Profinite rigidity of graph manifolds and JSJ decompositions of 3manifolds
Abstract
There has been much recent interest into those properties of a 3manifold determined by the profinite completion of its fundamental group. In this paper we give readily computable criteria specifying precisely when two orientable graph manifold groups have isomorphic profinite completions. Our results also distinguish graph manifolds among the class of all 3manifolds and give information about the structure of totally hyperbolic manifolds, and give control over the pro completion of certain graph manifold groups.
1 Introduction
Much attention has been paid recently to those properties of 3manifolds which can be deduced from the finite quotients of their fundamental groups; or, from another viewpoint, from the structure of their lattice of finitesheeted coverings. Having assembled these finite quotients into the profinite completion of the fundamental group, this amounts to the study of ‘profinite invariants’ of the 3manifold. A profinite invariant may be defined as some property of a group such that, whenever is a group with property and , then also has property . One may restrict attention to a particular class of groups (for example fundamental groups of compact orientable 3manifolds) and require both and to be from that class. For the rest of the paper we restrict our attention to this class of groups. By abuse of language one also refers to a profinite invariant of a 3manifold.
The strongest profinite invariant one could hope for is the isomorphism type of the group itself. In this case one refers to a group whose profinite completion determines the isomorphism type of the group (again, among compact orientable 3manifold groups) as ‘profinitely rigid’. The author showed recently [Wil15] that, apart from some limited examples due to Hempel [Hem14], closed Seifert fibre spaces are profinitely rigid. Bridson and Reid [BR15] and Boileau and Friedl [BF15] have shown that the figureeight knot complement, along with a handful of other knot complements, are profinitely rigid. Bridson, Reid and Wilton [BRW16] have also proved that the fundamental groups of oncepunctured torus bundles are profinitely rigid. To date these are the only known examples of profinitely rigid manifolds in the literature.
An important property of a 3manifold is its geometry: whether it admits one of the eight Thurston geometries and, if so, which one. This problem is more naturally stated only for closed manifolds, as then any geometry is unique. Wilton and Zalesskii [WZ17] have shown that the geometry of a 3manifold is a profinite invariant. The aspect of this theorem distinguishing between Seifert fibred or hyperbolic manifolds on the one hand and Sol or nongeometric manifolds on the other may be interpreted as the statement ‘triviality of the JSJ decomposition is a profinite invariant’.
In this paper we investigate graph manifolds and, more generally, the JSJ decomposition. We obtain in particular criteria determining precisely when two graph manifold groups can have isomorphic profinite completion. The exact statement requires a good deal of terminology and notation which is inappropriate in an Introduction, hence we will give a rather loose paraphrasing of the result. For the precise statement see Theorem 10.9.
Theorem A.
Let , be closed orientable graph manifolds with JSJ decompositions , respectively. Suppose . Then the graphs are isomorphic, such that corresponding vertex groups have isomorphic profinite completions. Furthermore:

If is not a bipartite graph, then is profinitely rigid and so .

If is a bipartite graph then there is an explicit finite list of numerical equations defined in terms of which admit a solution if and only if .
In particular, for a given there only finitely many (up to homeomorphism) such that .
We also use our analysis to distinguish graph manifolds from mixed or totally hyperbolic manifolds, which may be seen as an extension of [WZ17] stating that the profinite completion ‘sees’ which geometries are involved in the geometric decomposition of an aspherical 3manifold.
Theorem B.
Let be a mixed or totally hyperbolic 3manifold and let be a graph manifold. Then and do not have isomorphic profinite completions.
While our techniques give strong control over the Seifert fibre spaces involved in a 3manifold, it is more difficult to detect the hyperbolic pieces in a satisfactory way; we can only detect the presence of a hyperbolic piece, without giving any information about it. When there are no Seifertfibred pieces at all, we can extract some information about the configuration of the hyperbolic pieces.
Theorem C.
Let , be aspherical manifolds with and with JSJ decompositions , all of whose pieces are hyperbolic. Then the graphs and have equal numbers of vertices and edges and equal first Betti numbers.
This paper is structured as follows. In Sections 2, 3, and 5 we recall background material and the elements of profinite BassSerre theory. In Section 4 we prove some results about certain actions on profinite trees which, while known to experts, have not appeared in detail in the literature. Section 6 forms the technical core of the paper, proving many useful lemmas and ultimately the first part of Theorem 1. Theorem 1 is proved in Section 7, and Theorem 1 is proved in Section 8. In Section 9 we describe how, in certain situations, we may obtain a pro analogue of our JSJ decomposition theorem. Finally in Section 10 we complete the proof of Theorem 1.
Conventions.
For notational convenience, we will adopt the following conventions.

From Section 3 onwards, profinite groups will generally be given Roman letters ; the handful of discrete groups which appear will usually either be groups with standard symbols or the fundamental group of a space, so we will not reserve a particular set of symbols for them.

Profinite graphs will be given capital Greek letters , and abstract graphs will be

A finite graph of (profinite) groups will be denoted where is a finite graph and will be an edge or vertex group.

There is a divergence of notation between profinite group theorists, for whom denotes the adic integers, and manifold theorists who use the same symbol for the cyclic group of order . We will follow the former convention and use or to denote the cyclic group.

