DECIDING IF A HYPERBOLIC GROUP SPLITS OVER A GIVEN QUASICONVEX SUBGROUP JOSEPH PAUL MACMANUS

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DECIDING IF A HYPERBOLIC GROUP SPLITS OVER A

GIVEN QUASICONVEX SUBGROUP

JOSEPH PAUL MACMANUS

Abstract. We present an algorithm which decides whether a given quasicon-

vex residually ﬁnite subgroup Hof a hyperbolic group Gis associated with a

splitting. The methods developed also provide algorithms for computing the

number of ﬁltered ends ˜e(G, H)of Hin Gunder certain hypotheses, and give

a new straightforward algorithm for computing the number of ends e(G, H)

of the Schreier graph of H. Our techniques extend those of Barrett via the

use of labelled digraphs, the languages of which encode information on the

connectivity of ∂G −ΛH.

0. Introduction

The study of decision problems within group theory is almost as old as the deﬁ-

nition of an abstract group itself, dating back to Dehn’s classical word, conjugacy,

and isomorphism problems. The classical theorems of Novikov–Boone [30,6] and

Adian–Rabin [1,32] state that these problems — among many, many others — turn

out to be undecidable in the class of ﬁnitely presented groups. Thus, if we wish to

search for eﬀective solutions to group theoretical problems, one must restrict their

scope to some “nice” subclass of groups. One such subclass is the class of hyper-

bolic groups. Introduced by Gromov in his landmark essay [18], these are groups

whose Cayley graphs possess geometric properties reminiscent of negative curva-

ture. Indeed, within this class many problems become decidable. For example, the

class of hyperbolic groups has uniformly solvable word and conjugacy problems [9,

ch. III.H], and more recently it was shown that one can distinguish isomorphism

classes of hyperbolic groups [38,12].

Another algorithmic problem which has received attention in recent years is that

of splitting detection. In the language of Bass–Serre theory [39], a group Gsplits

over a subgroup Hif Gadmits a minimal simplicial action on a tree Twithout

inversions, and Hstabilises an edge in this action. This problem could be traced

back to the algorithm of Jaco–Oertel [21] which decides if a given closed irreducible

3-manifold Mis Haken, or equivalently if π1(M)splits over an inﬁnite surface

group. Splittings over ﬁnite subgroups are called ﬁnite splittings, and a celebrated

theorem of Stallings [40,41] states that a ﬁnitely generated group admits a ﬁnite

splitting if and only if it has more than one geometric end (cf. [37]). This provides

a powerful link between the coarse geometry of a group and its splitting properties.

Returning to the realm of hyperbolic groups, we have the following unpublished

result due to Gerasimov [16].

Theorem 0.1 (Gerasimov).There is an algorithm which, upon input of a presen-

tation of a hyperbolic group G, will compute the number of ends of G.

Mathematical Institute, Andrew Wiles Building, Observatory Quarter, Univer-

sity of Oxford, Oxford, OX2 6GG, UK

E-mail address:macmanus@maths.ox.ac.uk.

Date: First draft: 22 October 2022. Final version: 28 May 2024.

arXiv:2210.09973v5 [math.GR] 28 May 2024

2 DECIDING IF A HYPERBOLIC GROUP SPLITS OVER A GIVEN SUBGROUP

In particular, one can eﬀectively detect ﬁnite splittings of hyperbolic groups.

This result was later generalised by Diao–Feighn to ﬁnite graphs of ﬁnitely gener-

ated free groups [13], by Dahmani–Groves [11] to relatively hyperbolic groups, and

by Touikan [43] to ﬁnitely presented groups with a solvable word problem and no

2-torsion. Also worthy of mention is the algorithm by Jaco–Letscher–Rubinstein for

computing the prime decomposition of a closed orientable 3-manifold [20], as well

as the classical description of the Grushko decomposition of a one-relator groups

[27, Prop. II.5.13].

Finite splittings aside, the next logical step is to detect splittings over two-

ended (i.e. virtually cyclic) subgroups. This was achieved for (relatively) hyper-

bolic groups independently by Barrett [3] and Touikan [43] using quite distinct

approaches. Note that Touikan’s algorithm here only applies in the torsion-free

case.

Theorem 0.2 (Barrett, Touikan).There is an algorithm which, upon input of

a presentation of a hyperbolic group G, will decide if Gsplits over a two-ended

subgroup.

