Centralizers of automorphisms permuting free generators

Olga Macedońska; Witold Tomaszewski

doi:10.1515/math-2019-0007

Article Open Access

Centralizers of automorphisms permuting free generators

Olga Macedońska and Witold Tomaszewski

Published/Copyright: February 17, 2019

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$Open Mathematics$

From the journal Open Mathematics Volume 17 Issue 1

Abstract

By σ ∈ S_km we denote a permutation of the cycle-type k^m and also the induced automorphism permuting subscripts of free generators in the free group F_km. It is known that the centralizer of the permutation σ in S_km is isomorphic to a wreath product Z_k ≀ S_m and is generated by its two subgroups: the first one is isomorphic to Zkm , the direct product of m cyclic groups of order k, and the second one is S_m. We show that the centralizer of the automorphism σ ∈ Aut(F_km) is generated by its subgroups isomorphic to Zkm and Aut(F_m).

Keywords: permutation; centralizer; automorphism

MSC 2010: 20E05; 20E36; 20F28

1 Introduction

This paper was inspired by a question of Vitaly Sushchanskyy who asked about the structure of centralizers of automorphisms permuting free generators in a free group.

Another motivation of the present paper are numerous papers in which the authors investigate centralizers of finite subgroups of both Aut(F_n) and Out(F_n). K. Vogtmann’s paper [1] is the vast survey on this and related topics with many references. We present here a few results. If G is a finite subgroup of Aut(F_n), then E = F_n ⋊ G is a finite extension of F_n. In general, if E is given by the short exact sequence 1 → F_n → E → K → 1, then conjugation action of E on F_n induces a homomorphism θ: K → Out(F_n). There exists a connection between the group of outer automorphisms of E and the centralizer of θ(K). For example, it is known (see in [2, 3]) that Out(E) is finite if and only if the centralizer of θ(K) is finite. D. Boutin in [2] and M. Pettet in [3] describe finite extensions E of a free group F_n for which the group Out(E) is finite. In the present paper, if an automorphism of F_n is induced by a permutation σ of the cycle-type k^m (m cycles of length k), then the centralizer of σ in Aut(F_n) is finite if and only if σ is a long cycle (e.g. it is the cycle without fixed points). Since the automorphism induced by the permutation σ of the cycle-type k^m has no fixed points, there are no non-trivial inner automorphisms in the centralizer C(σ). Hence the subgroup C(σ) < Aut(F_n) is isomorphic to its image in Out(F_n). Y. Algom-Kfir and C. Pfaff study in [4] a centralizer of a subgroup of Out(F_n) cyclically generated by lone axis fully irreducible outer automorphisms. The centralizer of such group is infinite cyclic group. It follows from the main theorem of the present paper that the centralizer of an automorphism induced by a permutation of a cycle-type k^m is either finite cyclic or infinite non-abelian. S. Krstič proves in [5] (Theorem 2) that the centralizer of a finite subgroup of Aut(F_n) is finitely presentable.

Let σ be a permutation of the cycle-type k^m (m cycles of length k). We use the same letter σ for the induced automorphism of the free group F_km with standard basis X = {x₁, …, x_km}. The automorphism σ acts on the basis X by permuting subscripts of generators, which we write as:

σ:xi→xσ(i)orxiσ=xσ(i). (1)

We describe here the structure of the centralizer C(σ) ⊆ Aut(F_km) of the automorphism σ induced by the permutation σ.

It is known that the centralizer of σ in S_km is isomorphic to the wreath product Z_k ≀ S_m of a cyclic group of order k and a symmetric group S_m (see for example [6, 7] for details) and it is generated by subgroups isomorphic to Zkm and S_m (in fact C(σ) ≅ Zkm ⋊ S_m). We show that the centralizer of the automorphism induced by σ in the free group F_km has a similar structure and is generated by subgroups isomorphic to Zkm and Aut(F_m) but unfortunately in this case Zkm is not a normal subgroup of Aut(F_m).

2 Notation

Since the structure of the centralizer C(σ) does not depend on the contents of cycles of the permutation σ, we shall assume:

σ=(1,…,k)(k+1,…,2k)…((m−1)k+1,…,mk). (2)

The elements of the basis X of the free group F_km can be written as the entries of the k×m-matrix with columns denoted by 𝓧_i so that the columns correspond to the σ-orbits.

