Cubulating hyperbolic free-by-cyclic groups: the general case

Let $\Phi:F\rightarrow F$ be an automorphism of the finite-rank free group $F$. Suppose that $G=F\rtimes_\Phi\mathbb Z$ is word-hyperbolic. Then $G$ acts freely and cocompactly on a CAT(0) cube complex.Comment: 36 pages, 11 figures. Version 2 contains minor corrections. Accepted to GAF

Automatic groups and amalgams

AbstractThe objectives of this paper are twofold. The first is to provide a self-contained introduction to the theory of automatic and asynchronously automatic groups, which were invented a few years ago by J.W. Cannon, D.B.A. Epstein, D.F. Holt, M.S. Paterson and W.P. Thurston. The second objective is to prove a number of new results about the construction of new automatic and asynchronously automatic groups from old ones by means of amalgamated products

CAT(0) spaces with polynomial divergence of geodesics

We construct a family of finite 2-complexes whose universal covers are CAT(0) and have polynomial divergence of desired degree. This answers a question of Gersten, namely whether such CAT(0) complexes exist

Automatic structures, rational growth and geometrically finite hyperbolic groups

We show that the set $SA(G)$ of equivalence classes of synchronously automatic structures on a geometrically finite hyperbolic group $G$ is dense in the product of the sets $SA(P)$ over all maximal parabolic subgroups $P$. The set $BSA(G)$ of equivalence classes of biautomatic structures on $G$ is isomorphic to the product of the sets $BSA(P)$ over the cusps (conjugacy classes of maximal parabolic subgroups) of $G$. Each maximal parabolic $P$ is a virtually abelian group, so $SA(P)$ and $BSA(P)$ were computed in ``Equivalent automatic structures and their boundaries'' by M.Shapiro and W.Neumann, Intern. J. of Alg. Comp. 2 (1992) We show that any geometrically finite hyperbolic group has a generating set for which the full language of geodesics for $G$ is regular. Moreover, the growth function of $G$ with respect to this generating set is rational. We also determine which automatic structures on such a group are equivalent to geodesic ones. Not all are, though all biautomatic structures are.Comment: Plain Tex, 26 pages, no figure

Subgroups Of Word Hyperbolic Groups In Dimension 2

. If G is a word hyperbolic group of cohomological dimension 2, then every subgroup of G of type FP2 is also word hyperbolic. Isoperimetric inequalities are defined for groups of type FP2 and it is shown that the linear isoperimetric inequality in this generalized context is equivalent to word hyperbolicity. A sufficient condition for hyperbolicity of a general graph is given along with an application to `relative hyperbolicity'. Finitely presented subgroups of Lyndon's small cancellation groups of hyperbolic type are word hyperbolic. Finitely presented subgroups of hyperbolic 1-relator groups are hyperbolic. Finitely presented subgroups of free Burnside groups are finite in the stable range. 1. Introduction. In this article we shall prove the following results. Theorem 5.4. If G is a word hyperbolic group of cohomological dimension 2 and H is a finitely presented subgroup (or more generally a subgroup of type FP 2 ), then H is word hyperbolic. Theorem 7.9. A finitely presented su..

A Cohomological Characterization Of Hyperbolic Groups

A finitely presented group G is word hyperbolic i# H 2 (#) (G, ## ) = 0

Homological Dehn Functions And The Word Problem

. Homological Dehn functions over R and over Z are introduced to measure minimal fillings of integral 1-cycles by (real or integral) 2-chains in the Cayley 2complex of a finitely presented group. If the group G is the fundamental group of a finite graph of finitely presented vertex-groups Hv and finitely generated edge-groups, then there is a formula for an isoperimetric function (for genus 0 fillings) for G in terms of the real homological Dehn function for G and the (genus 0) Dehn functions for the Hv . Hierarchies of groups are introduced in which an isoperimetic function is determined by a formula in terms of the real homological Dehn function. In such a hierarchy the word problem is one of homological algebra. All 1-relator groups are in such a hierarchy. Applications are given to the generalized word problem (a.k.a. membership or Magnus problem) and to a homolgical determination of distortion. 1. Introduction. It is well-known in Riemannian geometry that under appropr..