1 Introduction
The concept of prime localization of a totally disconnected locally compact group G was introduced by Reid in [8]. Let p be prime. A local p-Sylow subgroup of G is a maximal pro-p subgroup of a compact open subgroup of G. The p-localizationG(p) of G is defined as the commensurator CommG(S) of a local p-Sylow subgroup S of G, equipped with the unique group topology which makes the inclusion of S into G(p)=CommG(S) continuous and open. We refer the reader to [8] for general properties of prime localization and its applications, of which we highlight the scale function introduced by Willis in [11].
For a set Ω of cardinality d∈ℕ≥3 and F≤F′≤Sym(Ω), the Burger–Mozes group U(F) and the Le Boudec group G(F,F′) act on a suitably colored d-regular tree Td with local action prescribed by F and F′. Lederle’s colored Neretin group N(F) consists of almost automorphisms of Td associated to U(F).
For a large family of the above groups, we determine local p-Sylow subgroups in terms of a p-Sylow subgroup of F. Let T⊆Td denote a finite subtree. For H≤Aut(Td) we let HT denote the pointwise stabilizer of T in H.
Proposition 10.
Let F≤Sym(Ω) and let F(p)≤F be a p-Sylow subgroup. Then U(F(p))T is a p-Sylow subgroup of U(F)T if and only if so is F(p)ω≤Fω for all ω∈Ω.
After collecting criteria and examples for the above situation, we determine general subgroups of their p-localization which we use to identify the latter as a group of the same type in certain cases. Recall that U(F)=G(F,F). In the following, F^ denotes the maximal subgroup of Sym(Ω) preserving the partition F\Ω setwise.
Theorem 17.
Let F≤F′≤F^≤Sym(Ω) and let F(p)≤F be a p-Sylow subgroup of F. Assume that we have F\Ω=F(p)\Ω and NFω′(F(p)ω)=F(p)ω for all ω∈Ω. Then G(F,F′)(p)=G(F(p),F′).
Theorem 18.
Let F≤Sym(Ω) and let F(p)≤F be a p-Sylow subgroup. Assume that we have F\Ω=F(p)\Ω and NF^ω(F(p)ω)=F(p)ω for all ω∈Ω. Then N(F)(p)=N(F(p)).
2 Preliminaries
In order to provide concise definitions of Burger–Mozes-type groups, we adopt Serre’s graph theory notation, see [9]: A graph consists of a vertex setV and an edge setE, together with a fixed-point-free involution of E denoted by e↦e¯ and maps o,t:E→V, providing the origin and terminus of an edge e such that o(e¯)=t(e) and t(e¯)=o(e). For x∈V, let E(x):={e∈E∣o(e)=x} be the set of edges issuing from x. Let Ω be a set of cardinality d∈ℕ≥3 and Td=(V,E) the d-regular tree. A legal coloringl of Td is a map l:E→Ω such that for every x∈V the map lx:E(x)→Ω, y↦l(y) is a bijection, and l(e)=l(e¯) for all e∈E. Given T⊆Td and H≤Aut(Td), we let HT denote the fixator of T in H.
Burger–Mozes groups
The universal groups introduced by Burger–Mozes in [2, Section 3.2] provide an equally rich and manageable class of groups acting on trees. The map
σ:Aut(Td)×V→Sym(Ω),(g,x)↦lgx∘g∘lx-1
captures the local permutation σ(g,x) of an automorphism g∈Aut(Td) at x∈V. To every permutation group F≤Sym(Ω) we associate a universal group acting on Td locally like F.
Definition 1.
Let F≤Sym(Ω). Set
U(F):={g∈Aut(Td)∣σ(g,x)∈F for all x∈V}.
Passing to a different legal coloring amounts to passing to a conjugate of U(F) in Aut(Td) which justifies omitting explicit reference to the legal coloring. For example, U(Sym(Ω))=Aut(Td) whereas U({id}) is the discrete cocompact subgroup generated by the color-preserving inversions of the edges in E(x) for a given vertex x∈V.
Remark 2.