For us, a graph manifold will be required to be nongeometric, i.e. not a single Seifert fibre space or a Solmanifold, hence not a torus bundle. A totally hyperbolic manifold will be a 3manifold all of whose JSJ pieces are hyperbolic, and a mixed manifold will be a manifold with both Seifertfibred and hyperbolic JSJ pieces. All 3manifolds will be orientable.

The symbols , , will denote ‘normal subgroup of finite index’, ‘open normal subgroup’, ‘normal subgroup of index a power of ’ respectively; similar symbols will be used for not necessarily normal subgroups.

For two elements of a group, will denote . That is, conjugation will be a right action.
The author would like to thank Marc Lackenby for carefully reading this paper, and is grateful to Henry Wilton for discussions regarding Lemma 6.6 and for drawing the author’s attention to [GDJZ15]. The author was supported by the EPSRC and by a Lamb and Flag Scholarship from St John’s College, Oxford.
2 Preliminaries
2.1 Profinite completions
We first briefly recall the notion of a profinite group and a profinite completion. For more detail, the reader is referred to [RZ00b].
Definition 2.1.
A profinite group is an inverse limit of a system of finite groups. If all these finite groups are from a particular class of finite groups, the limit is said to be a pro group.
Given a group , its profinite completion is the profinite group defined as the inverse limit of all finite quotient groups of :
The pro completion is the pro group defined as the inverse limit of those finite quotients of which are in the class .
In this paper, we will only refer to those classes comprising all finite groups whose orders are only divisible by primes from a certain set of primes . The term ‘pro’ is then replaced by ‘pro’. When is the set of all primes, we of course get the full profinite completion. We further abbreviate ‘pro’ to ‘pro’. The pro completion is usually denoted .
Typically, we only take the profinite completion of a residually finite group; then the natural map will be injective. A (topological) generating set for a profinite group is a subset such that the subgroup abstractly generated by is dense in .
The profinite completion was determined by the inverse system of all finite quotients. In fact, by a standard argument (see [DFPR82]) two finitely generated groups have the same profinite completion if and only if the sets of isomorphism types of finite groups which can arise as finite quotients of the two groups are the same.
For a profinite group where the are finite, the kernels of the maps form a basis for the topology at 1, and so they are open subgroups of . A deep result of Nikolov and Segal [NS07] shows that for a finitely generated profinite group, all finite index subgroups are open; that is, the group structure determines a unique topology giving the structure of a profinite group. Thus any abstract isomorphism of finitely generated profinite groups is a topological isomorphism also.
The profinite completion of the integers plays, as one might expect, a central role in much of the theory. It is the inverse limit
of all finite cyclic groups. The pro completion, denoted , is the inverse limit of those finite cyclic groups whose orders only involve primes from . Because of the Chinese Remainder Theorem, these finite cyclic groups split as products of cyclic groups of prime power order, and these splittings are natural with respect to the quotient maps . It follows that the profinite completion of splits as the direct product, over all primes , of the rings of adic integers
Similarly
This splitting as a direct product, which of course is not a feature of itself, means that ringtheoretically behaves less well than . In particular, there exist zerodivisors, which are precisely those elements which vanish under some projection to . The rings themselves do not have any zerodivisors, and injects into each (for every natural number and prime , some power of does not divide ), so no element of is a zerodivisor in . Furthermore, the group of units of is rather large, being the inverse limit of the multiplicative groups of the rings . In particular is a profinite group much larger than .
Elements of profinite groups may be raised to powers with exponents in . For if is a profinite group and , the map extends to a map by the universal property of , which by continuity extends to a map (that is, is a free profinite group on the element 1). The image of under this map is then denoted . This operation has all the expected properties; see Section 4.1 of [RZ00b] for more details.
As we are discussing , this seems a fitting place to include the following easy lemma.
Lemma 2.2.
If , , and , then . In particular if then .
Proof.
Since is an integer congruent to 0 modulo , there exists such that . Since is not a zerodivisor in , it follows that . ∎
2.2 The profinite topology
Whenever profinite properties of groups are discussed, it is usually necessary to have some control over subgroup separability. Here we recall prior results that will be used heavily, and often without comment, in the sequel.
Definition 2.3.
The (full) profinite topology on a discrete group is the topology induced by the map . A subset of is separable if and only if it is closed in the profinite topology.
In particular, a group is residually finite if and only if the subset is separable. If every finiteindex subgroup of a subgroup is separable in , then it follows that the natural map is in fact an isomorphism. If this is the case we say that induces the full profinite topology on . Seifert fibred and hyperbolic 3manifold groups have very good separability properties; specifically they are LERF, meaning that every finitely generated subgroup is separable. In this case all finitely generated subgroups have .
