We describe a procedure which verifies that a group given by generators and relators is word-hyperbolic. This procedure always works with a group which is word-hyperbolic, provided there is sufficient memory and time devoted to the problem. If the group is not word-hyperbolic, the procedure continues indefinitely. We also describe a procedure which computes the thinness of geodesic triangles in the Cayley graph of a word-hyperbolic group. Again this procedure is bound to work, given sufficient memory and time.
We examine residual properties of word-hyperbolic groups, adapting a method introduced by Darren Long to study the residual properties of Kleinian groups.
Let S be a closed surface of genus at least 2. We show that a finitely generated group G which is an extension of the fundamental group H of S is word hyperbolic if and only the orbit map of the quotient group G/H on the complex of curves is a quasi-isometric embedding.This in turn is equivalent to G/H being convex cocompact in the sense of Farb and Mosher.
The aim of this paper is to demonstrate that very many Dehn fillings on a cusped hyperbolic 3-manifold yield a 3-manifold which is irreducible, atoroidal and not Seifert fibred, and which has infinite, word hyperbolic fundamental group. We establish an extension of the Thurston-Gromov $2π$ theorem by showing that if each filling slope has length more than six, then the resulting 3-manifold has all the above properties. We also give a combinatorial version of the $2π$ theorem which relates to angled ideal triangulations. We apply these techniques by studying surgery along alternating links.
The study of word hyperbolic groups is a prominent topic in geometric group theory; however word hyperbolic groups are defined by a geometric condition which does not extend naturally to semigroups. We propose a linguistic definition. Roughly speaking a semigroup is word hyperbolic if its multiplication table is a context free language. For groups this definition is equivalent to the original geometric one. We also briefly consider word problems of semigroups.
This work is concerned with the stable norm in word hyperbolic groups as defined by Gromov. We give a short elementary proof of one of its basic property, that is existence of a computable uniform non null lower bound for stable norm in a word hyperbolic group.
We prove that the compressed word problem in a group that is hyperbolic relative to a collection of free abelian subgroups is solvable in polynomial time.
Formal languages based on the multiplication tables of finitely generated groups are investigated and used to give a linguistic characterization of word hyperbolic groups.
Thurston has claimed (unpublished) that central extensions of word hyperbolic groups by finitely generated abelian groups are automatic. We show that they are in fact biautomatic. Further, we show that every 2-dimensional cohomology class on a word hyperbolic group can be represented by a bounded 2-cocycle. This lends weight to the claim of Gromov that for a word hyperbolic group, the cohomology in every dimension is bounded.
This paper proves that any monoid presented by a confluent context-free monadic rewriting system is word-hyperbolic. This result then applied to answer a question asked by Duncan & Gilman by exhibiting an example of a word-hyperbolic monoid that does not admit a word-hyperbolic structure with uniqueness (that is, in which the language of representatives maps bijectively onto the monoid).
A longstanding question of Gromov asks whether every one-ended word-hyperbolic group contains a subgroup isomorphic to the fundamental group of a closed hyperbolic surface. An infinite family of word-hyperbolic groups can be obtained by taking doubles of free groups amalgamated along words that are not proper powers. We define a set of polygonal words in a free group of finite rank, and prove that polygonality of the amalgamating word guarantees that the associated square complex virtually contains a $π_1$-injective closed surface. We provide many concrete examples of classes of polygonal word…
This paper studies decision problems for semigroups that are word-hyperbolic in the sense of Duncan & Gilman. A fundamental investigation reveals that the natural definition of a `word-hyperbolic structure' has to be strengthened slightly in order to define a unique semigroup up to isomorphism. The isomorphism problem is proven to be undecidable for word-hyperbolic semigroups (in contrast to the situation for word-hyperbolic groups). It is proved that it is undecidable whether a word-hyperbolic semigroup is automatic, asynchronously automatic, biautomatic, or asynchronously biautomatic. (These…
We prove that, for a finitely generated group hyperbolic relative to virtually abelian subgroups, the generalised word problem for a parabolic subgroup is the language of a real-time Turing machine. Then, for a hyperbolic group, we show that the generalised word problem for a quasiconvex subgroup is a real-time language under either of two additional hypotheses on the subgroup. By extending the Muller-Schupp theorem we show that the generalised word problem for a finitely generated subgroup of a finitely generated virtually free group is context-free. Conversely, we prove that a hyperbolic gro…
We construct an example of a torsion free freely indecomposable finitely presented non-quasiconvex subgroup $H$ of a word hyperbolic group $G$ such that the limit set of $H$ is not the limit set of a quasiconvex subgroup of $G$. In particular, this gives a counterexample to the conjecture of G.Swarup that a finitely presented one-ended subgroup of a word hyperbolic group is quasiconvex if and only if it has finite index in its virtual normalizer.