The algorithm of Barrett, which is of particular interest to us, makes use of

Bowditch’s deep theorem on two-ended splittings of hyperbolic groups [7]. This

theorem states that a one-ended hyperbolic group Gwhich is not virtually Fuchsian

will admit such a splitting if and only if ∂G contains a cut pair. In fact, Barrett

applies this result to eﬀectively construct Bowditch’s canonical JSJ decomposition

of a hyperbolic group.

In this paper we will aim to expand on the techniques of [3], and apply them to

larger splittings. We will restrict our attention to quasiconvex subgroups, i.e. those

subgroups whose inclusion maps are quasi-isometric embeddings, since distorted

subgroups exhibit global geometry which is harder to understand on a local scale.

If a group Gsplits over a subgroup commensurable with H, we say His associated

with a splitting. Finding suﬃcient conditions for a subgroup to be associated to a

splitting is a problem which has received a great amount of interest (e.g. [34,35,

28,36,29]). Applying the results of [29] to the setting of quasiconvex subgroups of

hyperbolic groups, we are able to prove the following decidability result.

Theorem 0.3 (cf. 4.8).There is an algorithm which takes in as input a one-ended

hyperbolic group Gand generators of a quasiconvex, residually ﬁnite subgroup H.

This algorithm will then decide if His associated with a splitting, and will output

such a splitting if one exists.

It is possible to somewhat weaken the residual ﬁniteness assumption placed on H

in the theorem above and give a more general (but more involved) result. We will

postpone this more technical statement until Section 4.3 (see Theorems 4.6, 4.7).

In light of Stallings’ Theorem, it is a natural generalisation to deﬁne the number

of ends of a pair of groups (G, H)where H≤G. This deﬁnition was ﬁrst introduced

by Houghton [19] and later explored in more depth in the context of discrete groups

by Scott [33]. The number of ends of the pair (G, H), denoted e(G, H), can be

identiﬁed with the number of geometric ends of the quotient of the Cayley graph

of Gby the left action of H. This quotient graph is sometimes called the coset

graph or Schreier graph of H. It is not hard to show that if Gsplits over Hthen

e(G, H)≥2, but the converse does not hold. Our methods give a new proof of the

following theorem, originally due to Vonseel [44].

Theorem 0.4 (Vonseel, cf. 3.11).There is an algorithm which, upon input of a

one-ended hyperbolic group Gand generators of a quasiconvex subgroup H, will

output e(G, H).

DECIDING IF A HYPERBOLIC GROUP SPLITS OVER A GIVEN SUBGROUP 3

There is a competing notion of “ends” of a pair of a groups which goes by several

names in the literature. This idea was considered independently by Bowditch [8],

Kropholler–Roller [26], and Geoghegan [15], who refer to this invariant as coends,

relative ends, and ﬁltered ends respectively. See [36, ch. 2] for a discussion on the

equivalence of these three deﬁnitions. In this paper we will adopt the terminology

and notation of Geoghegan, and denote the number of ﬁltered ends of the pair

(G, H)by ˜e(G, H). This value appears to be more resilient to calculation without

extra hypotheses, but nonetheless we have some partial results. Recall that the

generalised word problem for a ﬁnitely generated group His the problem of, given

words w0,...wnin the generators of H, deciding whether w0∈ ⟨w1,...wn⟩H. We

then have the following statement.

Theorem 0.5 (cf. 3.12, 3.13).There is an algorithm which takes in as input a

one-ended hyperbolic group G, and generators of a quasiconvex subgroup H. This

algorithm will terminate if and only if ˜e(G, H)is ﬁnite, and if it terminates will

output the value of ˜e(G, H).

Furthermore, if one is also given a solution to the generalised word problem for

H, then there is an algorithm which decides whether ˜e(G, H)≥Nfor any given

N≥0.

We remark in Section 2.2 that ˜e(G, H)can be identiﬁed with the number of

components of ∂G −ΛH. Thus, the above algorithm allows us to decide if ∂G −

ΛHis disconnected. It’s also worth noting that if we know a priori that ˜e(G, H)

is ﬁnite, for example if His two-ended, then ˜e(G, H)is fully computable. We

will also see that the value of ˜e(G, H)is computable if His free. It does not

seem possible to decide in general if ˜e(G, H) = ∞using our machinery for an

arbitrary quasiconvex subgroup, without assuming further hypotheses. We discuss

this limitation in Section 3.4.

Acknowledgements. My thanks go to Panos Papazoglou for suggesting this prob-

lem, and for many helpful discussions. I’m also grateful to Sam Hughes and Ric

Wade for their detailed feedback, to Michah Sageev and Henry Wilton for fruitful

exchanges, and to Thomas Delzant for pointing me towards Vonseel’s work. Finally,

I thank the referees for their helpful suggestions.

