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

Gareth Wilkes
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 gG1, 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=A1BA2 or G=A1B, iterated wreath products are used to construct an explicit finite p-group P and a map GP 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 “simpler” 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=VXEX consists of a set VX of vertices and a set EX of edges. Each edge y has an opposite edge y¯, and has endpoints o(y) and t(y) with o(y¯)=t(y) and y¯¯=y.

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

1. a connected graph X,

2. a group Gx for each xVXEX, with Gy=Gy¯ for yEX,

3. monomorphisms fy:GyGt(y) for all yEX.

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+XEX such that, for all yEX, exactly one of y and y¯ is in E+X. Define a function

ϵ:EX{0,1}byϵ(y)={0ifyE+X,1ifyE+X.

The fundamental group G=π1(X,G) is then defined to be the group obtained from the free product of the Gx (for xVXEX) and the free group generated by letters sy (for yEX), subject to the following relations:

1. g=sy1-ϵ(y)fy(g)syϵ(y)-1 for yEX and gGy,

2. sy=1 for all yET, and sy¯=sy for all yEX.

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=(Gysy1-ϵ(y)Gt(y)syϵ(y)-1Gt(y)),

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

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

VX~=xVXG/Gx,EX~=yEXG/Gy.

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

o(gy~)=gsy-ϵ(y)o(y)~,t(gy~)=gsy1-ϵ(y)t(y)~.

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

G(gx~):-stabG(gx~)=gGxg-1.

Conversely [5, Section I.5.4], an action of G on a tree X~ gives rise to a graph of groups (X,G) 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 “HpG” means “H is a normal subgroup of G with index a power of p”.

## Lemma 2.

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

## Proof.

Let gG. If gH, then there is nothing to prove. If gH, then, by assumption, there is UpH such that gU. Consider

V=gGgUg-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 gV. 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)P(1)P(k)

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,G) be a graph of finite p-groups with fundamental group G=π1(X,G). Suppose there is a chief series (Gx(k))k0 of Gx for each xX, such that the following two properties hold.

1. For all k and all yEX, we have fy(Gy(k))=fy(Gy)Gt(y)(k).

2. For each k, there exists a family of injections ϕx(k):γ(k)(Gx)𝔽p (for xX) such that the diagrams

commute for all yEX.

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 Φ:GP restricting to an injection on all the (finitely many) Gx. Taking intersections of Φ(Gx) with a chief series (P(k))0kn 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 ϕx(k) arbitrarily at one point of each connected component of the graph Yk={xXγ(k)(Gx)1}, whereupon the maps ϕx(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 ϕx(k) consistently on a maximal forest T of Yk. For the remaining edges yYkE+X, one may again define ϕy(k) so that condition II holds. The only remaining cases of condition II that must be satisfied are for the edges y¯0 for y0E+XT. 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

γ(k)(Gy0)fy¯0γ(k)(Go(y0))fy¯1-1γ(k)(Gy1)fy1γ(k)(Gt(y1))γ(k)(Gt(y0))fy0-1γ(k)(Gy0)

is the identity. This condition may be seen as the analogue in our context for condition [2, ()] 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,G).

## 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,G). Then (X,G) satisfies the conditions of Theorem 4 if and only if there exists a chief series (G(z)(k))k0 for each stabiliser G(z) of zX~ such that the following conditions hold.

1. For all zEX~, we have G(z)(k)=G(z)G(t(z))(k), and for each zX~ and each gG, we have gG(z)(k)g-1=G(gz)(k).

2. For each k, there exists a family of injections ψz(k):γ(k)(G(z))𝔽p for zX~ such that the diagrams

commute for all zEX, and such that, for all zX~ and all gG, the diagram

commutes where ζg denotes left conjugation by g.

## Proof.

Suppose we have chief series (Gx(k))k0 for the graph of groups (X,G) satisfying conditions I and II of Theorem 4. For gx~X~ define a chief series

G(gx~)(k)=gGx(k)g-1

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

γ(k)(G(gx~))ζg-1γ(k)(G(x~))=γ(k)(Gx)ϕx(k)𝔽p.

This map is again well-defined under replacing g by gh for hGx because the conjugation action of Gx on itself induces the identity map on the γ(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 ((*) ‣ 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 syϵ(y)-1.

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

Gx(k)=G(x~)(k),ϕx(k)=ψx~(k):γ(k)(Gx)𝔽p

for xX. Conditions I and II now follow from conditions (I’) and (II’) via expression ((*) ‣ 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))k0 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 ϕx(0) in condition II define, by the universal property of the fundamental group of a graph of groups, a homomorphism Φ:G𝔽p whose restriction to each Gx is the composite

GxGx/Gx(1)=γ(0)(Gx)ϕx(0)𝔽p.

Let H=kerΦ. 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 zX~ 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 gG{1}. In the graph of groups (X,G), 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))k0 of Gz has length at most N for all zZ. We may take the quotient of each Gx by Gx(N) to obtain a new graph of groups (X,G/G(N)). There is a natural map of fundamental groups

Φ:G=π1(X,G)π1(X,G/G(N))G

Then Φ(g) is non-trivial in G, for it is given by a reduced word – the same reduced word as in G since Gz(N)=1 for all zZ. But G is residually p-finite by the first part of the theorem since the chief series for all G/G(N) have length at most N. Therefore, there is some map Ψ:GP for a finite p-group P such that ΨΦ(g)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.

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