f
=
id
.
With regard to (ii) (a), by the same argument as in (i), we know that
G
Γ
−
Δ
is characteristic.
Thus
Z
(
Aut
(
G
Γ
)
)
≅
Z
(
Aut
(
G
Δ
)
)
×
Z
(
Aut
(
G
Γ
−
Δ
)
)
=
Z
(
Aut
(
Z
/
p
k
Z
)
)
≅
(
Z
/
p
k
Z
)
∗
.
With regard to (ii) (b), with the same strategy as in (i), it is possible to show that, for
f
∈
Z
(
Aut
(
G
Γ
)
)
, we have
f
(
v
)
=
v
.
For
w
i
there exists
z
i
∈
⟨
v
⟩
such that
f
(
w
i
)
=
w
i
z
i
.
Our goal is to show that
z
i
=
1
.
If
ord
(
w
i
)
=
p
i
k
i
and
p
i
≠
p
, then
z
i
=
1
.
If
ord
(
w
i
)
=
∞
or
ord
(
w
i
)
=
p
s
, then let
l
∈
Aut
(
G
Γ
)
be the local automorphisms of
G
w
i
that maps
w
i
to
w
i
−
1
.
We have
f
∘
l
(
w
i
)
=
f
(
w
i
−
1
)
=
w
i
−
1
z
i
−
1
and
l
∘
f
(
w
i
)
=
l
(
w
i
z
i
)
=
w
i
−
1
z
i
.
Thus
z
i
=
z
i
−
1
, and therefore
z
i
2
=
1
.
Furthermore,
p
d
=
ord
(
z
i
)
∣
2
.
By assumption,
p
≠
2
, and therefore
ord
(
z
i
)
=
1
, which shows again that
f
=
id
.
With regard to (ii) (c), let
l
t
denote the map induced by
v
↦
v
t
in
Z
/
p
k
Z
.
We note that
l
t
∈
Aut
(
Z
/
p
k
Z
)
if and only if
p
∤
t
.
Furthermore,
Aut
(
Z
/
p
k
Z
)
=
{
l
t
∣
1
≤
t
<
p
k
,
p
∤
t
}
.
Due to the condition on the orders, dominated transvections are only well-defined if they have the form
ρ
w
v
p
k
−
i
, where the order of 𝑤 is
p
i
.
Analogously to before, a central element in
Aut
(
G
Γ
)
acts as the identity on
G
Γ
−
Δ
.
So, to compute the center, it is enough to compute when
ρ
w
v
p
k
−
i
and
l
t
commute.
A quick calculation shows that these commute if and only if
w
v
p
j
−
i
=
w
v
k
p
j
−
i
or in other words
p
j
−
1
≡
k
p
j
−
i
mod
p
j
.
The minimal non-trivial solution for 𝑡 is precisely
p
i
+
1
and other solutions are obtained by adding 𝑝 to 𝑡; hence one dominated transvection of this form reduces the center to
Z
/
p
j
−
i
Z
=
⟨
p
i
⟩
≤
Z
/
p
j
Z
.
Note that the generator
p
i
corresponds to the local automorphism
l
p
i
+
1
.
If there are multiple vertices, then since
p
x
∣
p
y
for all
x
<
y
, it suffices to consider the vertex group with the largest exponent; hence we are done.
With regard to (iii) (a), let us first consider the right-angled Coxeter group case.
In this case, we have
Z
(
Aut
(
G
Δ
)
)
=
{
1
}
, and hence the center lies completely in
T
Z
.
Clearly, every automorphism in
T
Z
commutes with every inner automorphism and every local automorphism as well.
Moreover, an element 𝑓 in
T
Z
has the form
w
↦
w
v
and
w
′
↦
w
′
for some partition
V
(
Γ
−
Δ
)
=
W
∪
W
′
and
w
∈
W
,
w
′
∈
W
′
.
If there exists a graph automorphism 𝜎 which maps some
w
∈
W
to a
w
′
∈
W
′
(or the other way around), then 𝑓 is not central since 𝑓 and 𝜎 do not commute.
Moreover, if there exists a dominated transvection
ρ
x
y
of two vertices
x
,
y
∈
W
, then 𝑓 is also not central since it does not commute with
ρ
x
y
.
However, 𝑓 commutes with all other dominated transvections and all other graph automorphisms, hence is central.
Now we consider the general case.
