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Publicly Available Published by De Gruyter May 7, 2019

A criterion for residual 𝑝-finiteness of arbitrary graphs of finite 𝑝-groups

Gareth Wilkes EMAIL logo
From the journal Journal of Group Theory

Abstract

We establish conditions under which the fundamental group of a graph of finite p-groups is necessarily residually p-finite. The technique of proof is independent of previously established results of this type, and the result is also valid for infinite graphs of groups.

Introduction

Residual properties of graphs of groups have long been a subject of study and have, for instance, been particularly important in relation to 3-manifold groups [3, 8, 1, 7]. Any study of such properties almost inevitably involves a reduction to the study of graphs of finite groups. The fundamental group of any finite graph of finite groups is well known to be residually finite [5, Proposition II.2.6.11]. However, the situation for properties of residual p-finiteness is rather more subtle. Throughout the paper, let p be a prime.

Definition 1.

A group G is residually p-finite if, for any g∈Gβˆ–1, there exists a homomorphism from G to a finite p-group whose kernel does not contain g, or, equivalently, if there exists a normal subgroup of G with index a power of p which does not contain g.

It is emphatically not the case that the fundamental group of any finite graph of finite p-groups is residually p-finite, and one must impose a condition on the graph of groups for this to be the case.

Higman [4] studied this problem in the case of amalgamated free products and proved that the existence of chief series for the p-groups satisfying a certain compatibility condition (see condition I of Theorem 4) is necessary and sufficient for an amalgamated free product of p-groups to be residually p-finite. A similar criterion for HNN extensions was proved by Chatzidakis [2].

In both of these papers, the strategy of the proof is to use wreath products in the following manner. For G=A1βˆ—BA2 or G=A1βˆ—B, iterated wreath products are used to construct an explicit finite p-group P and a map Gβ†’P whose restriction to the Ai is an injection. This map then has free kernel which is a normal subgroup of index a power of p. Since free groups are residually p-finite, this implies (see Lemma 2) that the original fundamental group of the graph of groups is also residually p.

Since finite graphs of groups can be constructed step-by-step as iterated amalgamated free products and HNN extensions, these two papers together could be applied to prove that a condition on chief series implies that a finite graph of p-groups is residually p-finite. However, such arguments cannot access infinite graphs of groups.

The purpose of this note is to give a new proof of a chief series condition for residual p-finiteness. There are no wreath products, and we will analyse all graphs of groups directly rather than building them up from one-edge graphs of groups (i.e. amalgams and HNN extensions). In particular, our proof is also valid for infinite graphs of groups. The criterion studied is given in Theorem 4.

The scheme of the proof is to use the language of Bass–Serre theory to give a reformulation of our criterion in terms of an action on the tree dual to the graph of groups. In this formulation, we can pass to an index p normal subgroup given by a graph of groups which is β€œsimpler” in a certain sense. This process concludes with a free group, which is well known to be residually p-finite, thus proving the theorem.

Graphs of groups and trees

For this paper, we will use the notions of graphs and graphs of groups as given by Serre [5, Section I.5.3]. We recall these notions and set up notation as follows. A graph X=V⁒XβŠ”E⁒X consists of a set VX of vertices and a set EX of edges. Each edge y has an opposite edge yΒ―, and has endpoints o⁒(y) and t⁒(y) with o⁒(yΒ―)=t⁒(y) and yΒ―Β―=y.

A graph of groups (X,Gβˆ™) consists of the following data:

  1. a connected graph X,

  2. a group Gx for each x∈V⁒XβˆͺE⁒X, with Gy=GyΒ― for y∈E⁒X,

  3. monomorphisms fy:Gyβ†ͺGt⁒(y) for all y∈E⁒X.

We fix a maximal subtree T of X, which exists by Zorn’s lemma if X is infinite. Choose also an orientation E+⁒X of X – that is, a subset E+⁒XβŠ†E⁒X such that, for all y∈E⁒X, exactly one of y and yΒ― is in E+⁒X. Define a function

Ο΅:E⁒Xβ†’{0,1} by ϡ⁒(y)={0if⁒y∈E+⁒X,1if⁒yβˆ‰E+⁒X.

