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.
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.
A group G is residually p-finite if, for any , 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  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 .
In both of these papers, the strategy of the proof is to use wreath products in the following manner. For or , iterated wreath products are used to construct an explicit finite p-group P and a map whose restriction to the 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 consists of a set VX of vertices and a set EX of edges. Each edge y has an opposite edge , and has endpoints and with and .
A graph of groups consists of the following data:
a connected graph X,
a group for each , with for ,
monomorphisms for all .
We fix a maximal subtree T of X, which exists by Zorn’s lemma if X is infinite. Choose also an orientation of X – that is, a subset such that, for all , exactly one of y and is in . Define a function
The fundamental group is then defined to be the group obtained from the free product of the (for ) and the free group generated by letters (for ), subject to the following relations:
for and ,
for all , and for all .
All the groups 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 is not necessarily contained in , but in a conjugate of it; and note that the map is equal to the composition
where the final map is left conjugation by .
The Bass–Serre tree of G dual to is the tree with vertex and edge sets
For , define to be the coset viewed as an element of . The adjacency maps in are
There is a natural (left-)action of G on with quotient graph X and with point stabilisers
Conversely [5, Section I.5.4], an action of G on a tree gives rise to a graph of groups whose Bass–Serre tree is G-isomorphic to .
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 “” means “H is a normal subgroup of G with index a power of p”.
Let . If H is residually p-finite, then G is residually p-finite.
Let . If , then there is nothing to prove. If , then, by assumption, there is such that . Consider
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 . This completes the proof. ∎
We proceed now to the main theorem. The criterion for residual p-finiteness is stated in terms of chief series.
A chief series for a finite p-group P is a sequence
of normal subgroups of P such that each successive quotient
is either trivial or of order p and such that for some n. The length of the chief series is the smallest n such that .
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.
Let be a graph of finite p-groups with fundamental group . Suppose there is a chief series of for each , such that the following two properties hold.
For all and all , we have .
For each k, there exists a family of injections (for ) such that the diagrams
commute for all .
Then G is residually p-finite.
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 restricting to an injection on all the (finitely many) . Taking intersections of with a chief series of P yields chief series of the satisfying the conditions of the theorem.
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].
We note that in the case that X is a tree, condition II follows from condition I: one may choose arbitrarily at one point of each connected component of the graph , whereupon the maps for the remaining x in that component of may be uniquely defined by forcing condition II to hold.
If X is not a tree, one may still define the consistently on a maximal forest T of . For the remaining edges , one may again define so that condition II holds. The only remaining cases of condition II that must be satisfied are for the edges for . Take an edge path in from to . Then condition II is easily seen to be satisfied if and only if the composite map
is the identity. This condition may be seen as the analogue in our context for condition [2, ()] given by Chatzidakis for HNN extensions.
Let G be a group. Let G act on a Bass–Serre tree dual to a graph of finite p-groups . Then satisfies the conditions of Theorem 4 if and only if there exists a chief series for each stabiliser of such that the following conditions hold.
For all , we have , and for each and each , we have .
For each k, there exists a family of injections for such that the diagrams
commute for all , and such that, for all and all , the diagram
commutes where denotes left conjugation by g.
of . This is well-defined (that is, it is invariant under replacing g by gh for ) because is normal in . Further define the map to be the composition
This map is again well-defined under replacing g by gh for because the conjugation action of on itself induces the identity map on the – 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 followed by a conjugation by .
for . Conditions I and II now follow from conditions (I’) and (II’) via expression ((*) ‣ Graphs of groups and trees) of the maps as a composition of an inclusion of edge stabilisers and a conjugacy. ∎
Proof of Theorem 4.
Suppose first that the chief series 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 , then G is free, hence is residually p-finite. Suppose . The maps in condition II define, by the universal property of the fundamental group of a graph of groups, a homomorphism whose restriction to each is the composite
Let . The group G acts on its Bass–Serre tree as in Lemma 5. Consider the action of H on . The point stabilisers for are by construction the groups . The chief series now automatically satisfy conditions (I’) and (II’), and all have length at most . Hence, by Lemma 5, the graph of groups decomposition of H dual to its action on is equipped with chief series of length at most 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 . In the graph of groups , 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 of has length at most N for all . We may take the quotient of each by to obtain a new graph of groups . There is a natural map of fundamental groups
Then is non-trivial in , for it is given by a reduced word – the same reduced word as in G since for all . But is residually p-finite by the first part of the theorem since the chief series for all have length at most N. Therefore, there is some map for a finite p-group P such that . This attests that G is residually p-finite. ∎
The author was supported by a Junior Research Fellowship from Clare College, Cambridge.
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