Clearly, vertex groups of order
p
l
for
p
≠
2
do not play a role since
w
↦
w
v
is not well-defined if
o
(
w
)
=
p
l
and
p
≠
2
.
Moreover, we can map every vertex 𝑤 of order
2
n
to
w
v
and obtain an element of
T
Z
.
Within the collection of vertices in
V
(
Γ
−
Δ
)
of the same order, the same restrictions as above apply about the existence of graph automorphisms and dominated transvections.
However, we also need to consider dominated transvections between vertices 𝑤 and
w
′
in
V
(
Γ
−
Δ
)
for
ord
(
w
)
≠
ord
(
w
′
)
if their respective stars satisfy the necessary condition.
Let
ord
(
w
)
=
2
n
>
2
m
=
ord
(
w
′
)
.
Suppose
f
∈
Z
(
Aut
(
G
Γ
)
)
and
f
(
w
)
=
w
v
,
f
(
w
′
)
=
w
′
v
.
Then an easy computation shows that 𝑓 commutes with
ρ
w
w
′
2
n
−
m
, but it does not commute with
ρ
w
′
w
, leading to the additional condition on the stars in the lemma.
However, if such a dominated transvection does not exist, then 𝑓 is central as in the right-angled Coxeter group case.
With regard to (iii) (b), since the order of 𝑣 is
2
k
, we now obtain local automorphisms of
G
Δ
.
However, since the order of 𝑣 is
2
k
, all local automorphisms are of the form
(
v
↦
v
2
s
+
1
)
for some
s
∈
N
.
An easy calculation shows that a dominated transvection
ρ
w
v
l
commutes with the local automorphism
v
↦
v
2
s
+
1
if and only if
ord
(
v
l
)
≤
2
(since
w
v
l
=
w
v
l
(
2
s
+
1
)
if and only if
v
2
s
l
=
1
).
Hence, applying the exact same arguments as in part (iii) (a), we obtain a factor
(
Z
/
2
Z
)
m
i
in the center.
To answer the question of whether or not some local automorphism can be central, we refer to part (ii), since these arguments show exactly in which cases a local automorphism can be central.
The assumption that
p
≠
2
was merely used to show that
Z
(
Aut
(
G
Γ
)
)
∩
T
Z
is trivial; the other arguments work even if
p
=
2
.
Hence we obtain the desired statement.
∎
Remark 7.14
If the graph Δ has more vertices than just one, similar methods can be applied to calculate the center in any given case.
More precisely, one can determine the center of
Aut
(
G
Δ
)
with Proposition 7.9 and then follow the arguments above to determine the center.
We summarize the results regarding the center of the automorphism group of a right-angled Coxeter group in the following corollary.
Corollary 7.15
Let
W
Γ
be a right-angled Coxeter group.
Let Δ be the induced subgraph of Γ generated by the vertices
v
∈
V
(
Γ
)
such that
st
(
v
)
=
V
(
Γ
)
.
Furthermore, we denote the vertices of Δ by
V
(
Δ
)
=
{
v
1
,
…
,
v
n
}
and
V
(
Γ
)
−
V
(
Δ
)
=
{
w
1
,
…
,
w
m
}
.
The center
Z
(
Aut
(
W
Γ
)
)
is non-trivial if and only if
n
=
1
,
m
≥
1
and there exists a vertex
w
j
∈
V
(
Γ
)
−
V
(
Δ
)
such that
st
(
w
j
)
⊈
st
(
w
i
)
and
st
(
w
i
)
⊈
st
(
w
j
)
for all
i
∈
{
1
,
…
,
m
}
,
i
≠
j
.
Moreover, let Ω be a subset of
V
(
Γ
)
−
V
(
Δ
)
defined as follows:
Ω
:
=
{
w
j
|
st
(
w
j
)
⊈
st
(
w
i
)
and
st
(
w
i
)
⊈
st
(
w
j
)
for all
i
∈
{
1
,
…
,
m
}
,
i
≠
j
}
.
Then Ω is preserved under the action of
Isom
(
Γ
−
Δ
)
and
Z
(
Aut
(
W
Γ
)
)
≅
(
Z
/
2
Z
)
l
,
where 𝑙 is equal to the cardinality of the set of orbits under this action.
An example of a graph satisfying the condition on the stars is the Frucht graph introduced in [11]; see Figure 7.
Hence the automorphism group of the right-angled Coxeter group associated to the join of the Frucht graph with a single vertex has non-trivial center.