The fundamental group G=Ο€1⁒(X,Gβˆ™) is then defined to be the group obtained from the free product of the Gx (for x∈V⁒XβˆͺE⁒X) and the free group generated by letters sy (for y∈E⁒X), subject to the following relations:

  1. g=sy1-ϡ⁒(y)⁒fy⁒(g)⁒syϡ⁒(y)-1 for y∈E⁒X and g∈Gy,

  2. sy=1 for all y∈E⁒T, and sy¯=sy for all y∈E⁒X.

All the groups Gx inject into G under this construction, and we identify them with their images in G. Note that, for an edge y of X, the group Gy is not necessarily contained in Gt⁒(y), but in a conjugate of it; and note that the map fy is equal to the composition

(*)fy=(GyβŸΆβŠ†sy1-ϡ⁒(y)Gt⁒(y)syϡ⁒(y)-1⟢Gt⁒(y)),

where the final map is left conjugation by syϡ⁒(y)-1.

The Bass–Serre tree of G dual to (X,Gβˆ™) is the tree X~ with vertex and edge sets

V⁒X~=βŠ”x∈V⁒XG/Gx,E⁒X~=βŠ”y∈E⁒XG/Gy.

For x∈X, define x~ to be the coset 1β‹…Gx viewed as an element of X~. The adjacency maps in X~ are

o⁒(g⁒y~)=g⁒sy-ϡ⁒(y)⁒o⁒(y)~,t⁒(g⁒y~)=g⁒sy1-ϡ⁒(y)⁒t⁒(y)~.

There is a natural (left-)action of G on X~ with quotient graph X and with point stabilisers

G⁒(g⁒x~):-stabG⁑(g⁒x~)=g⁒Gx⁒g-1.

Conversely [5, Section I.5.4], an action of G on a tree X~ gives rise to a graph of groups (X,Gβˆ™) whose Bass–Serre tree is G-isomorphic to X~.

Results

For the proof of the main theorem, we will make use of the following standard fact. We include a proof for completeness. Note that there is no requirement that G be finitely generated. The notation β€œH◁pG” means β€œH is a normal subgroup of G with index a power of p”.

Lemma 2.

Let H◁pG. If H is residually p-finite, then G is residually p-finite.

Proof.

Let g∈G. If gβˆ‰H, then there is nothing to prove. If g∈H, then, by assumption, there is U◁pH such that gβˆ‰U. Consider

V=β‹‚g∈Gg⁒U⁒g-1.

Since U is normal in H and H has finite index in G, there are only finitely many subgroups in this intersection. All are normal in H of p-power index, so the intersection V also has p-power index in H, and hence in G. By construction, V is normal in G and gβˆ‰V. This completes the proof. ∎

We proceed now to the main theorem. The criterion for residual p-finiteness is stated in terms of chief series.

Definition 3.

A chief series for a finite p-group P is a sequence

P=P(0)β‰₯P(1)β‰₯β‹―β‰₯P(k)β‰₯β‹―

of normal subgroups of P such that each successive quotient

γ(k)⁒(P)=P(k)/P(k+1)

is either trivial or of order p and such that P(n)=1 for some n. The length of the chief series is the smallest n such that P(n)=1.

Remark.

This differs slightly from the usual definition of a chief series in that the sequence does not terminate. This is a purely formal difference enabling us to state the next theorem in its greatest generality.

Theorem 4.

Let (X,Gβˆ™) be a graph of finite p-groups with fundamental group G=Ο€1⁒(X,Gβˆ™). Suppose there is a chief series (Gx(k))kβ‰₯0 of Gx for each x∈X, such that the following two properties hold.