We prove that the generalised word problem of a finitely generated subgroup of a finitely generated virtually free group is context-free, that a hyperbolic group must be virtually free if it has a torsion-free quasiconvex subgroup of infinite index with context-free generalised word problem, and that, for any hyperbolic group, the generalised word problem of a torsion-free quasiconvex subgroup is recognised by a real-time Turing machine.
This article studies the properties of word-hyperbolic semigroups and monoids, i.e. those having context-free multiplication tables with respect to a regular combing, as defined by Duncan & Gilman. In particular, the preservation of word-hyperbolicity under taking free products is considered. Under mild conditions on the semigroups involved, satisfied e.g. by monoids or regular semigroups, we prove that the semigroup free product of two word-hyperbolic semigroups is again word-hyperbolic. Analogously, with a mild condition on the uniqueness of representation for the identity element, satisfied…
The Diophantine problem for a monoid $M$ is the decision problem to decide whether any given system of equations has a solution in $M$. In this note, we give a simple example of a context-free, word-hyperbolic, finitely presented, special monoid $M$ with trivial group of units, and such that the Diophantine problem is undecidable in $M$. This answers two questions asked by Garreta & Gray in 2019, and shows that the decidability of the Diophantine problem in hyperbolic groups, as proved by Dahmani & Guirardel, does not generalise to word-hyperbolic monoids.
For each positive integer $k$ we present an example of Coxeter system $(G_k,S_k)$ such that $G_k$ is a word-hyperbolic Coxeter group, for any two generating reflections $s,t\in S_k$ the product $st$ has finite order, and the Coxeter graph of $S_k$ has negative inertia index equal to $k$. In particular, $G_k$ cannot be embedded into pseudo-orthogonal group $O(n,k-1)$ for any $n$ as a reflection group with generating set being reflections w.r.t. $(G_k,S_k)$
A result of Gersten states that if $G$ is a hyperbolic group with integral cohomological dimension $\mathsf{cd}_{\mathbb{Z}}(G)=2$ then every finitely presented subgroup is hyperbolic. We generalize this result for the rational case $\mathsf{cd}_{\mathbb{Q}}(G)=2$. In particular, our result applies to the class of torsion-free hyperbolic groups $G$ with $\mathsf{cd}_{\mathbb{Z}}(G)=3$ and $\mathsf{cd}_{\mathbb{Q}}(G)=2$ discovered by Bestvina and Mess.
We investigate which free constructions (amalgamated products and HNN-extensions) over word hyperbolic groups produce groups that are again word hyperbolic. A complete answer is obtained for the case when the amalgamated subgroups are virtually cyclic. The results are applied, in particular, to show that a ${\bf Q}$-completion of a torsion-free hyperbolic group has solvable word problem and conjugacy problem.
We are interested in the Guivarc'h inequality for admissible random walks on finitely generated relatively hyperbolic groups, endowed with a word metric. We show that for random walks with finite super-exponential moment, if this inequality is an equality, then the Green distance is roughly similar to the word distance, generalizing results of Blach{è}re, Ha{ï}ssinsky and Mathieu for hyperbolic groups [4]. Our main application is for relatively hyperbolic groups with respect to virtually abelian subgroups of rank at least 2. We show that for such groups, the Guivarc'h inequality with respect t…
We show that word-hyperbolic groups satisfy linear isoperimetric functions for all homotopy types of surface diagrams. This generalises the linear isoperimetric functions for disc and annular diagrams.
We study the word and conjugacy problems in lacunary hyperbolic groups (briefly, LHG). In particular, we describe a necessary and sufficient condition for decidability of the word problem in LHG. Then, based on the graded small-cancellation theory of Olshanskii, we develop a general framework which allows us to construct lacunary hyperbolic groups with word and conjugacy problems highly controllable and flexible both in terms of computability and computational complexity. As an application, we show that for any recursively enumerable subset $\mathcal{L} \subseteq \mathcal{A}^*$, where $\mathca…
We show that if $G$ is a non-elementary torsion-free word hyperbolic group then there exists another word hyperbolic group $G^*$, such that $G$ is a subgroup of $G^*$ but $G$ is not quasiconvex in $G^*$.