Unfortunately graph manifold groups are not in general LERF; in fact in a sense a generic graph manifold group is not LERF. See [BKS87], [NW01]. However, those subgroups of primary concern are well behaved in the profinite topology. In particular:
Theorem (Hamilton [Ham01]).
Let be a Haken 3manifold. Then the abelian subgroups of are separable in (and thus induces the full profinite topology on them).
Theorem 2.4 (Wilton and Zalesskii [Wz10]).
Let be a closed, orientable, irreducible 3manifold, and let be the graph of spaces corresponding to the JSJ decomposition. Then the vertex and edge groups are closed in the profinite topology on , and induces the full profinite topology on them.
2.3 Seifert fibre spaces and graph manifolds
In the literature there are two inequivalent definitions of what is meant by a ‘graph manifold’. In all cases, a graph manifold is an irreducible manifold whose JSJ decomposition consists only of Seifertfibred pieces. The JSJ decomposition is by definition minimal, so the fibrings of adjacent pieces of the decomposition never extend across the union of those pieces. Some authors additionally require that a graph manifold is not geometric; i.e., it is not itself Seifertfibred and is not a Sol manifold. Hence the JSJ decomposition is nontrivial and the graph manifold is not finitely covered by a torus bundle. In this paper we do impose this constraint. Furthermore we shall deal only with orientable graph manifolds.
At the other end of the spectrum, a manifold whose JSJ decomposition is nontrivial and has no Seifertfibred pieces at all will be called ‘totally hyperbolic’. When the JSJ decomposition has at least one Seifertfibred piece and at least one hyperbolic piece the manifold is called ‘mixed’.
We now briefly recall those facts about fundamental groups of Seifert fibre spaces which will be necessary. For a more full account, see [Sco83] or [Bri07].
The fundamental group of a Seifert fibre space has a short exact sequence
where is the base orbifold and is the homotopy class of a regular fibre. The subgroup generated by this fibre (which may be finite in general) is normal in . Moreover, either it is central in (when is orientable) or is contained in some index 2 subgroup of in which it is central (when is nonorientable). The orbifold fundamental group is a Fuchsian group. The Seifert fibre spaces arising in the JSJ decomposition of a graph manifold have boundary consisting of at least one incompressible torus comprised of fibres; in this case the subgroup generated by is infinite cyclic and is a free product of cyclic groups.
The base orbifold has either positive, zero, or negative Euler characteristic. If the Euler characteristic is negative, then there is a unique maximal normal cyclic subgroup of , called the (canonical) fibre subgroup.
No positive characteristic base orbifolds can occur when the boundary is incompressible and nonempty. The Seifert fibre spaces with boundary which have Euclidean base orbifold are precisely and the orientable bundle over a Klein bottle. The first of these can never arise in a JSJ decomposition of anything other than a torus bundle and we will henceforth ignore it. The second has two maximal normal cyclic subgroups (‘fibre’ subgroups), one central (with quotient the infinite dihedral group) and one not central (with quotient , here being the fundamental group of a Möbius band). We call the first of these fibre subgroups the canonical fibre subgroup. We refer to such pieces of the JSJ decomposition of a graph manifold as ‘minor’, and to those with base orbifold of negative characteristic as ‘major’ (the terms ‘small Seifert fibre space’ and ‘large Seifert fibre space’ being already entrenched in the literature with a rather different meaning).
By definition, all pieces of the JSJ decomposition of a graph manifold are Seifert fibre spaces, and by minimality of the decomposition the fibres of two adjacent Seifert fibred pieces do not match, even up to isotopy. Thus the canonical fibre subgroups of adjacent Seifert fibre spaces intersect trivially in the fundamental group of the graph manifold; if one piece is minor, neither of its two fibre subgroups intersect the fibre subgroup of the adjacent piece nontrivially.
Note that two minor pieces can never be adjacent; for each of these having only one boundary component, the whole graph manifold would then be just two minor pieces glued together. Each has an index 2 cover which is a copy of , so our graph manifold would be finitely covered by a torus bundle, and would thus be either a Euclidean, Nil or Sol manifold; but we required our graph manifolds to be nongeometric.
Many of these properties still hold in the profinite completion; for instance, when the base orbifold is of negative Euler characteristic, Theorem 6.4 of [Wil15] guarantees that we still have a unique maximal procyclic subgroup which is either central or is central in an index 2 subgroup. We may directly check that the profinite completion of the Klein bottle group also still has just two maximal normal procyclic subgroups, one of which is central. Hence our notion of (canonical) fibre subgroup, as a maximal procyclic group with the above property, carries over to the profinite world.
Seifert fibre spaces are wellcontrolled by their profinite completions.
Theorem 2.5 (Wilkes [Wil15]).
Let be a closed orientable Seifert fibre space. Then:

