Polynomial-time proofs that groups are hyperbolic

Abstract

It is undecidable in general whether a given finitely presented group is word hyperbolic. We use the concept of pregroups, introduced by Stallings (1971), to define a new class of van Kampen diagrams, which represent groups as quotients of virtually free groups. We then present a polynomial-time procedure that analyses these diagrams, and either returns an explicit linear Dehn function for the presentation, or returns fail, together with its reasons for failure. Furthermore, if our procedure succeeds we are often able to produce in polynomial time a word problem solver for the presentation that runs in linear time. Our algorithms have been implemented, and when successful they are many orders of magnitude faster than KBMAG, the only comparable publicly available software.

Introduction

The Dehn function of a finitely presented group is linearly bounded if and only if the group is hyperbolic. We describe a new, polynomial-time procedure for proving that a group defined by a finite presentation is hyperbolic, by establishing such a linear upper bound on its Dehn function. Our procedure returns a positive answer significantly faster than other methods, and in particular always terminates in low degree polynomial time, although sometimes it will terminate with fail even when the input group is hyperbolic. Our approach has the added advantage that it can sometimes be carried out by hand, which can enable one to prove the hyperbolicity of infinite families of groups.

A finitely generated group is (word) hyperbolic if its Cayley graph is negatively curved as a geometric metric space; that is, if its geodesic triangles are uniformly slim. There are several good sources, such as Alonso et al. (1991), for an introduction to and development of the basic properties of hyperbolic groups. Another useful reference for the specific properties that we need in this paper is Holt et al. (2017, Chapter 6). In particular, hyperbolic groups are finitely presentable, and they admit a Dehn algorithm. Furthermore, for groups that are defined by a finite presentation, this last condition implies that the group is hyperbolic.

We use Lyndon and Schupp (1977, Chapter V) as reference for the theory of van Kampen diagrams over group presentations, and its application to groups defined by presentations that satisfy various small cancellation hypotheses. The arguments used in the proofs of these results can be formulated in terms of the assignment of curvature to the vertices, edges and faces of reduced van Kampen diagrams. The idea is to show that, under appropriate conditions, the curvature in those parts of the diagrams that are not close to the boundary is non-positive. For example, one specific conclusion of Theorem 4.4 of Lyndon and Schupp (1977) is that groups with presentations that satisfy $C^{\prime }(\lambda )$ for $\lambda \le 1/6$, or $T(4)$ together with $C^{\prime }(\lambda )$ for $\lambda \le 1/4$, have Dehn algorithms. It is not assumed in these results that the group presentations in question are finite, but we shall be working only with finite presentations in this paper, in which case these conditions imply that the group is hyperbolic.

The algorithmic methods developed in this paper involve the assignment of curvature to van Kampen diagrams in the manner described above, such that the total curvature of every diagram is 1. A serious limitation of methods that rely on small cancellation conditions is that they are unlikely to be satisfied in the presence of short defining relators such as powers $x^n$, where n is small and x is a generator. However, such relators are present in many of the most interesting group presentations. Our methods use the theory of pregroups, developed by Stallings (1971). This theory enable us to remove short relators, replacing them with certain other relators of length three (the pregroup relators), which we then ignore when considering generalisations of small cancellation.

Our general aim is to assign the curvature in such a way that vertices and edges have zero curvature, faces labelled by pregroup relators have non-positive curvature, and other faces that are not close to the boundary of the diagram have curvature that is bounded above by some constant $-\epsilon <0$. Our principal theoretical result is Theorem 5.9, which states roughly that, if we can assign curvature in this manner to all reduced diagrams over the presentation, then the group has a Dehn function that is bounded above by a linear function that we can specify explicitly in terms of ε and various basic parameters of the defining presentation. So the group is hyperbolic.

Although we cannot expect such methods to work for all hyperbolic groups, we can explore a variety of methods of assigning curvature, which we call curvature distribution schemes. In this paper we restrict attention to a single such scheme, which we call RSym. In Theorem 6.13, we apply Theorem 5.9 to calculate an explicit linear bound on the Dehn function that is satisfied in the event that RSym succeeds in assigning curvature in the required fashion. As a simple remedy in examples in which RSym does not succeed and some, but not all, interior faces of a diagram end up with zero or positive curvature, we could try to transfer some of the negative curvature from those faces that already have it to those that do not. This process is hard to implement in a computer algorithm, but it can often be done by hand, which significantly increases the applicability of the methods.

The principal algorithmic challenge is to prove that RSym succeeds on a sufficiently large set of reduced diagrams over the input presentation. Our main algorithm, RSymVerify, which is described in Section 7, attempts to achieve this. It is technically complicated and involves a detailed study of the possible neighbourhoods of interior faces in diagrams over the presentation.

For hyperbolic groups, the word problem is solvable in linear time by a Dehn algorithm. However, the current best results require as a preprocessing step the computation of the set S of all words of length up to 8δ that are trivial in the group, where geodesic triangles in the Cayley graph are δ-thin. Given a linear bound λn on the Dehn function $\mathrm{D}(n)$ of a finitely-presented group G, it is therefore theoretically possible to use brute force to test all such words for triviality in G, and hence to construct the set S. However, this requires time and space that are exponential in both δ and λ, and so is completely impractical. We instead devise an additional polynomial-time test, which, if satisfied, enables the polynomial-time construction of a linear time word problem solver. This additional test is also the basis of future joint work by the sixth author, which will give a polynomial-time construction of a quadratic time solver for the conjugacy problem, the second of Dehn's classic problems.