In this paper we obtain uniform positive lower bounds on stable commutator length in word-hyperbolic groups and certain groups acting on hyperbolic spaces (namely the mapping class group acting on the complex of curves, and an amalgamated free product acting on the Bass-Serre tree). If G is a word hyperbolic group which is delta hyperbolic with respect to a symmetric generating set S, then there is a positive constant C depending only on delta and on |S| such that every element of G either has a power which is conjugate to its inverse, or else the stable commutator length is at least equal to …
We show that superreflexivity can be characterized in terms of bilipschitz embeddability of word hyperbolic groups. We compare characterizations of superreflexivity in terms of diamond graphs and binary trees. We show that there exist sequences of series-parallel graphs of increasing topological complexity which admit uniformly bilipschitz embeddings into a Hilbert space, and thus do not characterize superreflexivity.
The notions of stable and Morse subgroups of finitely generated groups generalize the concept of a quasiconvex subgroup of a word-hyperbolic group. For a word-hyperbolic group $G$, Kapovich provided a partial algorithm which, on input a finite set $S$ of $G$, halts if $S$ generates a quasiconvex subgroup of $G$ and runs forever otherwise. In this paper, we give various detection and decidability algorithms for stability and Morseness of a finitely generated subgroup of mapping class groups, right-angled Artin groups, toral relatively hyperbolic groups, and finitely generated groups discriminat…
In this note we prove the claim given in the title. A group G is noncommutatively slender if each map from the fundamental group of the Hawaiian Earring to G factors through projection to a canonical free subgroup. Graham Higman, in his seminal 1952 paper, proved that free groups are noncommutatively slender. Such groups were first defined by K. Eda. Eda has asked which finitely presented groups are noncommutatively slender. This result demonstrates that random finitely presented groups in the few- relator sense of Gromov are noncommutatively slender.
We introduce a natural equivalence relation on the space $\sH_0$ of horofunctions of a word hyperbolic group that take the value 0 at the identity. We show that there are only finitely many ergodic measures that are invariant under this relation. This can be viewed as a discrete analog of the Bowen-Marcus theorem. Furthermore, if $η$ is such a measure and $G$ acts on a space $(X,μ)$ by p.m.p. transformations then $η\times μ$ is virtually ergodic with respect to a natural equivalence relation on $\sH_0\times X$. This is comparable to a special case of the Howe-Moore theorem. These results are a…
We discuss the linearity and discreteness of amalgamated products of linear word-hyperbolic groups. In particular, we prove that the double of an Anosov group along a maximal cyclic subgroup is always linear, and we construct examples of such groups which do not admit any discrete and faithful representation in rank 1. We also build new examples of non-linear word-hyperbolic groups, elaborating on a previous work of Canary--Stover--Tsouvalas.
We prove that the class of finitely presented inverse monoids whose Schützenberger graphs are quasi-isometric to trees has a uniformly solvable word problem, furthermore, the languages of their Schützenberger automata are context-free. On the other hand, we show that there is a finitely presented inverse monoid with hyperbolic Schützenberger graphs and an unsolvable word problem.
We consider non-elementary representations of two generator free groups in $PSL(2,\mathbb{C})$, not necessarily discrete or free, $G = < A, B >$. A word in $A$ and $B$, $W(A,B)$, is a palindrome if it reads the same forwards and backwards. A word in a free group is {\sl primitive} if it is part of a minimal generating set. Primitive elements of the free group on two generators can be identified with the positive rational numbers. We study the geometry of palindromes and the action of $G$ in $\HH^3$ whether or not $G$ is discrete. We show that there is a {\sl core geodesic} $Ł$ in the convex hu…
The computational complexity of the word problem in HNN-extension of groups is studied. HNN-extension is a fundamental construction in combinatorial group theory. It is shown that the word problem for an ascending HNN-extension of a group H is logspace reducible to the so-called compressed word problem for H. The main result of the paper states that the word problem for an HNN-extension of a hyperbolic group H with cyclic associated subgroups can be solved in polynomial time. This result can be easily extended to fundamental groups of graphs of groups with hyperbolic vertex groups and cyclic e…
We prove that any word hyperbolic group which is virtually compact special (in the sense of Haglund and Wise) is conjugacy separable. As a consequence we deduce that all word hyperbolic Coxeter groups and many classical small cancellation groups are conjugacy separable. To get the main result we establish a new criterion for showing that elements of prime order are conjugacy distinguished. This criterion is of independent interest; its proof is based on a combination of discrete and profinite (co)homology theories.