If has the geometry , , , Nil, or then is profinitely rigid.

If has the geometry , and is therefore a surface bundle over a hyperbolic surface with periodic monodromy , those orientable 3manifolds with the same profinite completion as are precisely the surface bundles over with monodromy , for coprime to the order of .
The nonrigid examples of geometry were found by Hempel [Hem14].
For Seifert fibre spaces with boundary, such as those arising in the JSJ decomposition of a graph manifold, we naturally require some conditions on the boundary.
Definition 2.6.
Let be an orientable 2orbifold with boundary, with fundamental group
where the boundary components of are represented by the conjugacy classes of the elements together with
Then an exotic automorphism of of type is an automorphism such that and for all , where denotes conjugacy in . Similarly, let be a nonorientable 2orbifold with boundary, with fundamental group
where the boundary components of are represented by the conjugacy classes of the elements together with
Let be the orientation homomorphism of . Let . Then an exotic automorphism of of type with signs is an automorphism such that and for all where for all .
Remark.
The reader may find the term ‘exotic automorphism of ’ a little jarring as the automorphism really acts on . This name was chosen to emphasize that this notion depends on an identification of the group as the fundamental group of a specific orbifold, with specific elements representing boundary components. For example, an exotic automorphism of a threetimes punctured sphere is a rather different thing from an exotic automorphism of a oncepunctured torus: even though both groups are free of rank 2, there are different numbers of boundary components to consider.
For the same reason, there is a canonical map to a cyclic group of order 2 giving the orientation homomorphism this is not a characteristic quotient of a free group, but is uniquely defined when an identification with an orbifold group is chosen.
Theorem 2.7 (Wilkes [Wil15]).
Let be Seifert fibre spaces with boundary components . Suppose is an isomorphism of group systems
Then:

If is a minor Seifert fibre space, then .

If is a major Seifert fibre space, then the base orbifolds of , may be identified with the same orbifold such that splits as an isomorphism of short exact sequences
where is some invertible element of and is an exotic automorphism of of type .
Hence if is a surface bundle over the circle with fibre a hyperbolic surface with periodic monodromy , then is also such a surface bundle with monodromy where is congruent to modulo the order of .
Remark.
Since this precise theorem statement does not appear in [Wil15] in quite this form, we should comment on how it arises, and in particular the point that is rather than just .
First we remark that this theorem follows immediately from the case of orientable base orbifold. Note that preserves the canonical index 2 subgroup given by the centraliser of the fibre, so the corresponding 2fold covers and are related in the above fashion, for some labelling of the boundary components of each. The result then follows, noting that an automorphism of the base which induces an exotic automorphism on the index 2 orientation subgroup is itself an exotic automorphism.
So consider the case of orientable base orbifold. Let . The fact that an isomorphism of fundamental groups gives an isomorphism of short exact sequences is the content of Theorem 6.4 of [Wil15]. The profinite groups in question are then central extensions, given by cohomology classes
This latter cohomology group is an abelian group of exponent dividing the order of the monodromy. The statement concerning surface bundles is equivalent to the statement that
To see this equivalence, one could write out presentations of the groups and hence compute the cohomology classes directly from the characterisation in Section 6.2 of [Wil15]. See also Section 5 of [Hem14], where the computation of the fibre invariants of a surface bundle with monodromy is carried out. Note that raising the monodromy to a power actually raises the fibre invariants , and hence the cohomology class, of the space to a power inverse to modulo the order of  hence the order of and in the last part of the theorem.
From the proof of Theorem 6.7 of [Wil15], the automorphism of is an exotic automorphism of type for some , and the action of such a on cohomology is multiplication by . Let be a central extension of by corresponding to the cohomology class . Since the action of on cohomology is multiplication by , and noting that this action is contravariant, we have
and hence we have a short exact sequence of isomorphisms
Precomposing this with the short exact sequence from the theorem there is an isomorphism
which, since the map on gives a covariant map on cohomology, says that the cohomology class representing is in fact . Hence
as was claimed.
Definition 2.8.
If are as in the latter case of the above theorem, we say that is a Hempel pair of scale factor , where . Note that is only welldefined modulo the order of , which may be taken to be the lowest common multiple of the orders of the cone points of . Note that a Hempel pair of scale factor is a pair of homeomorphic Seifert fibre spaces.
3 Groups acting on profinite graphs
3.1 Profinite graphs
We state here the definitions and basic properties of profinite graphs for convenience. The sources for this section are [RZ01], Section 1 or [RZ00a], Sections 1 and 2. See these references for proofs and more detail.
Definition 3.1.
An abstract graph is a set with a distinguished subset and two retractions . Elements of are called vertices, and elements of are called edges. Note that a graph comes with an orientation on each edge.
If an abstract graph is in addition a profinite space (that is, an inverse limit of finite discrete topological spaces), is closed and are in addition continuous, then is called a profinite graph. Note that may not be closed.
A morphism of graphs , is a function such that for each . In particular, sends vertices to vertices, but may not send edges only to edges. A morphism of profinite graphs , is a map which is a morphism of abstract graphs and is continuous.
A profinite graph may equivalently be described as an inverse limit of finite abstract graphs , and . If happens to be closed, we can choose the inverse system so that the transition maps send edges to edges, and then . If is another finite graph, any morphism of onto factors through some .
A path of length in a graph is a morphism into of a finite graph consisting of edges and vertices (with some choice of orientations on the edges) such that the endpoints of each edge are . Note that such a morphism into a profinite graph is automatically continuous.
An abstract graph is pathconnected if any two vertices lie in the image of some path in . A profinite graph is connected if any finite quotient graph is pathconnected. Equivalently a connected profinite graph is the inverse limit of pathconnected finite graphs.
A pathconnected profinite graph is connected. If the set of edges of a connected profinite graph is closed, then each vertex must have an edge incident to it.
Example 3.2 (A connected profinite graph with a vertex with no edge incident to it).
Let be the onepoint compactification of two copies of the natural numbers. Let , and define by for , , and , for . This is the inverse limit of the system of finite line segments of length , where the maps collapse the final edge to the vertex . Hence is connected, but no edge is incident to .
Example 3.3 (A connected graph whose proper connected subgraphs are finite).
Consider the Cayley graphs with the natural maps coming from the natural group homomorphisms. Then is the Cayley graph of with respect to the generating set (see below). If is a connected proper subgraph of , then the image of in some is not all of ; hence it is a line segment in . In each , the preimage of this line segment is disjoint copies of ; by connectedness is precisely one of these. That is, is an isomorphism for all . Thus is just a line segment of finite length.
Example 3.4 (Cayley graph).
Let be a profinite group and be a closed subset of , possibly containing 1. The Cayley graph of with respect to is the profinite graph , where , and .
The Cayley graph is the inverse limit of the Cayley graphs of finite quotients of , with respect to the images of the set in these quotients. The set generates topologically if and only if its image in each finite quotient is a generating set; hence is connected if and only if every finite Cayley graph is connected, if and only if generates .
3.2 Profinite trees
As one might expect from the reduced importance of paths in the theory of profinite graphs, a ‘no cycles’ condition does not give a good definition of ‘tree’. Instead a homological definition is used. Let denote the finite field with elements for a prime. The following definitions and statements can be found in Section 2 of [RZ00a]. For more on profinite modules and chain complexes, see Chapters 5 and 6 of [RZ00b].
Definition 3.5.
Given a profinite space where the are finite spaces, define the free profinite module on to be
the inverse limit of the free modules with basis . Similarly for a pointed profinite space define
These modules satisfy the expected universal property, that a map from to a profinite module (respectively, a map from to sending to 0) extends uniquely to a continuous morphism of modules from the free module to .
Definition 3.6.
Let be a profinite graph. Let be the pointed profinite space with distinguished point the image of . Consider the chain complex
where the map is the evaluation map and sends the image of an edge in to . Then define and .
Proposition 3.7.
Let be a profinite graph.