X=x1xk+1…x(m−1)k+1x1σxk+1σ…x(m−1)k+1σx1σ2xk+1σ2…x(m−1)k+1σ2⋮⋮⋱⋮x1σk−1x2σk−1…xmσk−1=x1xk+1…x(m−1)k+1x2xk+2…x(m−1)k+2⋮⋮⋱⋮xkx2k…xmk=(X1,X2,…,Xm). (3)

Note that the subscripts of the generators in columns are the elements of orbits of the permutation σ. Note also, that the first row of this matrix defines the other rows by applying the automorphism σ.

3 Properties of automorphisms commuting with σ

If A := {a₁, a₂, …, a_km} is another basis in F_km, then the action of σ on a_i is defined by its action on X. We are interested in the case, when the basis A is also a union of σ-orbits.

Definition 1

The basis A := {a₁, a₂, …, a_km} in F_km is called a σ-basis if it can be ordered so that

σ:ai→aσ(i),(aiσ=aσ(i)), (4)

that is for σ of the form (2) the σ-basis A can be written as

A=a1ak+1…a(m−1)k+1a1σak+1σ…a(m−1)k+1σa1σ2ak+1σ2…a(m−1)k+1σ2⋮⋮⋱⋮a1σk−1a2σk−1…a(m−1)k+1σk−1=a1ak+1…a(m−1)k+1a2ak+2…a(m−1)k+2⋮⋮⋱⋮aka2k…amk=(A1,A2,…,Am). (5)

Example 1

Let σ = (1, 2, 3)(4, 5, 6), n = 6, X = {x₁, x₂, x₃, x₄, x₅, x₆}. Take another basis in F₆: A = {x₁x₄, x₂x₅, x₃x₆, x₄, x₅, x₆}.

X=x1 x4x2 x5x3 x6,A=x1x4 x4x2x5 x5x3x6 x6

We formulate now a criterion for an automorphism α ∈ Aut(F_n) to centralize the automorphism σ ∈ Aut(F_n).

Proposition 1

An automorphism α ∈ Aut(F_n, α : x_i → a_i, X → A commutes with the automorphism σ if and only if A is a σ-basis.

Proof

Let α : x_i → a_i. Then xiασ=aiσ and xiσα=(xσ(i))α=aσ(i). So the equality ασ = σα holds if and only if (a_i)^σ = a_σ(i) as required. □

Corollary 1

An automorphisms α is in C(σ) if and only if it maps each orbit {x, x^σ, …, x^{σ^k−1}} onto some orbit {a, a^σ, …, a^{σ^k−1}}.

Example 2

An automorphism δ, acting as σ in only one σ-orbit commutes with σ, hence it is the σ-automorphism and is in C(σ).

4 m-tuples and Nielsen transformations

We remind now some useful notions from [8, Chapter 3].

By a basic m-tuple we mean any ordered set of free generators in a free group F_m with the basis (w₁, …, w_i, …, w_j, …, w_m). The following m-tuples are called elementary m-tuples:

(i) (w1,...wi−1,...wj,...wm), replaced wi with wi−1, (ii)(w1,...wiwj,...wj,...wm), replaced wi with wiwj(or wjwi),(iii) (w1,...wj,...,wi,...wm), switched wi and wj. (6)

The basic m-tuples in F_m form a group with the unity element X = (x₁, x₂, …, x_m) and with the following multiplication:

(a1,a2,...,am)⋅(b1,b2,...,bm)=(a1(bν),a2(bν),...,am(bν)). (7)

Example 3

(x1x3,x2x3,x3−1)⋅(b2b3−1,b3b1,b1)=(b2b3−1b1,b3b12,b1−1).

(x1x3,x2x3,x3−1)⋅(x2x3−1,x3x1,x1)=(x2x3−1x1,x3x12,x1−1).

Each m-tuple (b₁, b₂, …, b_m) defines an automorphism β : x_i → b_i which acts on m-tuple (a₁, a₂, …, a_m) by multiplication from the right, as in (7),

(a1,a2,...,am)β=(a1(bν),a2(bν),...,am(bν))=Xαβ.

To define the notion of a Nielsen transformation, we recall that in matrix theory, to switch i-th and j-th rows in a matrix M, we multiply M from the left by the identity matrix with switched i-th and j-th rows. The action of the Nielsen transformations is similar, namely:

Each m-tuple (a₁, a₂, …, a_m) defines a Nielsen transformation N_α acting on another m-tuple (b₁, b₂, …, b_m) by multiplication from the left, as in

Nα(b1,b2,...,bm)=(a1(bν),a2(bν),...,am(bν)).