Let F≤Sym(Ω). Elements of U(F) are readily constructed: Given v,w∈V(Td) and τ∈F, define g:B(v,1)→B(w,1) by setting g(v)=w and σ(g,v)=τ. Given a collection of permutations (τω)ω∈Ω such that τ(ω)=τω(ω) for all ω∈Ω, there is a unique extension of g to B(v,2) such that σ(g,vω)=τω, where vω∈S(v,1) is the unique vertex with l(v,vω)=ω. Then proceed iteratively.
Recall that Aut(Td) is a totally disconnected locally compact group with the permutation topology for its action on V. The following collection of properties is implicit in [2, Section 3.2] and elaborated in [5, Proposition 4.6].
Proposition 3 ([2, Section 3.2]).
Let F≤Sym(Ω). Then the group U(F) is
locally permutation isomorphic to F,
edge-transitive if and only if F is transitive, and
discrete in Aut(Td) if and only if F is semiregular.
As a consequence of the above, U(F) is a (compactly generated) totally disconnected locally compact group in its own right for the subspace topology of Aut(Td). For future reference, we also state the following, see [5, Section 4.1].
Proposition 4.
If F≤Sym(Ω), then U(F) satisfies Tits’ Independence Property.
Le Boudec groups
In [1], Le Boudec introduces groups acting on Td locally like F≤Sym(Ω)almost everywhere. The precise definition reads as follows.
Definition 5.
Let F≤Sym(Ω). Set
G(F):={g∈Aut(Td)∣the set {x∈V∣σ(g,x)∉F} is finite}.
Notice that U(F) is a subgroup of G(F). We equip G(F) with the unique group topology making the inclusion U(F)↣G(F) continuous and open. It exists essentially due to the fact that G(F) commensurates a compact open subgroup of U(F), see [1, Lemma 3.2]. We state explicitly that this topology differs from the subspace topology of Aut(Td), see e.g. Proposition 8 below. However, it entails that G(F) is locally isomorphic to U(F).
Given g∈G(F), a vertex v∈V with σ(g,v)∉F is a singularity. The local action at singularities is restricted as follows.
Lemma 6 ([1, Lemma 3.3]).
Let F≤Sym(Ω) and g∈G(F) with a singularity v∈V. Then σ(g,v) preserves the partition F\Ω of Ω into F-orbits setwise.
For F≤Sym(Ω), the maximal subgroup of Sym(Ω) which preserves the partition F\Ω=⊔i∈IΩi setwise is F^:=∏i∈ISym(Ωi).
Definition 7.
Let F≤F′≤F^≤Sym(Ω). Set G(F,F′):=G(F)∩U(F′).
We remark that G(F,F)=U(F) and G(F,F^)=G(F).
Proposition 8.
Let F⪇F′≤F^≤Sym(Ω) and b∈V(Td). Then G(F,F′)b is non-compact and residually discrete.
Proof.
The vertex stabilizer G(F,F′)b can be written as the (strictly) increasing union G(F,F′)b=⋃n∈ℕKn of the open sets Kn, consisting of the elements of G(F,F′)b whose singularities are contained in B(b,n). Hence it is non-compact.
As to residual discreteness, an identity neighborhood basis of G(F,F′)b consisting of open normal subgroups is given by (G(F,F′)B(b,n))n∈ℕ.
∎
Le Boudec groups enjoy many interesting properties, see [1, Introduction].
Lederle groups
As before, we consider the d-regular tree Td=(V,E) equipped with a legal coloring and a base vertex b∈V. Further, let F≤Sym(Ω). In [6], Lederle introduces an intriguing, locally isomorphic version of U(F) resembling Neretin’s group [7] and thereby generalizes Neretin’s construction.
Towards a precise definition, we recall the following from [6, Section 3.2]. A finite subtree T⊆Td is complete if it contains b and all its non-leaf vertices have valency d. We denote the set of leaves of T by L(T)⊆V(Td). Given a leaf v∈L(T), let Tv denote the subtree of Td spanned by v and those vertices outside T whose closest vertex in T is v. Then define Td\T:=⊔v∈L(T)Tv, a forest of |L(T)| trees.