is connected if and only if

The homology groups are functorial, and if , then
Definition 3.8.
A profinite graph is a pro tree, or simply tree, if is connected and . If is a nonempty set of primes then is a tree if it is a tree for every . We say simply ‘profinite tree’ if consists of all primes.
Note that a finite graph is a tree if and only if it is an abstract tree. It follows immediately from Proposition 3.7 that an inverse limit of finite trees is a tree. However this is not the only source of trees, and the notion of tree is not independent of the prime .
Example 3.9.
Let be the Cayley graph of with respect to the generator 1, written as the inverse limit of cycles of length . For a finite set ,
so for each , as this is now the simplicial homology of the realisation of as a topological space. Thus the map is just multiplication by . In particular, induces the zero map on homology. It follows that the inverse limit of these groups is trivial, i. e.
so that is a tree.
Now let be the Cayley graph of the adic integers with respect to 1, where is another prime. Again is the inverse limit of the cycles of length , and the maps induced on homology by for are multiplication by . If these are the zero map, so again
so that is a tree. On the other hand, if , multiplication by is an isomorphism from to itself, so that
and is not a tree for .
It transpires (Theorem 1.4.3 of [RZ01]) that the Cayley graph of any free pro group with respect to a free basis (that is, a pro group satisfying the appropriate universal property in the category of pro groups) is a tree. Note that the examples above show that a profinite group acting freely on a profinite tree need not be free profinite. For the subgroup acts freely on the above Cayley graph, but is not a free object in the category of all profinite groups (for instance, if is another prime, the map does not extend to a map ). It turns out (Theorem 3.1.2 of [RZ01]) that such a group will instead be projective in the sense of category theory. A pro group acting freely on a pro tree is a free pro group however (Theorem 3.4 of [RZ00a]), as in the category of pro groups, any projective group is free pro. This is one of the ways in which the pro theory is more amenable than the general theory.
Some of the topological properties of abstract trees do carry over well to the world of profinite trees:
Proposition 3.10 (Propositions 1.5.2, 1.5.6 of [Rz01]).
Let be a tree.