For example if n = 2, the Nielsen transformation (x₁x₂, x₁) changes the basis (a, b) into (x₁x₂, x₁)·(a, b) = (ab, a).

SinceNαNβX=Nα(b1,b2,...,bn)=(a1(bν),a2(bν),...,an(bν))=Xαβ,

the group of Nielsen transformations and the group of automorphisms Aut(F_n) are anti-isomorphic [8] (sec 3.2, (11)) and the following equality holds

Nα1Nα2...NαkX=Xα1α2...αk.

Nαk...Nα2Nα1X=Xαk...α2α1. (8)

The elementary m-tuples (6) define the so called elementary automorphisms and elementary Nielsen transformations.

5 SE-automorphisms

Let F^m,F^mσ,⋯,F^mσk−1 be the free subgroups generated by the elements of the rows in the matrix (3), then

Fkm=F^m∗F^mσ∗⋯∗F^mσk−1,〈x1xk+1…x(m−1)k+1〉〈x1σxk+1σ…x(m−1)k+1σ〉⋮⋮⋱⋮〈x1σk−1xk+1σk−1…x(m−1)k+1σk−1〉=F^mF^mσ⋮F^mσk−1.

If the same elementary automorphism τ acts on every row (in every F^mσi ), then it defines a so-called simultaneous elementary automorphism which we address as a SE-automorphism in F_km [9].

Note that if the same τ acts on every row, then it acts on the row of columns, which generate the free group Fm~ of rank m

Fm~:=〈X1,X2,…,Xm〉.

So the elementary automorphism τ in Fm~ induces the above SE-automorphism in F_km. We can write this as

Corollary 2

An elementary automorphism τ in the free m-generator group Fm~ = 〈𝓧₁, 𝓧₂, …, 𝓧_m〉 defines the SE-automorphism in Fkm=F^m∗⋯∗F^mσk−1 acting as τ in each of k factors F^mσi.

There are three types (a), (b), (c) of the elementary automorphisms τ in Fm~ , defining the SE-automorphisms in Fkm=F^m∗F^mσ∗⋯∗F^mσk−1 :

for a fixed i and all x ∈ 𝓧_i, x^τ = x⁻¹. while the elements in other orbits are fixed, X^τ = (𝓧₁, 𝓧₂, …, Xi−1 , …, 𝓧_j, …, 𝓧_m),
for fixed i, j on each level in (3), for all x ∈ 𝓧_i y ∈ 𝓧_j: x^τ = xy (or yx), while 𝓧_j and other orbits are fixed, so

X^τ = (𝓧₁, 𝓧₂}, …, 𝓧_i𝓧_j, …, 𝓧_j, …, 𝓧_m) or

X^τ = (𝓧₁, 𝓧₂, …, 𝓧_j𝓧_i, …, 𝓧_j, …, 𝓧_m),
for fixed i, j, and all x ∈ 𝓧_i y ∈ 𝓧_j on each level: x^τ = y, y^τ = x. Other orbits are fixed, so X^τ = (𝓧₁, 𝓧₂, …, 𝓧_j, …, 𝓧_i, …, 𝓧_m).

Proposition 2

The SE-automorphisms (a), (b), (c) in Aut(F_km) generate a subgroup in the centralizer C(σ) isomorphic to Aut(F_m).

Proof

Since by [8] (Theorem 3.2 on page 131) the elementary automorphisms generate the full automorphism group, it follows by Corollary 2 that the group generated by all SE-automorphisms in Fkm=F^m∗F^mσ∗⋯∗F^mσk−1 is isomorphic to the group Aut( Fm~ ) and hence to Aut(F_m).

Each elementary automorphism in Fm~ has one of the forms (a),(b), (c), changes the basis X into a σ-basis X^τ, and by Proposition 1 is in the centralizer of σ, which finishes the proof. □

The subgroup generated by the SE-automorphisms is the proper subgroup in the centralizer C(σ), since it does not contain automorphisms δ_i, which maps each generator a in the i−th column to a^σ (vertical permutations in i−th columns). So we have to consider also this type of σ-automorphisms mentioned in Example 4.

(d) for a fixed i and all y ∈ 𝓧_i, y^δ_i := y^σ, that is δ_i acts on 𝓧_i as a cyclic permutation of order k, while the other orbits are fixed,

Xδi:=X1,X2,…,Xiσ,…,Xj,…,Xm.

Proposition 3

The automorphisms of the type (d) form a subgroup isomorphic to the direct product Zkm of m cyclic groups of order k.