  1. For all kβˆˆβ„• and all y∈E⁒X, we have fy⁒(Gy(k))=fy⁒(Gy)∩Gt⁒(y)(k).

  2. For each k, there exists a family of injections Ο•x(k):Ξ³(k)⁒(Gx)β†ͺ𝔽p (for x∈X) such that the diagrams

    commute for all y∈E⁒X.

Then G is residually p-finite.

Remark.

The converse to Theorem 4 holds if the graph X is finite. In this case, if G is residually p-finite, then there is a finite p-group P and a map Ξ¦:Gβ†’P restricting to an injection on all the (finitely many) Gx. Taking intersections of Φ⁒(Gx) with a chief series (P(k))0≀k≀n of P yields chief series of the Gx satisfying the conditions of the theorem.

Remark.

If X is finite, then the conditions of the theorem are also sufficient for G to be conjugacy p-separable – this is equivalent to residual p-finiteness by [6, Theorem 4.2].

Remark.

We note that in the case that X is a tree, condition II follows from condition I: one may choose Ο•x(k) arbitrarily at one point of each connected component of the graph Yk={x∈X∣γ(k)⁒(Gx)β‰ 1}, whereupon the maps Ο•x(k) for the remaining x in that component of Yk may be uniquely defined by forcing condition II to hold.

If X is not a tree, one may still define the Ο•x(k) consistently on a maximal forest T of Yk. For the remaining edges y∈Yk∩E+⁒X, one may again define Ο•y(k) so that condition II holds. The only remaining cases of condition II that must be satisfied are for the edges yΒ―0 for y0∈E+⁒Xβˆ–T. Take an edge path y1,…,ym in Yk from o⁒(y0) to t⁒(y0). Then condition II is easily seen to be satisfied if and only if the composite map

Ξ³(k)⁒(Gy0)β†’fyΒ―0Ξ³(k)⁒(Go⁒(y0))β†’fyΒ―1-1Ξ³(k)⁒(Gy1)β†’fy1Ξ³(k)⁒(Gt⁒(y1))β†’β‹―β†’Ξ³(k)⁒(Gt⁒(y0))β†’fy0-1Ξ³(k)⁒(Gy0)

is the identity. This condition may be seen as the analogue in our context for condition [2, (βˆ—β£βˆ—)] given by Chatzidakis for HNN extensions.

The proof of Theorem 4 proceeds most smoothly if we translate conditions I and II of Theorem 4 into the language of the Bass–Serre tree dual to the graph of groups (X,Gβˆ™).

Lemma 5.

Let G be a group. Let G act on a Bass–Serre tree X~ dual to a graph of finite p-groups (X,Gβˆ™). Then (X,Gβˆ™) satisfies the conditions of Theorem 4 if and only if there exists a chief series (G⁒(z)(k))kβ‰₯0 for each stabiliser G⁒(z) of z∈X~ such that the following conditions hold.

  1. For all z∈E⁒X~, we have G⁒(z)(k)=G⁒(z)∩G⁒(t⁒(z))(k), and for each z∈X~ and each g∈G, we have g⁒G⁒(z)(k)⁒g-1=G⁒(gβ‹…z)(k).

  2. For each k, there exists a family of injections ψz(k):Ξ³(k)⁒(G⁒(z))β†ͺ𝔽p for z∈X~ such that the diagrams

    commute for all z∈E⁒X, and such that, for all z∈X~ and all g∈G, the diagram

    commutes where ΞΆg denotes left conjugation by g.

Proof.

Suppose we have chief series (Gx(k))kβ‰₯0 for the graph of groups (X,Gβˆ™) satisfying conditions I and II of Theorem 4. For g⁒x~∈X~ define a chief series

G⁒(g⁒x~)(k)=g⁒Gx(k)⁒g-1

of G⁒(g⁒x~). This is well-defined (that is, it is invariant under replacing g by gh for h∈Gx) because Gx(k) is normal in Gx. Further define the map ψg⁒x~(k) to be the composition

Ξ³(k)⁒(G⁒(g⁒x~))β†’ΞΆg-1Ξ³(k)⁒(G⁒(x~))=Ξ³(k)⁒(Gx)β†’Ο•x(k)𝔽p.