More precisely, the isometry group of this graph is trivial; thus the center of the automorphism group of this right-angled Coxeter group is in fact isomorphic to
(
Z
/
2
Z
)
12
.
8 Applications
In this section, we provide two applications of our results.
The first application is to automatic continuity and the second one is to the stable rank of the reduced group
C
∗
-algebra.
We start this section by discussing the question of automatic continuity and giving some examples and non-examples.
We then prove Corollary I, which we show to be “optimal” in the case of right-angled Artin groups.
Afterwards, we move on to the second application and prove Corollary J.
Given a map
φ
:
L
→
G
between two topological groups 𝐿 and 𝐺, the automatic continuity problem is the following: assuming 𝜑 is a group homomorphism on the level of groups, find conditions on 𝐿, 𝐺 or 𝜑 which imply that 𝜑 is continuous.
Here we focus on the case where 𝐿 is an arbitrary locally compact Hausdorff group and 𝐺 has the discrete topology.
By definition, a discrete group 𝐺 is called lcH-slender if every algebraic homomorphism
φ
:
L
→
G
, where 𝐿 is a locally compact Hausdorff group, is continuous.
Important examples of lcH-slender groups are right-angled Artin groups [9, 20], torsion free hyperbolic groups [6], torsion free CAT(0) groups [10] and torsion free Helly groups [19].
An algebraic characterization of lcH-slender abelian groups was given in [8] and was generalized for lcH-slender groups in [10].
Torsion freeness is a necessary condition for lcH-slenderness.
This follows from the fact that if 𝐺 has a torsion element, then 𝐺 has an element of order 𝑝, where 𝑝 is a prime number and we can always define a discontinuous group homomorphism
φ
:
∏
N
Z
/
p
Z
↠
Z
/
p
Z
using a vector space argument (see the proof of [8, Theorem C]).
Thus if 𝐺 is infinite, then this discontinuous group homomorphism cannot interact with the entire structure of the group since its image is finite.
This leads to the following question.
Question 8.1
Under which algebraic conditions on the discrete group 𝐺 is every algebraic epimorphism
φ
:
L
↠
G
continuous?
The next lemma shows that if the center of 𝐺 is not well-behaved in the sense of automatic continuity, then there exists a discontinuous algebraic epimorphism from a locally compact Hausdorff group to 𝐺.
Lemma 8.2
Let 𝐺 be a group.
Assume that there exists a discontinuous group homomorphism
φ
1
:
L
→
Z
(
G
)
, where 𝐿 is a locally compact Hausdorff group and
Z
(
G
)
is the center of 𝐺.
Then
φ
:
L
×
G
→
G
defined as
φ
(
l
,
g
)
=
φ
1
(
l
)
⋅
g
is a discontinuous surjective group homomorphism, where 𝐺 has the discrete topology and
L
×
G
has the product topology.
Proof
The map 𝜑 is a group homomorphism since
φ
1
(
L
)
is contained in the center of 𝐺.
Furthermore, we have
φ
(
{
1
}
×
G
)
=
G
; hence 𝜑 is surjective.
We have
φ
|
L
=
φ
1
and this map is by assumption discontinuous; therefore, 𝜑 is also discontinuous.
∎
Corollary 8.3
Let 𝐺 be a group.
If the center of 𝐺 has a non-trivial finite order element, or contains ℚ as a subgroup or the 𝑝-adic integers
Z
p
for a prime 𝑝, then there exists a discontinuous algebraic epimorphism from a locally compact Hausdorff group to 𝐺.
Proof
It was proven in [10] that a group 𝐻 is lcH-slender if and only if 𝐻 is torsion free, contains no ℚ and no
Z
p
as a subgroup.
Hence there exists a discontinuous algebraic homomorphism from a locally compact Hausdorff group to
Z
(
G
)
.
Thus, by Lemma 8.2, there exists a discontinuous epimorphism from a locally compact Hausdorff group to 𝐺.
∎
Corollary 8.4
Let
G
Γ
be a graph product of finitely generated abelian groups.
If the center
Z
(
Aut
(
G
Γ
)
)
is non-trivial, then there exist a locally compact group 𝐿 and a discontinuous surjective homomorphism
φ
:
L
↠
Aut
(
G
Γ
)
.
Proof
This follows from the previous corollary since the center of the automorphism group is finite by Lemma 7.3.