Note The σ-automorphism (a), (b), (c) act in rows of the basis-matrix, while (d) permutes vertically elements from different rows.

Our goal now is to show that the SE-automorphisms (a), (b), (c), and (d) generate the centralizer of σ in Aut(F_km). To proceed we shall use the Nielsen transformations on the m-tuples of orbits.

6 Transformations for m-tuples of σ-orbits

Let F_km be a free group with the σ-basis A which is the union of m σ-orbits.

A=Xα=A1,A2,…,Ai,…,Am.

Note that the Nielsen transformations of this m-tuples into

(𝓐₁, 𝓐₂, …, Ai−1 , …, 𝓐_j, …, 𝓐_m),
(𝓐₁, 𝓐₂, …, 𝓐_i𝓐_j, …, 𝓐_j, …, 𝓐_m) or

(𝓐₁, 𝓐₂, …, 𝓐_j𝓐_i, …, 𝓐_j, …, 𝓐_m),
(𝓐₁, 𝓐₂, …, 𝓐_j, …, 𝓐_i, …, 𝓐_m),
(𝓐₁, 𝓐₂, …, Aiσ , …, 𝓐_j, …, 𝓐_m),

define the ES-automorphisms in F_km. (For convenience we shall call the automorphisms and transformations (d) also elementary)

Corollary 3

Since the set of σ-bases in F_n is closed under the above Nielsen transformations on m-tuples of σ-orbits the equality (8) holds:

Nτk...Nτ2Nτ1A=Aτk...τ2τ1,

where the elementary automorphisms τ correspond to the elementary Nielsen transformations.

7 Transformation of a σ-basis A to the standard basis X

We have to show that each σ-basis A written as (5) can be transformed to the σ-basis X written as (3) by the sequence N_τ₁, …, N_{τ_k} of elementary Nielsen transformations, applied to the m-tuples of orbits, which are the columns in the matrices. The proof is similar to that for Theorem 3.1 in [8] and uses the following terminology for the words in the free group F_n = 〈 x₁, x₂, …, x_n〉, n = km:

A word is freely reduced if no cancellations in it is possible.
An x-length |a| of a word a is a number of xi±1 in its freely reduced form.
The x-length of a basis A is the sum of x-lengths of all its elements.
A major initial (major terminal) segment of a word has a minimally bigger length then half of the word’s length.
A major initial (major terminal) segment of a word a ∈ A is isolated if no word b ∈ A has such an initial (or terminal) segment. Note that otherwise |a⁻¹b| < |b| (or |ba⁻¹| < |b| ).
A subset S in the free group F_n is Nielsen-reduced if the major initial and major terminal segments of each a ∈ S are isolated and for an element a of even length either its left or its right half is isolated.
The basis X in F_km is Nielsen-reduced and has the minimal x-length, which is equal to n = km.

Since by Definition 1, a σ-basis A in general case has a form

A=w1w2…wmw1σw2σ…wmσ⋮⋮⋱⋮w1σk−1w2σk−1…wmσk−1=(A1,A2,A3,…,Am), (9)

and the permutation σ has no fixed points, it follows that every σ-orbit {wi,wiσ,…,wiσk−1} is a Nielsen reduced set.

Theorem 1

For each σ-basis A in F_km there is a sequence of Nielsen transformations N_τ₁, N_τ₂, …, N_{τ_r} of types in {(a), (b), (c), (d)}, such that

Nτr…Nτ1A=X.

Proof

Let A = (𝓐₁, 𝓐₂, …, 𝓐_m) be the freely reduced σ-basis in F_km written as the matrix (9). Since elements in columns have different first (last) letters x_i, we have that the subsets in each column are Nielsen-reduced, while the whole basis A need not be Nielsen-reduced.

Assume first, that the x-length of the σ-basis A is equal km. Then its entries are of the form xiε , ε = ±1. Since (xiε)σ=(xiσ)ε, all elements in a σ-orbit have the same value of ε. Then by the Nielsen-transformations of the type (a) we can eliminate all ε = −1. Now by permuting (cyclically) elements in the columns by transformations (d), we get the proper order inside each column. Then the permutations of the columns (transformations (c)) lead to the basis X, as required.
Let now the x-length of the σ-basis A be greater than km. Then by Lemma 3.1 in [8] the basis A is not Nielsen-reduced. It follows that there is a word a = uv in some column 𝓐_i in (9) which has a major (say initial) segment u (|u| > |v|), equal to the initial segment of some word b = uw in some other column 𝓐_j in (9). We apply the Nielsen transformations of the type (d) to these columns to get the top elements in i-th and j-th columns respectively:

uv and uw

Then the i-th and j-th columns consist of elements uv, (uv)^σ, … and uw, (uw)^σ, … respectively. Now we change 𝓐_i to Ai−1 by transformation (a) and apply transformation (b) to change 𝓐_j to Ai−1 𝓐_j. We get the top elements in i-th and j-th columns

(uv)−1 and v−1w.

