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A characterization of cocompact hyperbolic and finite-volume hyperbolic groups in dimension three. (English) Zbl 0761.57008
The main result of the paper states that a group $$G$$ is a cocompact group of isometries of the hyperbolic 3-space $$H^ 3$$ iff it has a finite generating set $$C$$ and associated Cayley graph $$\Gamma=\Gamma(G,C)$$ such that the word metric on $$\Gamma$$ is quasi-isometric with the standard hyperbolic metric on $$H^ 3$$. The authors prove a similar result for groups of finite volume where they use the “augmented Cayley graph” instead of the Cayley graph of a group. These results are similar to a result of M.Gromov [Riemann surfaces and related topics, Proc. 1978 Stony Brook Conf., Ann. Math. Stud. 97, 183-213 (1981; Zbl 0467.53035)], but without the added assumption that the group be the fundamental group of a convex path space.

##### MSC:
 57M50 General geometric structures on low-dimensional manifolds 57M05 Fundamental group, presentations, free differential calculus 30C62 Quasiconformal mappings in the complex plane 57M15 Relations of low-dimensional topology with graph theory 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
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##### References:
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