Let H≤Aut(Td). Given finite complete subtrees T,T′⊆Td with |L(T)| equal to |L(T′)|, a forest isomorphism φ:Td\T→Td\T′ such that for every v∈L(T) there is hv∈H with φ|Tv=hv|Tv is an H-honest almost automorphism of Td. Two H-honest almost automorphisms of Td given by φ:Td\T1→Td\T1′ and ψ:Td\T2→Td\T2′ are said to be equivalent if there exists a finite complete subtree T⊇T1∪T2 with φ|Td\T=ψ|Td\T. Notice that for any finite complete subtree T⊇T1 there is a unique finite complete subtree T′⊇T1′ and representative φ′:Td\T→Td\T′ of φ; analogously for T1′. Hence we may pick a finite complete subtree T⊇T1′∪T2 and representatives of φ and ψ with codomain and domain equal to Td\T respectively, thus allowing for a composition of equivalence classes of H-honest almost automorphisms. Now Lederle’s colored Neretin groups (original notation ℱ(U(F))) can be defined as follows.
Definition 9.
Let F≤Sym(Ω). Set
N(F):={[φ]∣φ is a U(F)-honest almost automorphism of Td}.
Observe that N(F)∩Aut(Td)=G(F). As before, there exists a unique group topology on N(F) such that the inclusion U(F)↣N(F) is open and continuous. This is essentially due to the fact that N(F) commensurates a compact open subgroup of U(F), see [6, Proposition 2.24]. Overall, given F≤Sym(Ω), we have the following continuous and open injections:
U(F)↣G(F)↣N(F).
3 Local Sylow subgroups
This section is concerned with determining local Sylow subgroups of the Burger–Mozes-type groups. Throughout, Ω denotes a set of cardinality d∈ℕ≥3 and p is a prime. We consider the d-regular tree Td=(V,E) with a fixed legal coloring and base vertex b∈V. Furthermore, T denotes a finite subtree of Td.
Note that it suffices to consider U(F): Any local Sylow subgroup of U(F) is also a local Sylow subgroup of G(F,F′) and N(F) by definition of the topologies.
In a sense, the following proposition provides local p-Sylow subgroups of U(F) in the case where the operations of taking a p-Sylow subgroup and taking point stabilizers commute for F. It is the basis of all subsequent statements about the p-localization of Burger–Mozes-type groups and amends [8, Lemma 4.2].
Proposition 10.
Let F≤Sym(Ω) and let F(p)≤F be a p-Sylow subgroup. Then U(F(p))T is a p-Sylow subgroup of U(F)T if and only if so is F(p)ω≤Fω for all ω∈Ω.
Proof.
First, assume that the tree T consists of a single vertex b∈V. The sphere S(b,k)⊂V of radius k around b∈V is, via the given legal coloring, in natural bijection with
Pk:={w=(ω1,…,ωk)∈Ωk∣ωi+1≠ωi for all i∈{1,…,k-1}}.
The restriction of U(F) to S(b,k) yields a subgroup of Sym(S(b,k)) of cardinality given by
|U(F)b|S(b,1)|=|F|
and
|U(F)b|S(b,k+1)|=|U(F)b|S(b,k)|⋅∏w∈Pk|Fωk|.
The maximal powers of p dividing |U(F)b|S(b,k)| and |U(F(p))b|S(b,k)| are hence equal for all k∈ℕ0 if and only if F(p)ω≤Fω is a p-Sylow subgroup for all ω∈Ω.
Similarly, when T is not a single vertex, the size of the restriction of U(F)T to a sufficiently larger subtree is a product of the |Fω| involving allω∈Ω.
∎
For transitive F≤Sym(Ω), it suffices to check the above criterion for one choice of a p-Sylow subgroup F(p) of F and all ω∈Ω. We now identify classes of permutation group and values of p to which Proposition 10 applies. For the symmetric and alternating groups we have the following, complete description.
Proposition 11.
Let F=Sym(Ω) or F=Alt(Ω) and let F(p)≤F be a p-Sylow subgroup. Further, let ps (s∈N0) be the maximal power of p dividing d. Then F(p)ω≤Fω is a p-Sylow subgroup for all ω∈Ω if and only if either
F=Alt(Ω) and (d,p)=(3,2).