Every connected profinite subgraph of is a tree

Any intersection of subtrees of is a (possibly empty) subtree
It follows that for every subset of a tree there is a unique smallest subtree of containing . If consists of two vertices then this smallest subtree is called the geodesic from to and is denoted . Note that if a profinite tree is pathconnected, hence is also an abstract tree, then will coincide with the usual notion of geodesic, a shortest path from to . However there is no requirement for a geodesic to be a path. For instance, our above analysis of the connected subgraphs of the profinite tree shows that if then either or the geodesic is the entire tree .
3.3 Group actions on profinite trees
The theory of profinite groups acting on profinite trees is less tractable than the classical theory, but still parallels it in many respects. In this section we will recall results from the unpublished book [RZ01], and prove others which will be of use. The theory was originally developed in [GR78], [Zal89], [ZM89a], and [ZM89b], and the pro version of the theory may be found in published form in [RZ00a].
Definition 3.11.
A profinite group is said to act on a profinite graph if acts continuously on the profinite space in such a way that
for all . For each , the stabiliser will be denoted . For subsets , , the set of points in fixed by every element of will be denoted .
Note that given an edge , we cannot have without fixing both endpoints of , as for each . In particular, the qualification ‘without inversion’ applied to group actions in the classical theory of [Ser03] is here subsumed in the definition.
If acts on , the quotient space is a welldefined profinite graph. As one might expect, such an action may be represented as an ‘inverse limit of finite group actions on finite graphs’. More precisely,
Definition 3.12.
Let a profinite group act on a profinite graph . A decomposition of as an inverse limit of finite graphs is said to be a decomposition if acts on each in such a way that all quotient maps and transition maps are equivariant.
Indeed, the following is true
Lemma 3.13 (Lemma 1.2.1 of [Rz01]).
Let a profinite group act on a profinite graph . Then the graphs form an inverse system and
Proof.
The first statement is clear. By the universal property of inverse limits, we have a natural continuous surjection
and it remains to show that this is injective. If are identified in every , then for all there is some such that . Thus the closed subsets
of are all nonempty, and the collection is closed under finite intersections; so by compactness of , their intersection is nonempty, so there is some such that ; but the intersection of all is trivial, so . ∎
Definition 3.14.
If is a profinite group acting on a profinite tree , the action is said to be:

faithful, if the only element of fixing every vertex of is the identity;

irreducible, if no proper subtree of is invariant under the action of ;

free if for all .
Note that a group acts freely on its Cayley graph with respect to any closed subset , with quotient the ‘bouquet of circles’ on the pointed profinite space , i. e. the profinite graph with vertex space .
Faithful and irreducible actions are the most important actions; indeed, given a group acting on a profinite tree , we can quotient by the kernel of the action (i. e. those group elements fixing all of ) and then pass to a minimal invariant subtree to get a faithful irreducible action. Such a subtree exists by:
Proposition 3.15 (Proposition 1.5.9 of [Rz01]; Lemma 3.11 of [RZ00a]).
Let a profinite group act on a pro tree . Then there exists a minimal invariant subtree of , and if , it is unique.
Theorem 3.16 (Theorem 3.1.5 of [Rz01]).
Suppose a pro group acts on a tree . Then the set of fixed points under the action of is either empty or a subtree of .
Theorem 3.17 (Theorem 3.1.7 of [Rz01]).
Any finite group acting on a profinite tree fixes some vertex.
Proposition 3.18.
Let act on a profinite tree , and let . If fixes a vertex for some which is not a zerodivisor, then also fixes some vertex.
Proof.
Let be the subtree of fixed by ; it is nonempty by assumption. Consider the action of on . The closed (normal) subgroup of generated by acts trivially on , so there is a quotient action of on . Now this quotient group is . Using the splitting , we find
where is the projection onto each factor. No is zero because is not a zerodivisor in . Closed subgroups of are finite index or trivial (see Proposition 2.7.1 of [RZ00b]), so is a direct product of finite cyclic groups.
The subtree fixed by a direct product of groups is the intersection of the subtrees fixed by each group. Any finite product of finite cyclic groups fixes some vertex by Theorem 3.17, so the subtrees fixed by each finite cyclic group are a collection of nonempty closed subsets of with the finite intersection property. By compactness, the intersection of all of them is nonempty; but this is the subtree fixed by , which is the same as the subtree fixed by . So fixes some vertex of (hence of ). ∎
Remark.
The condition that is not a zerodivisor is necessary. For instance, (written multiplicatively with generator ) acts freely on its Cayley graph, but if for nonzero, then fixes no vertex but is the identity. Recall however that no element of is a zero divisor in , so that the Proposition applies in particular when .
Proposition 3.19 (Proposition 3.2.3(b) of [Rz01]).
Let be an abelian profinite group acting faithfully and irreducibly on a profinite tree. Then acts freely and for some set of primes .
4 Acylindrical actions
Actions on profinite trees are particularly malleable when the action is acylindrical.
Definition 4.1.
Let a profinite group act on a profinite tree . The action is acylindrical if the stabiliser of any injective path of length greater than is trivial.
For instance an action with trivial edge stabilisers is 0acylindrical. In [WZ17] Wilton and Zalesskii exploited the fact that if edge groups are malnormal in the adjacent vertex groups then the action on the standard graph is 1acylindrical.
We now prove some results about acylindrical actions on profinite trees. The following lemma is taken from the Appendix to [HZ12] and will be used to remove some of the pathologies associated with profinite graphs; we reproduce it here for completeness.
Lemma 4.2.
Let be a profinite graph in which there are no paths longer than edges for some integer . Then the connected components of (as a profinite graph) are precisely the path components (that is, the connected components as an abstract graph). In particular if is connected then it is pathconnected.
Proof.
First define the composition of two binary relations on a set to be the relation that if and only if there exists such that and ; inductively define . Further define to be the relation that if and only if . Let be the identity relation.
For an abstract graph define (where could be vertices), and set . If then there is some path of length at most containing and ; if there is a path of edges containing and then . This discrepancy is due to our convention that graphs are oriented, so we may need to include vertices in addition to the edges in a path of length to get a chain of related elements of . The pathcomponents of are then the equivalence classes of the equivalence relation .
Now in our profinite graph , we have for some as there is a uniform bound on the length of paths in . One can show that the continuity of the maps and compactness of imply that , and all , are closed compact subsets of . In particular, the equivalence classes of are closed subsets of ; that is, the pathcomponents of are closed. The quotient profinite graph has no edges, hence its maximal connected subgraphs are points. Thus connected components of (as a profinite graph) are contained in, hence equal to, a pathcomponent of . ∎
Proposition 4.3.
Let a profinite group act irreducibly on a profinite tree . Then either is a single vertex or it contains paths of arbitrary length.
Proof.
Assume that there is a bound on the lengths of paths in ; then by the previous result is pathconnected, hence is a tree when considered as an abstract graph. Consider two geodesics of maximal length in this tree. These must intersect in at least their midpoint (a vertex if the length is even or an edge if odd); for otherwise one can easily construct a longer path. Hence the intersection of all such maximal geodesics is nonempty. This intersection is a profinite subtree of invariant under the action of , which must permute these maximal geodesics. Hence if the action of is irreducible, is equal to the intersection of its maximal geodesics; hence is merely a path of length . But then the midpoint of this path is fixed under the action of ; hence this middle vertex (or edge) is the whole of . If fixes an edge, then by our conventions that graphs are oriented, then it fixes each endpoint. Hence is a single vertex. ∎
Corollary 4.4.
Let a profinite group act acylindrically on a profinite tree for some . Suppose fixes two vertices . Then are in the same pathcomponent of .
Proposition 4.5.
Let a profinite group act acylindrically on a profinite tree . Let be a closed abelian subgroup of . Then either for some set of primes or fixes some vertex of .
Proof.
Let be a minimal invariant subtree for the action of on . Then acts irreducibly on , so by Proposition 4.3 either is a point (whence fixes a point of ) or contains paths of arbitrary length. By Proposition 3.19, if is not a projective group then the action is not faithful, so some nontrivial element of fixes , hence fixes paths of arbitrary length. But this is impossible by acylindricity. ∎
The following concept will be useful later.
Definition 4.6.
Given a profinite group and a copy of contained in it, the restricted normaliser of in is the closed subgroup
where is a generator of .
Note that this is a closed subgroup of , containing the centraliser as an index 1 or 2 subgroup. We deal with the restricted normaliser to avoid certain technicalities in later proofs; in particular the centraliser may not be of finite index in the full normaliser. There is a continuous homomorphism from the full normaliser to . The centraliser is the kernel of this map, and the reduced normaliser is the preimage of the unique order 2 subgroup which acts nontrivially on each .
Proposition 4.7.
Let a profinite group act acylindrically on a profinite tree .