∎
The question above regarding automatic continuity of epimorphisms from a locally compact Hausdorff group to a given discrete group was partially answered in [19].
Here we recall a weaker version of this result which is suitable for our investigations.
Theorem 8.5
Theorem 8.5 ([19])
Let
φ
:
L
↠
G
be an epimorphism from a locally compact Hausdorff group 𝐿 to a discrete group 𝐺.
If every torsion subgroup of 𝐺 is finite, 𝐺 does not contain ℚ or the 𝑝-adic integers
Z
p
as a subgroup and 𝐺 has no non-trivial finite normal subgroups, then 𝜑 is continuous.
It is known that, for a given finite group 𝐺, there always exists a discontinuous surjective group homomorphism from the compact group
∏
N
G
into 𝐺; see [25, Example 4.2.12].
In general, it may be difficult to find a discontinuous surjective group homomorphism.
Nevertheless, we conjecture the following result.
Conjecture 8.6
Let 𝐺 be a group.
Assume that every torsion subgroup in 𝐺 is finite, 𝐺 does not contain ℚ or the 𝑝-adic integers
Z
p
as a subgroup for any prime 𝑝.
The following statements are equivalent.
Every epimorphism from a locally compact Hausdorff group 𝐿 to 𝐺 is continuous.
The group 𝐺 has no non-trivial finite normal subgroups.
Corollary 8.3 implies that if 𝐺 has a non-trivial finite normal subgroup 𝑁 such that
N
⊆
Z
(
G
)
, then there exists a discontinuous epimorphism from a locally compact Hausdorff group to 𝐺.
Furthermore, if 𝐺 has a non-trivial finite normal subgroup 𝑁 such that
G
≅
N
×
M
, then there exists a discontinuous epimorphism
φ
:
∏
N
N
×
M
↠
G
.
Moreover, the centralizer of a non-trivial finite normal subgroup
N
⊴
G
, defined by
Z
G
(
N
)
=
{
g
∈
G
∣
g
n
=
n
g
for all
n
∈
N
}
, has finite index in 𝐺; therefore, the same construction as in Lemma 8.2 shows that there exists a surjective, discontinuous group homomorphism
φ
:
∏
N
N
×
Z
G
(
N
)
↠
⟨
N
,
Z
G
(
N
)
⟩
,
where the subgroup
⟨
Z
G
(
N
)
,
N
⟩
has finite index in 𝐺.
Thus if 𝐺 has a non-trivial finite normal subgroup, there exists a surjective, discontinuous group homomorphism from a locally compact Hausdorff group to a finite index subgroup of 𝐺.
Recall that a group 𝐺 is called complete if 𝐺 is centerless and every automorphism of 𝐺 is inner.
For example, the group
Sym
(
n
)
is complete for
n
≠
2
,
6
.
The notion of completeness goes back to Hölder [17], where he studied decompositions of a group 𝐺.
More precisely, given a group 𝐺 and a fixed normal subgroup 𝑁 of 𝐺, one can ask the question how the group 𝑁 is involved in the decompositions of a group 𝐺 in smaller pieces where one piece is equal to the group 𝑁.
Hölder proved that if 𝑁 is complete, then any short exact sequence
{
1
}
→
N
→
G
→
M
→
{
1
}
splits and 𝐺 is isomorphic to the direct product
N
×
M
.
Hence if 𝐺 has a non-trivial complete finite normal subgroup, then there exists a discontinuous epimorphism from a locally compact Hausdorff group to 𝐺.
Given a graph product
G
Γ
of cyclic groups, Corollary 4.3 implies that torsion subgroups in
Aut
(
G
Γ
)
are finite. Moreover, by Lemma 4.4, we also know that ℚ is not a subgroup of
Aut
(
G
Γ
)
.
Additionally,
Aut
(
G
Γ
)
is finitely generated by Theorem 4.1, hence countable and therefore cannot contain the uncountable group
Z
p
as a subgroup.
Hence Theorem 8.5 immediately implies the following result.
Corollary 8.7
Let
φ
:
L
↠
Aut
(
G
Γ
)
be an abstract group epimorphism from a locally compact Hausdorff group 𝐿 to the automorphism group of a graph product of cyclic groups
G
Γ
.
If
Aut
(
G
Γ
)
has no non-trivial finite normal subgroups, then 𝜑 is continuous.
In the special case where
G
Γ
is a right-angled Artin group, we obtain a precise characterization of automatic continuity.