Proof.
If p>d, then F(p) is trivial and so is any p-Sylow subgroup of Fω. Now assume p≤d and consider the following diagram of subgroups of F and indices.
For every ω∈Ω we have
[F:Fω]=|F⋅ω|=d
and
[F(p):F(p)ω]=|F(p)⋅ω|=prω
for some rω∈ℕ0. Note that p∤k by definition. Now examine the equation d⋅[Fω:F(p)ω]=k⋅prω.
If F(p) is trivial, then F=Alt(Ω) and p is even, hence (iii). Now assume that F(p) is non-trivial. Then there is ω∈Ω such that rω≥1. Thus, if p∤d, then p∣[Fω:F(p)ω] and hence F(p)ω is not a p-Sylow subgroup of Fω. We conclude that the condition s≥1 is necessary. Note that the biggest prω (ω∈Ω) which occurs is given by the biggest power of p which is smaller than or equal to d due to the iterated wreath product structure of F(p). As p∤k, we conclude (ii).
Conversely, suppose s≥1 and ps+1≥d. If p is odd, or F=Sym(Ω) and p is even, then F(p) is a direct product of s-fold iterated wreath products and the maximum power of p dividing [F(p):F(p)ω] and [F:Fω] is ps in both cases. The same index assertions hold for F=Alt(Ω) and p even.
∎
For a general permutation group F≤Sym(Ω) and ω∈Ω we have
|F(p)⋅ω|=|F(p)||F(p)ω|=|F(p)|⋅[Fω:F(p)ω]|Fω|=[Fω:F(p)ω][F:F(p)]⋅|F⋅ω|
by the Orbit-Stabilizer Theorem. In particular, we conclude the following.
Proposition 12.
Let F≤Sym(Ω) and let F(p)≤F be a p-Sylow subgroup. Assume that F\Ω=F(p)\Ω. Then F(p)ω≤Fω is a p-Sylow subgroup for all ω∈Ω. ∎
Proposition 13.
Let |Ω|=pn and F≤Sym(Ω) transitive. Also, let F(p)≤F be a p-Sylow subgroup. Then so is F(p)ω≤Fω for all ω∈Ω and F(p) is transitive.
Proof.
In this case, the above equation reads
|F(p)⋅ω|=[Fω:F(p)ω][F:F(p)]⋅pn.
As always, |F(p)⋅ω| is a power of p and bounded by |Ω|=pn. Since p does not divide [F:F(p)], the above implies that p does not divide [Fω:F(p)ω].
∎
4 Prime localizations
This section is concerned with the p-localizations of Burger–Mozes-type groups. Recall that for groups H≤G one defines the commensurator of H in G by
CommG(H):={g∈G∣[H:H∩gHg-1]<∞,[gHg-1:gHg-1∩H]<∞}.
The p-localization of a totally disconnected locally compact group G is defined as the commensurator CommG(S) of a local p-Sylow subgroup S of G, equipped with the unique group topology that makes the inclusion of the subgroup S into G(p):=CommG(S) continuous and open. Then the inclusion CommG(S)→G is continuous.
The following lemma due to Caprace–Monod [3, Section 4] and Caprace–Reid–Willis [4, Corollary 7.4] is crucial for the subsequent statements of this section. See also [10].
Lemma 14.
Let G be residually discrete, locally compact and totally disconnected. Further, let K≤G be compact. Then CommG(K)=⋃L≤oKNG(L).
Proof.
Every element of G which normalizes an open subgroup of K commensurates K because open subgroups of K have finite index in K given that K is compact.
Conversely, let g∈CommG(K) and consider H:=〈K,g〉. Then H is a compactly generated open subgroup of CommG(K) and hence a compactly generated, totally disconnected locally compact group in its own right. It inherits residual discreteness from CommG(K) which injects continuously into the residually discrete group G. By [3, Corollary 4.1], H has an identity neighborhood basis consisting of compact open normal subgroups. Hence g normalizes an open subgroup of K.