Let be closed subgroups of with , and suppose fixes a vertex of . Then fixes some vertex of .

Let be a subgroup of fixing some vertex of . Then the centraliser and the restricted normaliser both fix some vertex of .
Proof.
First let us prove part (i). If there exists such that is in ; hence fixes . Then by Proposition 3.18, there is some vertex fixed by . By Corollary 4.4, and lie in the same path component of , hence so do and . Thus acts on the path component of in , which is an abstract tree. Every element of fixes some vertex of ; hence by Section I.6.5, Proposition 26 of [Ser03], every finite set of elements of has a common fixed point in , hence in . Thus every finite intersection of the trees for is nonempty; by compactness of , their intersection is nonempty. That is, fixes a vertex of .
Now consider part (ii). The reduced normaliser has the centraliser as an index 2 subgroup, so by (i) it suffices to consider the centraliser of . Let be a generator of , and let . We will show that must fix a vertex of in the same path component of as ; the rest of the proof proceeds as for part (i).
Let . This is abelian so by Proposition 4.5 either fixes a vertex of or is a projective group . If it is projective, let generate . Then for some . Because , it follows that is not a zerodivisor in . So by Proposition 3.18, there is some vertex of fixed by , hence by . In either case, fixes a vertex ; since fixes and , they lie in the same path component of and we are done.
Note that in both parts of the Proposition, the vertex fixed by the subgroup in question is in the same pathcomponent of as , and moreover will be joined to it by a path of length less than . ∎
Remark.
The following profinite analogue of the abovequoted result in [Ser03] seems plausible, and would allow a strengthening of the above result. Let a profinite group act on a profinite tree , and suppose that every element of fixes a vertex of . Then fixes a vertex of . However the author does not know if this result is true.
5 Graphs of profinite groups
The theory of profinite graphs of groups can be defined for general profinite graphs ; we shall only consider finite graphs here as this considerably simplifies the theory and is sufficient for our needs. First we recall the notion of free profinite product, as developed in Section 9.1 of [RZ00b] and Chapter 4 of [RZ01].
Definition 5.1.
Given profinite groups a free profinite product of the consists of a profinite group and morphisms and which is universal with respect to this property. That is, for any other profinite group and morphisms there is a unique map such that .
The free profinite product exists and is unique, and will be denoted
Free profinite products are generally quite wellbehaved, for instance for discrete groups we have
Free products are a special case of a graph of groups in which all edge groups are trivial. We now move to the general definition.
Definition 5.2.
A finite graph of profinite groups consists of a finite graph , a profinite group for each , and two (continuous) monomorphisms for which are the identity when . We will often suppress the graph and refer to ‘the graph of groups ’.
Definition 5.3.
Given a finite graph of profinite groups , choose a maximal subtree of . A profinite fundamental group of the graph of groups with respect to consists of a profinite group , and a map
such that
and
and with universal with these properties. The profinite group will be denoted .
Note that in the category of discrete groups this is precisely the same as the classical definition as a certain presentation. The group so defined exists and is independent of the maximal subtree (see Section 5.2 of [RZ01]).
In the classical BassSerre theory, a graph of discrete groups gives rise to a fundamental group and an action on a certain tree whose vertices are cosets of the images of the vertex groups in and whose edge groups are cosets of the edge groups. Putting a suitable topology and graph structure on the corresponding objects in the profinite world and proving that the result is a profinite tree, is rather more involved than the classical theory; however the conclusion is much the same. We collate the various results into the following theorem.
Theorem 5.4 (See Section 5.3 of [Rz01]).
Let be a finite graph of profinite groups. Then there exists an (essentially unique) profinite tree , called the standard graph of , on which acts with the following properties. Set .