1. Preliminaries

In this section we recall the basic notions and tools we require. We begin with

a look at almost invariant sets and (ﬁltered) ends of pairs, before brieﬂy turning

towards hyperbolic groups and their quasiconvex splittings. Throughout this paper

we will assume a working knowledge of Bass–Serre theory, a good reference for

which is [39].

1.1. Almost invariant subsets and (ﬁltered) ends. We will need the idea of

an almost invariant subset. A very good introduction to the upcoming deﬁnitions

can be found in [34], which features many helpful examples. The reader should

note however that this paper contains an error, a correction of which can be found

in [36].

First, some notation. In what follows, Gwill be a ﬁnitely generated group and

H≤Ga ﬁnitely generated subgroup. If Zis a set upon which Hacts on the left,

then denote by H\Zthe quotient of Zby this action.

Deﬁnition 1.1. Let Uand Vbe two sets. Denote by U△Vthe symmetric diﬀer-

ence of Uand V, deﬁned as

U△V:= (U−V)∪(V−U).

4 DECIDING IF A HYPERBOLIC GROUP SPLITS OVER A GIVEN SUBGROUP

We say that two sets U,Vare almost equal if U△Vis ﬁnite.

Deﬁnition 1.2. Let Gact on the right on a set Z. We say that U⊂Zis almost

invariant if for all g∈G,Ug is almost equal to U.

Deﬁnition 1.3. We say a subset U⊂Gis H-ﬁnite or small, if Uprojects to a

ﬁnite subset of H\G. If Uis not H-ﬁnite, then we say Uis H-inﬁnite, or large.

To ease notation, given a subset X⊂Gwe will write X∗:= G−X.

Deﬁnition 1.4. We say that a subset X⊂Gis H-almost invariant if it is invariant

under the left action of H, and H\Xis almost invariant under the right action of

Gon H\G. We say that Xis non-trivial if both Xand X∗are H-inﬁnite.

Let Xand Ybe two non-trivial H-almost invariant subsets of G. We say that

Xand Yare equivalent, if X△Yis H-ﬁnite.

Deﬁnition 1.5. Let Xbe an H-almost invariant subset. Given g∈G, we say that

gX crosses Xif all of

gX ∩X, gX ∩X∗, gX∗∩X, gX∗∩X∗

are large. If there exists g∈Gsuch that gX crosses Xthen we say that Xcrosses

itself. If Xdoes not cross itself, we say it is almost nested. If one of the above

intersections is empty, we say Xis nested.

It is easy to see that if Xand Yare equivalent H-almost invariant sets, then X

crosses itself if and only if Ycrosses itself.

Example 1.6. Suppose a group Gsplits as an amalgam or HNN extension over

a subgroup H. Then one can construct a non-trivial nested H-almost invariant

subset X⊂Gas follows. Let Tbe the Bass-Serre tree of this splitting. We now

construct a G-equivariant map ϕ:G→V T , where V T denotes the set of vertices

of T. Given 1∈G, set ϕ(1) arbitrarily to some w∈V T . For each g∈G, set

ϕ(g) = gϕ(1). Since Gacts upon itself freely and transitively, ϕis well deﬁned for

all g∈G.

Given ϕas above, let e∈ET be the edge stabilised by Hwith endpoints u, v.

Deleting the interior of eseparates Tinto two components, Tuand Tvcontaining

uand vrespectively. Set X=ϕ−1(V Tu), then it is a simple exercise to check that

Xis a non-trivial nested H-almost invariant subset of G.

We can now state the following key theorem due to Scott–Swarup [35], which is

in some sense a converse to Example 1.6. Recall that a subgroup Hof Gis said to

be associated to a splitting if Gsplits over a subgroup commensurable with H.

Theorem 1.7 ([35, Thm. 2.8]).Let Gbe a ﬁnitely generated group, Ha ﬁnitely

generated subgroup, and Xan H-almost invariant subset of G. Suppose that Xis

almost nested, then His associated to a splitting.

There is a generalisation of the above, which will be important to us. Firstly,

we must further loosen our requirements for nesting. Denote by CommG(H)the

commensurator of Hin G. That is,

CommG(H) = {g∈G:|H:H∩Hg|<∞,|Hg:H∩Hg|<∞}.

Then we have the following deﬁnition.

Deﬁnition 1.8. Let X⊂Gbe H-almost invariant. We say that Xis semi-nested

if {g∈G:gX crosses X}is contained in CommG(H).

