The combinatorial structure of cocompact discrete hyperbolic groups
type: Item
instance of: scholarly article
title: The combinatorial structure of cocompact discrete hyperbolic groups
zbMATH Open document ID: 0606.57003
DOI: 10.1007/BF00146825
author: James W. Cannon
published in: Geometriae Dedicata
publication date: 1984-01-01T00:00:00Z
review text: Combinatorial group theory began with Dehn's study [\textit{M. Dehn}, Math. Ann. 69, 137-168 (1910); ibid. 71, 116-144 (1912); ibid. 72, 413-420 (1912)] of the fundamental group of the closed 2-dimensional manifold admitting a hyperbolic structure. At first Dehn freely used arguments from hyperbolic geometry, but rapidly he and others moved in the direction of results which could be stated and proved in purely combinatorial and algebraic terms. \textit{W. P. Thurston} [see ''Geometry and topology of 3-manifolds'', Princeton Univ., Notes (1978); ''On the geometry and dynamics of diffeomorphisms of surfaces. I'', preprint] has recently shown that large classes of groups of interest to topologists, while not obviously amenable to attack by standard methods of combinatorial group theory, nevertheless are discrete hyperbolic groups. His result suggests the value of a return to geometric considerations in combinatorial group theory. We show that Dehn's principal combinatorial theorems - in particular his solutions to the word and conjugacy problems for hyperbolic surface groups - have simple geometric reinterpretations, and that these solutions, as reinterpreted, are true for all cocompact, discrete hyperbolic groups. We also show that the global combinatorial structure of such groups is particularly simple in the sense that the Cayley group graphs have descriptions by linear recursion.
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20F10
Mathematics Subject Classification ID: 57N65
Mathematics Subject Classification ID: 57R19
Mathematics Subject Classification ID: 20F34
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 22E40
zbMATH DE Number: 3980027
zbMATH Keywords: word problem
zbMATH Keywords: Combinatorial group theory
zbMATH Keywords: conjugacy problems
zbMATH Keywords: cocompact, discrete hyperbolic groups
zbMATH Keywords: Cayley group graphs
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00146825
OpenAlex ID: W2028978432
description: scientific article; zbMATH DE number 3980027