A Characterization of Cocompact Hyperbolic and Finite-volume Hyperbolic Groups in Dimension Three
نویسندگان
چکیده
We show that a cocompact hyperbolic group in dimension 3 is characterized by certain properties of its word metric which depend only on the group structure and not on any action on hyperbolic space. We prove a similar theorem for finite-volume hyperbolic groups in dimension 3.
منابع مشابه
A Bowen type rigidity theorem for non-cocompact hyperbolic groups
We establish a Bowen type rigidity theorem for geometrically finite actions of the fundamental group of finite volume noncompact hyperbolic manifolds (with dimension at least 3). Mathematics Subject Classification (2000). 53C24, 30F40, 37F30.
متن کاملDeformation of finite-volume hyperbolic Coxeter polyhedra, limiting growth rates and Pisot numbers
A connection between real poles of the growth functions for Coxeter groups acting on hyperbolic space of dimensions three and greater and algebraic integers is investigated. In particular, a certain geometric convergence of fundamental domains for cocompact hyperbolic Coxeter groups with finite-volume limiting polyhedron provides a relation between Salem numbers and Pisot numbers. Several examp...
متن کاملInfinitely many hyperbolic Coxeter groups through dimension 19
We prove the following: there are infinitely many finite-covolume (resp. cocompact) Coxeter groups acting on hyperbolic space Hn for every n ≤ 19 (resp. n ≤ 6). When n = 7 or 8, they may be taken to be nonarithmetic. Furthermore, for 2 ≤ n ≤ 19, with the possible exceptions n = 16 and 17, the number of essentially distinct Coxeter groups in Hn with noncompact fundamental domain of volume ≤ V gr...
متن کاملHyperbolic Orbifolds of Minimal Volume
We provide a survey of hyperbolic orbifolds of minimal volume, starting with the results of Siegel in two dimensions and with the contributions of Gehring, Martin and others in three dimensions. For higher dimensions, we summarise some of the most important results, due to Belolipetsky, Emery and Hild, by discussing related features such as hyperbolic Coxeter groups, arithmeticity and consequen...
متن کاملOn Homological Coherence of Discrete Groups
We explore a weakening of the coherence property of discrete groups studied by F. Waldhausen. The new notion is defined in terms of the coarse geometry of groups and should be as useful for computing their K-theory. We prove that a group Γ of finite asymptotic dimension is weakly coherent. In particular, there is a large collection of R[Γ]-modules of finite homological dimension when R is a fin...
متن کامل