∎
Now, let F≤F′≤F^≤Sym(Ω). In the case of Proposition 10, the following proposition identifies certain subsets of the p-localization of G(F,F′) and thereby expands [8, Lemma 4.2] given that U(F)=G(F,F). We establish the following notation: Given partitions 𝒫:=(Pi)i∈I of V and ℋ=(Hj)j∈J of H≤Sym(Ω), let
Γ𝒫(ℋ):={g∈Aut(Td)∣for all i∈I there is j∈J such that σ(g,v)∈Hj for all v∈Pi}
denote the set of automorphisms of Td whose local permutations at the vertices of a given element of 𝒫 all come from the same element of ℋ.
Proposition 15.
Let F≤F′≤F^≤Sym(Ω) and F(p)≤F a p-Sylow subgroup such that F(p)ω≤Fω is a p-Sylow subgroup for all ω∈Ω. Set S:=U(F(p))b. Then
CommG(F,F′)(S)=〈U({id}),CommG(F,F′)b(S)〉
≥〈G(F(p),F′),{ΓV/L(NF(F(p))/F(p))∣L≤S open}〉.
Proof.
By Proposition 10, the group S is a local p-Sylow subgroup of U(F) and hence of G(F,F′). We first show that G(F,F′)(p) contains U({id}). Indeed, given g∈U({id}) we have gSg-1=U(F(p))g(b). Thus S∩gSg-1=U(F(p))(b,g(b)) which has finite index in both S=U(F)b and gSg-1=U(F(p))g(b) by the Orbit-Stabilizer Theorem. Since U({id}) acts vertex-transitively on Td, we conclude
CommG(F,F′)(S)=〈U({id}),CommG(F,F′)b(S)〉.
Now, the vertex stabilizer G(F,F′)b is residually discrete by Proposition 8. Hence, by Lemma 14, the commensurator CommG(F,F′)b(S) is the union of the normalizers in G(F,F′)b of open subgroups of S=U(F(p))b. For example, we may consider Ln:=U(F(p))B(b,n)≤oS for every n∈ℕ. The normalizer of Ln in G(F,F′)b contains those elements of G(F(p),F′)b all of whose singularities are contained in B(b,n). Taking the union over all n∈ℕ and using vertex-transitivity of G(F(p),F′) in the sense that G(F(p),F′)=〈G(F(p),F′)b,U({id})〉, we can conclude that CommG(F.F′)(S) contains G(F(p),F′) as a topological subgroup. Alternatively, use [1, Lemma 3.2].
As to ΓV/L(NF(F(p))), note that for all g,s∈Aut(Td) and v∈V we have
σ(gsg-1,v)=σ(g,sg-1v)σ(s,g-1v)σ(g-1,v)
=σ(g,sg-1v)σ(s,g-1v)σ(g,g-1v)-1.
Hence if g∈ΓV/L(NF(F(p))/F(p)), i.e. the coset σ(g,v)F(p)⊆NF(F(p)) is constant on L-orbits, then
gLg-1⊆U(F(p))
whence g∈CommG(F,F′)(S).
∎
Remark 16.
Whereas the next result provides conditions on F≤Sym(Ω) which ensure U(F)(p)=G(F(p),F) and we have U(F)(p)=U(F) for semiregular F by Proposition 3, it may happen that G(F(p),F)⪇U(F)(p)⪇U(F). Indeed, if for every ω∈Ω there is an element aω∈Fω such that for all λ∈Ω we have F(p)λ∩aωF(p)λaω-1={id}, then there is an element g∈U(F)B(b,1) such that for S:=U(F(p))B(b,1) we have S∩gSg-1={id} and therefore g∉U(F)(p): Choose the local permutation of g at the vertex v∈V(Td) to be aω whenever d(v,b)=d(v,bω)+1. If in addition NF(F(p))⪈F(p), the assertion holds by virtue of Proposition 15. For instance, these assumptions are satisfied for F=S6 and p=3.
Theorem 17.
Let F≤F′≤F^≤Sym(Ω) and let F(p)≤F be a p-Sylow subgroup of F. Assume that we have F\Ω=F(p)\Ω and NFω′(F(p)ω)=F(p)ω for all ω∈Ω. Then G(F,F′)(p)=G(F(p),F′).