There are certain families of words and word sequences (words in the generators of a two-generator group) that arise frequently in the Teichm{ü}ller theory of hyperbolic three-manifolds and Kleinian and Fuchsian groups and in the discreteness problem for two generator matrix groups. We survey some of the families of such words and sequences: the semigroup of so called {\sl good} words of Gehring-Martin, the so called {\sl killer} words of Gabai-Meyerhoff-NThurston, the Farey words of Keen-Series and Minsky, the discreteness-algorithm Fibonacci sequences of Gilman-Jiang and {\sl parabolic dust}…
In this paper, we introduce a geometric statistic called the "sprawl" of a group with respect to a generating set, based on the average distance in the word metric between pairs of words of equal length. The sprawl quantifies a certain obstruction to hyperbolicity. Group presentations with maximum sprawl (i.e., without this obstruction) are called statistically hyperbolic. We first relate sprawl to curvature and show that nonelementary hyperbolic groups are statistically hyperbolic, then give some results for products, for Diestel-Leader graphs and lamplighter groups. In free abelian groups, t…
In this paper we study the generic, i.e., typical, behavior of finitely generated subgroups of hyperbolic groups and also the generic behavior of the word problem for amenable groups. We show that a random set of elements of a nonelementary word hyperbolic group is very likely to be a set of free generators for a nicely embedded free subgroup. We also exhibit some finitely presented amenable groups for which the restriction of the word problem is unsolvable on every sufficiently large subset of words.
In this paper we study hyperbolicty of the universal group $U(P)$ of a pregroup $P$. Given a finitely generated group $G$ and a pregroup $P$ such that $G \simeq U(P)$, we provide a particular set of axioms on $P$ which ensure that $G$ is word-hyperbolic.
We provide a new method of constructing non-quasiconvex subgroups of hyperbolic groups by utilizing techniques inspired by Stallings' foldings. The hyperbolic groups constructed are in the natural class of right-angled Coxeter groups (RACGs for short) and can be chosen to be 2-dimensional. More specifically, given a non-quasiconvex subgroup of a (possibly non-hyperbolic) RACG, our construction gives a corresponding non-quasiconvex subgroup of a hyperbolic RACG. We use this to construct explicit examples of non-quasiconvex subgroups of hyperbolic RACGs including subgroups whose generators are a…
We introduce a quantitative characterization of subgroup alternatives modeled on the Tits alternative in terms of group laws and investigate when this property is preserved under extensions. We develop a framework that lets us expand the classes of groups known to have locally uniform exponential growth to include extensions of either word hyperbolic or right-angled Artin groups by groups with locally uniform exponential growth. From this, we deduce that the automorphism group of a torsion-free one-ended hyperbolic group has locally uniform exponential growth. Our methods also demonstrate that…
For relatively hyperbolic groups, we investigate conditions guaranteeing that the subgroup generated by two relatively quasiconvex subgroups $Q_1$ and $Q_2$ is relatively quasiconvex and isomorphic to $Q_1 \ast_{Q_1 \cap Q_2} Q_2$. The main theorem extends results for quasiconvex subgroups of word-hyperbolic groups, and results for discrete subgroups of isometries of hyperbolic spaces.
We consider groups defined by cyclic presentations where the defining word has length three and the cyclic presentation satisfies the T(6) small cancellation condition. We classify when these groups are hyperbolic. When combined with known results, this completely classifies the hyperbolic T(6) cyclically presented groups.
Gromov asked whether every one-ended word-hyperbolic group contains a hyperbolic surface group. We prove that every one-ended double of a free group has a hyperbolic surface subgroup if (1) the free group has rank two, or (2) every generator is used the same number of times in the amalgamating words. To prove this, we formulate a stronger statement on Whitehead graphs and prove its specialization by combinatorial induction for (1) and the characterization of perfect matching polytopes by Edmonds for (2).
We study properties of generic elements of groups of isometries of hyperbolic spaces. Under general combinatorial conditions, we prove that loxodromic elements are generic (i.e. they have full density with respect to counting in balls for the word metric) and translation length grows linearly. We provide applications to a large class of relatively hyperbolic groups and graph products, including right-angled Artin groups and right-angled Coxeter groups.
Let G be a word-hyperbolic group with given finite generating set, for which various standard structures and constants have been pre-computed. A (non-practical) algorithm is described that, given as input two lists A and B, each composed of m words in the generators and their inverses, determines whether or not the lists are conjugate in G, and returns a conjugating element should one exist. The algorithm runs in time O(m mu)$, where mu is an upper bound on the lengths of elements in the two lists. Similarly, an algorithm is outlined that computes generators of the centraliser of A, with the s…