This map is again well-defined under replacing g by gh for h∈Gx because the conjugation action of Gx on itself induces the identity map on the Ξ³(k)⁒(Gx) – a p-group cannot induce a non-trivial automorphism of either the trivial group or of a cyclic group of order p.

The parts of conditions (I’) and (II’) concerning invariance under G-conjugation hold by construction. The conditions on edges follow from conditions I and II of Theorem 4 together with the G-conjugation invariance, by recalling from ((*) β€£ Graphs of groups and trees) that the inclusion of an edge stabiliser into a vertex stabiliser is, up to a G-conjugacy, equal to the map fy followed by a conjugation by syϡ⁒(y)-1.

Conversely, given chief series for the point stabilisers G⁒(z) satisfying (I’) and (II’), we may define

Gx(k)=G⁒(x~)(k),Ο•x(k)=ψx~(k):Ξ³(k)⁒(Gx)→𝔽p

for x∈X. Conditions I and II now follow from conditions (I’) and (II’) via expression ((*) β€£ Graphs of groups and trees) of the maps fy as a composition of an inclusion of edge stabilisers and a conjugacy. ∎

Proof of Theorem 4.

Suppose first that the chief series (G(k))kβ‰₯0 all have length at most N for some N – this is automatic if X is finite. We prove the theorem by induction on N. If N=0, then G is free, hence is residually p-finite. Suppose N>0. The maps Ο•x(0) in condition II define, by the universal property of the fundamental group of a graph of groups, a homomorphism Ξ¦:G→𝔽p whose restriction to each Gx is the composite

Gxβ†’Gx/Gx(1)=Ξ³(0)⁒(Gx)β†ͺΟ•x(0)𝔽p.

Let H=ker⁑Φ. The group G acts on its Bass–Serre tree X~ as in Lemma 5. Consider the action of H on X~. The point stabilisers H⁒(z) for z∈X~ are by construction the groups G⁒(z)(1). The chief series H⁒(z)(k)=G⁒(z)(k+1) now automatically satisfy conditions (I’) and (II’), and all have length at most N-1. Hence, by Lemma 5, the graph of groups decomposition of H dual to its action on X~ is equipped with chief series of length at most N-1 satisfying conditions I and II. Therefore, by induction, H is residually p-finite. Since H is a normal subgroup of index p in G, it follows from Lemma 2 that G is also residually p-finite.

Now move to the general case. Let g∈Gβˆ–{1}. In the graph of groups (X,Gβˆ™), the element g is equal to a reduced word [5, Section I.5.2] which is supported on some finite subgraph Z of X. Let N be such that the chief series (Gz(k))kβ‰₯0 of Gz has length at most N for all z∈Z. We may take the quotient of each Gx by Gx(N) to obtain a new graph of groups (X,Gβˆ™/Gβˆ™(N)). There is a natural map of fundamental groups

Ξ¦:G=Ο€1⁒(X,Gβˆ™)β†’Ο€1⁒(X,Gβˆ™/Gβˆ™(N))≕Gβ€²

Then Φ⁒(g) is non-trivial in Gβ€², for it is given by a reduced word – the same reduced word as in G since Gz(N)=1 for all z∈Z. But Gβ€² is residually p-finite by the first part of the theorem since the chief series for all Gβˆ™/Gβˆ™(N) have length at most N. Therefore, there is some map Ξ¨:Gβ€²β†’P for a finite p-group P such that Ψ⁒Φ⁒(g)β‰ 1. This attests that G is residually p-finite. ∎


Communicated by John S. Wilson


Acknowledgements

The author was supported by a Junior Research Fellowship from Clare College, Cambridge.

References

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Received: 2018-10-31
Revised: 2019-04-08
Published Online: 2019-05-07
Published in Print: 2019-09-01

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