×

zbMATH — the first resource for mathematics

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)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] P. Buser and H. Karcher, Gromov’s almost flat manifolds, Astérisque 81 (1981). · Zbl 0459.53031
[2] James W. Cannon, The combinatorial structure of cocompact discrete hyperbolic groups, Geom. Dedicata 16 (1984), no. 2, 123 – 148. · Zbl 0606.57003
[3] Tim Bedford, Michael Keane, and Caroline Series , Ergodic theory, symbolic dynamics, and hyperbolic spaces, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1991. Papers from the Workshop on Hyperbolic Geometry and Ergodic Theory held in Trieste, April 17 – 28, 1989. · Zbl 0743.00040
[4] William J. Floyd, Group completions and limit sets of Kleinian groups, Invent. Math. 57 (1980), no. 3, 205 – 218. · Zbl 0428.20022
[5] M. Gromov, Hyperbolic manifolds, groups and actions, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 183 – 213.
[6] Mikhael Gromov, Infinite groups as geometric objects, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) PWN, Warsaw, 1984, pp. 385 – 392. · Zbl 0599.20041
[7] Mikhael Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53 – 73. · Zbl 0474.20018
[8] Bernard Maskit, Kleinian groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287, Springer-Verlag, Berlin, 1988. · Zbl 0627.30039
[9] G. D. Mostow, Strong rigidity of locally symmetric spaces, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1973. Annals of Mathematics Studies, No. 78. · Zbl 0265.53039
[10] Dennis Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 465 – 496.
[11] W. P. Thurston, The geometry and topology of \( 3\)-manifolds, Lecture Notes, Princeton Univ., 1978.
[12] P. Tukia and J. Väisälä, Quasiconformal extension from dimension \? to \?+1, Ann. of Math. (2) 115 (1982), no. 2, 331 – 348. · Zbl 0484.30017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.