Since |u| > |v|, we get |uw| > |vw| ≥ |v⁻¹w|. It diminishes the length of the j-th column at least for m and hence diminishes the x-length of the σ-basis A. Similarly we can isolate the halves of the entries. By repeating these steps we get the Nielsen reduced σ-basis of minimal length equal to km, which was considered in the part 1 of the proof. So the proof is complete. □

Example 4

Let σ = (1, 2, 5)(3, 4, 6). We transform

the σ-basis A = X^α = {x₁x₆, x₂x₃, x₅x₄}{x₄x₂x₃, x₆x₅x₄, x₃x₁x₆}

to the σ-basis X = {x₁, x₂, x₅}{x₃, x₄, x₆}.

A=x1x6x4x2x3x2x3x6x5x4x5x4x3x1x6→(d)x1x6x3x1x6x2x3x4x2x3x5x4x6x5x4→(a)(x1x6)−1x3x1x6(x2x3)−1x4x2x3(x5x4)−1x6x5x4→(b)(x1x6)−1x3(x2x3)−1x4(x5x4)−1x6→(d)(x1x6)−1x6(x2x3)−1x3(x5x4)−1x4→(b)x1−1x6x2−1x3x5−1x4→(a)x1x6x2x3x5x4→(d)x1x3x2x4x5x6=X.

8 The Main Theorem

Theorem 2

The centralizer C(σ) in Aut(F_km) for the automorphism σ with m orbits of order k is generated by the SE-automorphisms (a), (b), (c) and (d).

Proof

Let an automorphism α ∈ C(σ) map X → A,

A=Xα. (10)

By Theorem 1 the basis A can be changed into X by a sequence of transformations N_τ of the types (a), (b), (c), (d), then by (8) and (10)

X=Nτk...Nτ2Nτ1A=Aτk...τ2τ1=Xατk...τ2τ1,

and hence, for the corresponding SE-automorphisms τ we get

α=τ1−1τ2−1...τk−1

is a product of SE-automorphisms of types (a), (b), (c) and (d) as required. □

Corollary 4

By Propositions 2 and 3, for C(σ) in Aut(F_km) we have

C(σ)≃〈Zkm,Aut(Fm)〉,

which shows the similarity with the results from [6, 7] for permutation σ ∈ S_km and C(σ) ⊆ S_km but in the case of automorphism permuting generators Zkm is not a normal subgroup of C(σ).

Theorem 3

The centralizer C(σ) in Aut(F_km) for the automorphism σ with m cycles of order k can be generated by the automorphism cyclically permuting elements in the first σ-orbit (the type (d)), and in addition, by two automorphisms if m ≥ 4 and by three automorphisms for m = 2, 3. Moreover, C(σ) is finitely presented.

Proof

By Proposition 2, the SE-automorphisms in F_km, being the elementary automorphisms in the free group 〈 𝓧₁, 𝓧₂, …, 𝓧_m〉, generate the group

Aut(〈X1,X2,…,Xm〉)≅Aut(Fm).

It is shown in [10] (see [8], p.165), that the group Aut(F_m) can be generated by two automorphisms which map (for m = 4, 6, 8, …) the free generators as:

X1,X2,…,Xm→X2,X3,…,Xm,X1,X1,X2,…,Xm→X2−1,X1,X3,…,XmXm−1−1,Xm−1−1.

and for odd m ≥ 5 by two automorphisms:

X1,X2,…,Xm→X2−1,X3−1,…,Xm−1,X1−1,X1,X2,…,Xm→X2−1,X1,X3,…,XmXm−1−1,Xm−1−1.

For m = 2, 3 we need three additional automorphisms.

The centralizer C(σ) is finitely presented by Krstič’s Theorem (see [5], Theorem 2). □

Dedicated to the memory of Vitaly Sushchanskyy (1946-2016)

Acknowledgement

The authors are grateful to the Referee for the additional information.

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Received: 2018-03-01

Accepted: 2019-01-03

Published Online: 2019-02-17

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Keywords for this article

permutation; centralizer; automorphism

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