A nitely presented group G is called metabolic if H 2 (1) (G; A) = 0 for all normed abelian coeecient groups A; it is equivalent to require that a Cayley graph for G admit a thin combing. Metabolicity is stronger than hyperbolicity, and it is an open problem whether all hyperbolic groups are metabolic. We prove that if H is metabolic, then H ha;b]i F(a; b) is also metabolic, where F(a;b) is fre...

We give examples of direct products of three hyperbolic groups in which there cannot exist an algorithm to decide which finitely presented subgroups are isomorphic. In [1] Baumslag, Short and the present authors constructed an example of a biautomatic group G in which there is no algorithm that decides isomorphism among the finitely presented subgroups of G. The group G was a direct product of ...

We introduce an obstruction to the existence of a coarse embedding of a given group or space into a hyperbolic group, or more generally into a hyperbolic graph of bounded degree. The condition we consider is “admitting exponentially many fat bigons”, and it is preserved by a coarse embedding between graphs with bounded degree. Groups with exponential growth and linear divergence (such as direct...

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 (ie without this obstruction) are called statistically hyperbolic. We first relate sprawl to ...

The Coxeter simplex with symbol is a compact hyperbolic 4-simplex and the related Coxeter group Γ is a discrete subgroup of Isom(H 4). The Coxeter simplex with symbol is a spherical 3-simplex, and the related Coxeter group G is the group of symmetries of the regular 120-cell. Using the geometry of the regular 120-cell, Davis [3] constructed an epimorphism Γ → G whose kernel K was torsion-free, ...

The Cannon Conjecture from the geometric group theory asserts that a word hyperbolic group that acts effectively on its boundary, and whose boundary is homeomorphic to the 2-sphere, is isomorphic to a Kleinian group. We prove the following Criterion for Cannon’s Conjecture: A hyperbolic group G (that acts effectively on its boundary) whose boundary is homeomorphic to the 2-sphere is isomorphic ...

We study conjugacy classes of solutions to systems of equations and inequations over torsion-free hyperbolic groups, and describe an algorithm to recognize whether or not there are finitely many conjugacy classes of solutions to such a system. The class of immutable subgroups of hyperbolic groups is introduced, which is fundamental to the study of equations in this context. We apply our results...

We study the ergodic and rigidity properties of weakly hyperbolic actions. First, we establish ergodicity for C2 volume preserving weakly hyperbolic group actions on closed manifolds. For the integral action generated by a single Anosov diffeomorphism this theorem is classical and originally due to Anosov. Motivated by the Franks/Manning classification of Anosov diffeomorphisms on tori, we rest...

We give a criterion in terms of the boundary for the existence of a proper cocompact action of a word-hyperbolic group on a CAT(0) cube complex. We describe applications towards lattices and hyperbolic 3-manifold groups. In particular, by combining the theory of special cube complexes, the surface subgroup result of Kahn-Markovic, and Agol’s criterion, we find that every subgroup separable clos...

We obtain a criterion for quasiconvexity of a subgroup of an amalgamated free product of two word hyperbolic groups along a virtually cyclic subgroup. The result provides a method of constructing new word hyperbolic group in class (Q), that is such that all their finitely generated subgroups are quasiconvex. It is known that free groups, hyperbolic surface groups and most 3-dimensional Kleinian...