Corollary 8.8
Let
φ
:
L
↠
Aut
(
A
Γ
)
be an abstract group epimorphism from a locally compact Hausdorff group 𝐿 into the automorphism group of a right-angled Artin group
A
Γ
.
If Γ is not a clique, then 𝜑 is continuous.
If Γ is a clique, then
π
∘
φ
:
L
↠
Aut
(
A
Γ
)
≅
GL
n
(
Z
)
↠
PGL
n
(
Z
)
is continuous, where 𝜋 is the canonical projection.
Moreover, there exists a discontinuous epimorphism
φ
:
∏
N
Z
/
2
Z
×
GL
n
(
Z
)
↠
GL
n
(
Z
)
.
Proof
The first result follows immediately from Corollary 8.7 and Corollary C.
Furthermore, the group
PGL
n
(
Z
)
has no finite normal subgroups and also satisfies the properties of Theorem 8.5; therefore, every algebraic group epimorphism from a locally compact Hausdorff group 𝐿 to
PGL
n
(
Z
)
is continuous.
Now we give a discontinuous epimorphism from a locally compact Hausdorff group to
GL
n
(
Z
)
.
The center of
GL
n
(
Z
)
is equal to
{
I
,
−
I
}
.
Thus there exists a discontinuous group homomorphism
φ
1
:
∏
N
Z
/
2
Z
→
Z
/
2
Z
→
{
I
,
−
I
}
; for an explicit definition of this group homomorphism, see [24].
By Lemma 8.2, the group homomorphism
φ
:
∏
N
Z
/
2
Z
×
GL
n
(
Z
)
↠
GL
n
(
Z
)
,
where
φ
(
l
,
g
)
=
φ
1
(
l
)
⋅
g
, is a discontinuous surjective group homomorphism.
∎
For the second part of this section, we deal with reduced group
C
∗
-algebras.
Associated to a group 𝐺, there exists an associated reduced group
C
∗
-algebra which we denote by
C
r
∗
(
G
)
. For the precise definition and more information, see for example [14].
We say
C
r
∗
(
G
)
has stable rank1 if the group of invertible elements is dense in
C
r
∗
(
G
)
.
Recall that a group 𝐺 is called acylindrically hyperbolic if it acts acylindrically on a Gromov-hyperbolic space.
A good reference for reduced group
C
∗
-algebras of acylindrically hyperbolic groups is [14] and a good reference for acylindrical hyperbolicity of automorphism groups of graph products is [13].
We need not go into more detail here and instead move on to the proof of Corollary J.
Corollary J
Let
G
Γ
be a graph product of finitely generated groups.
If Γ is not a join, contains at least two vertices and
G
Γ
≇
Z
/
2
Z
∗
Z
/
2
Z
, then the stable rank of the reduced group
C
∗
-algebra of
Aut
(
G
Γ
)
is equal to 1.
In particular, if
G
Γ
is a right-angled Artin group and Γ is not a join and contains at least two vertices, then the stable rank of the reduced group
C
∗
-algebra of
Aut
(
G
Γ
)
is equal to 1.
Proof
First we invoke [13, Theorem 1.1] to see that
Aut
(
G
Γ
)
is an acylindrically hyperbolic group.
Then [14, Theorem 1.1] implies that the stable rank of
C
r
∗
(
Aut
(
G
Γ
)
)
is 1 if the amenable radical is trivial, which is in particular the case if there are no non-trivial finite normal subgroups.
Proposition E and Theorem G imply that, in the setting of Corollary J,
Aut
(
G
Γ
)
has no non-trivial finite normal subgroup, which proves Corollary J.
∎
Contrary to the automatic continuity situation, we do not know whether this result is “optimal” even for right-angled Artin groups.
In fact, we believe even the “easiest” case to be open, which is the following question.
Question 8.9
Let
n
≥
2
; is the stable rank of
C
r
∗
(
GL
n
(
Z
)
)
equal to 1?
Acknowledgements
We want to thank the referee for many helpful remarks.
We would like also to thank Samuel M. Corson, Dominic Enders and Linus Kramer for interesting discussions and remarks concerning automatic continuity.
Additionally, we would like to thank Daniel Keppeler, Luis Paris and Petra Schwer for many useful comments and remarks on an earlier version of this article.
On top of that, we want to thank Maria Gerasimova, Tim de Laat and Hannes Thiel for very helpful discussions regarding
C
∗
-algebras.
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