Here is a breakdown of the contents of the paper. In Section 2, we summarise the required properties of pregroups, and define a new kind of presentation, called a pregroup presentation, for a group G. It was shown by Rimlinger (1987) that a finitely generated group H is virtually free if and only if H is the universal group $U(P)$ of a finite pregroup P: see Theorem 2.14. Pregroup presentations enable us to view the group G as a quotient of a virtually free group $U(P)$, rather than just as a quotient of a free group, and hence to ignore any failures of small cancellation on the defining relators of $U(P)$.

In Section 3 we define coloured van Kampen diagrams over these new pregroup presentations, where the relators of the virtually free group $U(P)$ (which we collect in a set $V_P$) are coloured red, and the additional relators (which we collect in a set $\mathcal{R}$) are green. We show in Proposition 3.17 that, given any coloured van Kampen diagram Γ satisfying a certain technical condition, there exists a coloured van Kampen diagram ${\mathrm{\Gamma }}^{\prime }$, with the same boundary word as Γ, whose area is bounded by an explicit linear function of the number of green faces of Γ. Hence, to prove that a group is hyperbolic it suffices to prove a linear upper bound on the number of green faces appearing in any reduced coloured diagram of boundary length n.

In Section 4 we show that if we replace our presentation by a certain related presentation, then we can assume without loss of generality that each vertex of a coloured diagram is incident with at least two green faces. This property will be critical to our later curvature analysis, and is automatically satisfied by diagrams over free groups, since for them all faces are green.

Section 5 is devoted to the definition and general discussion of curvature distribution schemes. These provide an overall schema for the design of many possible methods for proving that a group given by a finite pregroup presentation is hyperbolic: since a pregroup presentation is a generalisation of a standard presentation, these methods apply to all finite presentations. As mentioned earlier, in Theorem 5.9 we characterise how these schema can produce explicit bounds on the Dehn function.

In Section 6 we present the RSym curvature distribution scheme mentioned earlier. For reasons of space and ease of comprehension, we restrict attention to this scheme in this paper, but our approach can be used to define many others. Theorem 6.13 gives an explicit bound on the Dehn function of the presentation when RSym succeeds on all coloured van Kampen diagrams of minimal coloured area.

In Section 7 we prove (see Theorem 7.20, Theorem 7.22) that under some mild and easily testable assumptions on the set $\mathcal{R}$ of green relators, one can test whether RSym succeeds on all of the (infinitely many) coloured van Kampen diagrams of minimal area. This test is carried out by our procedure RSymVerify (Procedure 7.19), which runs in time $O(|X{|}^5+r^3|X{|}^4|\mathcal{R}{|}^2)$, where X is the set of generators, and r is the length of the longest green relator. Our assumptions hold, for example, for all groups given as quotients of free products of free and finite groups. We also prove that without these assumptions, one can test whether RSym succeeds on all minimal diagrams in time polynomial in $|X|$ and $r|\mathcal{R}|$: our procedure to do this is called RSymIntVerify (Procedure 7.30).

In Section 8 we go on to consider the word problem. Whilst a successful run of RSymVerify or RSymIntVerify proves an explicit linear bound on the Dehn function, it is rarely practical to construct a set of Dehn rewrites. We present a low degree polynomial-time method to construct a word problem solver: see Theorem 8.6. The construction of the solver succeeds in many but not all examples in which RSym succeeds, and the solver itself runs in linear time: see Theorem 8.9 and Proposition 8.12.

In Section 9 we consider a variety of examples of finite group presentations, and show how RSym can be used by hand to prove that the groups are hyperbolic. In particular, we prove that RSym succeeds on groups satisfying any of a wide variety of small cancellation conditions, we use RSym to analyse two infinite families of presentations, and we discuss a range of possible future applications of RSym to problems concerning the hyperbolicity of finitely-presented groups.

Our procedures have been released as part of both the GAP (The GAP Group, 2019) and MAGMA (Bosma et al., 1997) computer algebra systems, and in Section 10 we present runtimes on a variety of examples, including some with very large numbers of generators and relations. Almost none of these examples could have been analysed using previously existing methods, due to the size of the presentations.

Since we have introduced many new terms and much new notation, we conclude with Appendix A, which contains lists of all new terms, notation, and procedures.

As far as we know, the only other publicly available software that can prove hyperbolicity of a group defined by an arbitrary finite presentation is the first author's KBMAG package (Holt, 1995) for computing automatic structures. Hyperbolicity is verified by proving that geodesic bigons in the Cayley graph are uniformly thin, as described in Holt (1996, Section 5). It was proved by Papasoglu (1995) that this property implies hyperbolicity, but it does not provide a useful bound on the Dehn function. An algorithm for computing the “thinness” constant for geodesic triangles in the Cayley graph of a hyperbolic group is described in Epstein and Holt (2001), but this is of limited applicability in practice, on account of its high memory requirements. Even on the simplest examples, the KBMAG programs involve far too many computational steps for them to be carried out by hand, and they can only be applied to individual presentations. The automatic structure does however provide a fast method (at worst quadratic time) of reducing words to normal form and hence solving the word problem in the group.

Shortly before submitting this paper, we became aware of a paper by Lysenok (2018), which explores similar concepts of redistributing curvature to prove hyperbolicity to those presented in Section 5 of this paper. His main theorem is similar to our Theorem 5.9, but the ideas are less fully developed.