If F does not fix a point of Ω and F\Ω=F(p)\Ω then p divides |Ω|. By Proposition 12 the same assumption implies that the point stabilizers in F(p) are p-Sylow subgroups of the respective point stabilizers in F. In the case F=F′, the theorem asks that these be self-normalizing.
Proof.
By Proposition 10 and Proposition 15 it suffices to show that
CommG(F,F′)b(U(F(p))b)=G(F(p),F′)b.
By Proposition 15, the group G(F(p),F′)b is a subgroup of said commensurator.
Suppose g∈CommG(F,F′)b(U(F(p))b)≤G(F,F′)b. Given that G(F,F′)b is residually discrete by Proposition 8, the element g normalizes an open subgroup L≤U(F(p))b by virtue of Lemma 14. If g has only finitely many local permutations in F′\F(p), then g∈G(F(p),F′)b. Otherwise, the above implies that there is n∈ℕ such that gU(F(p))B(b,n)g-1⊆L⊆U(F(p))b and g has a local permutation in F′\F(p) on S(b,n). Then construct h∈G(F(p),F′) with local permutations in F(p) on spheres of radius at least n and such that h-1g fixes B(b,n) pointwise as follows: Set h|B(b,n-1):=g and use the assumption F′\Ω=F\Ω=F(p)\Ω to extend h to all Td using F(p) only. Then h-1g has a local permutation in Fω′\F(p)ω for some ω∈Ω on S(b,n) and one obtains (h-1g)U(F(p))B(b,n)(h-1g)-1⊆L⊆U(F(p))b. However, this contradicts the assumption NFω′(F(p)ω)=F(p)ω for all ω∈Ω.
∎
Theorem 17 can be used to determine the p-localization of Lederle’s colored Neretin group N(F) under similar assumptions.
Theorem 18.
Let F≤Sym(Ω) and let F(p)≤F be a p-Sylow subgroup. Assume that F\Ω=F(p)\Ω and NF^ω(F(p)ω)=F(p)ω for all ω∈Ω. Then N(F)(p)=N(F(p)).
Proof.
By Proposition 10, the group S:=U(F(p))b is a local Sylow subgroup of N(F). Furthermore, by [6, Proposition 2.24], we have
N(F(p))≤CommN(F)(S).
Now, let g∈CommN(F)(S) and g:Td\T→Td\T′ a representative of g as an U(F)-honest almost automorphism. Given that F\Ω=F(p)\Ω, there exists a U(F(p))-honest almost automorphism h∈N(F(p))≤CommN(F)(S) with representative h:Td\T′→Td\T such that hg:Td\T→Td\T fixes the leaves of T and therefore extends to an automorphism of Td fixing T. Furthermore, on each connected component of Td\T, the automorphism hg∈N(F)∩Aut(Td) coincides with an element of U(F). Hence, using Proposition 4, we have hg∈U(F) and so
hg∈CommN(F)∩Aut(Td)(S)=CommG(F)(S)=G(F)(p)=G(F(p))≤N(F(p))
by Theorem 17. Given that h∈N(F(p)), we conclude g∈N(F(p)) as required.
∎
Proposition 15 suggests that Theorem 17 might hold as soon as F(p) is self-normalizing in F′. This is not the case as the following remark shows.
Remark 19.
Theorem 17 does not hold when the condition
NFω′(F(p)ω)=F(p)ω for all ω∈Ω
is replaced with
NF′(F(p))=F(p).
There are transitive, non-regular permutation groups F≤Sym(Ω) and primes p such that F\Ω=F(p)\Ω and NF(F(p))=F(p) for which F(p) is regular. In particular, NFω(F(p)ω)⪈F(p)ω. In this case, U(F(p))b is a local p-Sylow subgroup of U(F) by Proposition 12. However, U(F(p))b≅F(p) is finite and hence U(F)(p)=U(F)⪈G(F(p),F).
A small example of this situation is a certain F≅S4≤S8 and the prime p=2, namely put F:=〈(123)(456),(14)(25)(37)(68)〉. Here, F(2) is regular and self-normalizing in F of order eight.