The quotient graph is isomorphic to .

The stabiliser of a point is a conjugate of in , where is the quotient map.
Conversely (see Section 5.4 of [RZ01]) an action of a profinite group on a profinite tree with quotient a finite graph gives rise to a decomposition as a finite graphs of profinite groups. However no analogous result holds when the quotient graph is infinite.
In the classical theory one tacitly identifies each with its image in the fundamental group of a graph of groups. In general in the world of profinite groups the maps may not be injective, even for simple cases such as amalgamated free products. We call a graph of groups injective if all the maps are in fact injections.
Let be a finite graph of abstract groups. We can then form a finite graph of profinite groups by taking the profinite completion of each vertex and edge group of . We have not yet addressed whether the ‘functors’
and
on a graph of discrete groups yield the same result; that is, whether the order in which we take profinite completions and fundamental groups of graphs of groups matters. In general the two procedures do not give the same answer; we require some additional separability properties.
Definition 5.5.
A graph of discrete groups is efficient if is residually finite, each group is closed in the profinite topology on , and induces the full profinite topology on each .
Theorem 5.6 (Exercise 9.2.7 of [RZ00b]).
Let be an efficient finite graph of discrete groups. Then is an injective graph of profinite groups and
In our case of interest, the abovequoted Theorem 2.4 of Wilton and Zalesskii may be rephrased as ‘the JSJ decomposition of a 3manifold group is efficient’, and the profinite completion of our graph manifold group acts in a wellcontrolled fashion on a profinite tree.
6 The JSJ decomposition
Definition 6.1.
Let be an aspherical 3manifold with JSJ decomposition . The Seifert graph of is the full subgraph of spanned by those vertices whose associated 3manifold is a Seifert fibre space. It will be denoted .
In this section we analyse the JSJ decomposition of an aspherical manifold, and show that ‘the Seifertfibred part’ is a profinite invariant. Specifically we prove:
Theorem 6.2.
Let , be closed aspherical 3manifolds with JSJ decompositions , respectively. Assume that there is an isomorphism . Then there is an isomorphism such that for every .
If in addition , are graph manifolds then, after possibly performing an automorphism of , the isomorphism induces an isomorphism of JSJ decompositions in the following sense:

there is a graph isomorphism ;

restricts to an isomorphism for every .
The proof will consist of an analysis of the subgroups of together with the normalisers of their cyclic subgroups. We maintain the above notations for the rest of the section. Additionally let be the standard graph of a graph of profinite groups , and let be the projection. Let denote the canonical fibre subgroup of a vertex stabiliser which is a copy of the profinite completion of some Seifert fibre space group. By abuse of terminology, we will refer to vertices of as major, minor or hyperbolic when the corresponding vertex stabiliser is the profinite completion of a major or minor Seifert fibre space group or a cusped hyperbolic 3manifold.
We will first show that the action on is 4acylindrical.
Remark.
Wilton and Zalesskii use this fact in their paper [WZ10] for graph manifolds, as well as in [WZ17] and [HWZ12] more generally. Their proof does however contain a gap. Specifically, their version of Lemma 6.3 below only allows for conjugating elements in the original group , rather than its profinite completion. There is a similar problem in the hyperbolic pieces, which we deal with in Lemma 6.6 below.
Lemma 6.3.
Let be a hyperbolic 2orbifold, and let be curves representing components of . Let , and let be the closure in of . Then for , either or and .
Proof.
By conjugating by we may assume that ; drop the subscript on . Note that is torsionfree, so it is sufficient to pass to a finite index subgroup and show that .
Because is hyperbolic, it has some finitesheeted regular cover with more than two boundary components; then given any pair of boundary components, has a decomposition as a free product of cyclic groups, among which are the two boundary components. Let be the corresponding finite index normal subgroup of . Note that for some set of coset representatives of in (which give coset representatives of in ), each is the closure of the fundamental group of a component of ; so set where for some . Furthermore, if two boundary components of are covered by the same boundary component , then they must have been the same boundary component of ; that is, if , then .
Now the intersections of with are free factors; that is,
where is a free product of cyclic groups (unless