Informally, we relax our deﬁnition to allow crossings of Xby gX on the condition

that gH is “very close” to H. We then have the following useful result, due to Niblo–

Sageev–Scott–Swarup [29], which says that this relaxation still produces splittings.

DECIDING IF A HYPERBOLIC GROUP SPLITS OVER A GIVEN SUBGROUP 5

Theorem 1.9 ([29, Thm. 4.2]).Let Gbe a ﬁnitely generated group and Ha ﬁnitely

generated subgroup. Suppose that there exists a non-trivial H-almost invariant sub-

set X⊂Gwhich is semi-nested. Then Gsplits over a subgroup commensurable

with H.

This idea of “crossings” of almost invariant sets is much more rich than what is

presented here, and pertains to the idea of “compatible” splittings. The interested

reader should consult [35], which features many helpful examples, as a starting

point.

There is a natural way to “count” these H-almost invariant subsets, which pro-

vides a useful integer invariant of the subgroup. Let P(H\G)denote the power

set of H\G. Let F(H\G), denote the set of ﬁnite subsets. Under the operation of

symmetric diﬀerence △,P(H\G)can be seen as a Z2-vector space, and F(H\G)

a subspace. The quotient space E(H\G) := P(H\G)/F(H\G)can be identiﬁed

naturally with the set of H-almost invariant sets of G, modulo equivalence.

Deﬁnition 1.10. Let Gbe a group and H≤G. We deﬁne the number of ends

of the pair (G, H)as the rank of E(H\G)as a Z2-vector space. Denote by e(G) =

e(G, {1}), and say that e(G)is the number of ends of G.

There is also the following characterisation of ends of pairs, which will be helpful

later. This result motivates the earlier description that e(G, H)“counts” H-almost

invariant subsets.

Proposition 1.11 ([33, Lem. 1.6]).Let Gbe a group, Ha subgroup, and n≥0.

Then e(G, H)≥nif and only if there exists a collection of npairwise disjoint

H-almost invariant subsets of G.

There is a more geometric intuition for the above, which can be seen in the coset

graph.

Proposition 1.12 ([33, Lem. 1.1]).Let Γbe a Cayley graph of G, and H⊂G.

Then e(G, H)is equal to the number of ends of the coset graph H\Γ.

From the above it is then clear that this deﬁnition generalises the standard

geometric notion of ends. This also provides a helpful way of seeing that the

number of geometric ends of a ﬁnitely generated group is indeed independent of the

choice of generating set.

There is another competing, but equally interesting notion of ends of a pair

of groups, namely the idea of ﬁltered ends. This was considered independently

by Geoghegan [15, Ch. 14], Kropholler–Roller [26], and Bowditch [8]. We now

summarise the deﬁnition as it appears in [15]. We ﬁrst need the following technical

preliminaries relating to ﬁltrations.

Let Ybe a connected, locally ﬁnite cell complex. A ﬁltration K={Ki}of Yis

an ascending sequence of subcomplexes K1⊂K2⊂. . . ⊂Ysuch that SiKi=Y.

We say that this ﬁltration is ﬁnite if each Kiis ﬁnite. We call the pair (Y, K)a

ﬁltered complex, and a map f: (Y, K)→(X, L)between ﬁltered complexes is called

aﬁltered map if the following four conditions hold:

(1) ∀i,∃jsuch that f(Kj)⊂Li,

(2) ∀i,∃jsuch that f(Ki)⊂Lj,

(3) ∀i,∃jsuch that f(Y−Kj)⊂X−Li,

(4) ∀i,∃jsuch that f(Y−Ki)⊂X−Lj.

We say that a homotopy Htbetween two ﬁltered maps is a ﬁltered homotopy if Ht

is a ﬁltered map for each t. Fix a basepoint b∈Y, then a ﬁltered ray based at b is

a map γ: [0,∞)→Ywith γ(0) = b, which is ﬁltered with respect to the ﬁltration

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DECIDINGIFAHYPERBOLICGROUPSPLITSOVERAGIVENQUASICONVEXSUBGROUPJOSEPHPAULMACMANUSAbstract.Wepresentanalgorithmwhichdecideswhetheragivenquasicon-vexresiduallyfinitesubgroupHofahyperbolicgroupGisassociatedwithasplitting.Themethodsdevelopedalsoprovidealgorithmsforcomputingthenumberoffilteredends˜e(G,H)of...

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DECIDING IF A HYPERBOLIC GROUP SPLITS OVER A GIVEN QUASICONVEX SUBGROUP JOSEPH PAUL MACMANUS

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