The isomorphism problem for hyperbolic groups. I
type: Item
instance of: scholarly article
title: The isomorphism problem for hyperbolic groups. I
zbMATH Open document ID: 0868.57005
DOI: 10.2307/2118520
published in: Annals of Mathematics. Second Series
publication date: 1997-08-18T00:00:00Z
review text: In 1911 Dehn formulated three fundamental problems for groups which are the word, the conjugacy and the isomorphism problem (and solved them for the canonical presentations of the fundamental groups of closed surfaces). All three problems are unsolvable, in general, so one has to restrict to reasonable subclasses of groups; however, in contrast to the word problem, the isomorphism problem could be solved only for some rather isolated classes of groups so far. In the present paper, the isomorphism problem is solved for the class of torsion-free hyperbolic groups (as introduced and popularized by Gromov as a reasonable large but accessible class of groups), under the additional hypothesis that the hyperbolic group does not admit an essential small action on a real tree. This hypothesis is equivalent to the fact that the outer automorphism group of the hyperbolic group is finite. (A second part of the paper is announced giving a solution of the isomorphism problem for general torsion-free hyperbolic groups). The presented algorithm gives also an effective computation of the (finite) outer automorphism group of such groups, and permits to decide if the outer automorphism group of a general torsion-free hyperbolic group is finite. In combination with some other results, the algorithm implies solutions of the homeomorphism problem for closed hyperbolic manifolds, for closed negatively curved manifolds of dimension \(\geq 5\) and for ``rigid weakly geometric'' 3-manifolds, as well as solutions of the conjugacy problem for pseudo-Anosov automorphisms of surfaces and for irreducible automorphisms of free groups. The main tools of the algorithm include Makanin's theorem on the decidability of the Diophantine theory of free semigroups with paired alphabet, canonical representatives which are special kinds of quasi-geodesics introduced and used in a paper by Rips and the author to solve equations over hyperbolic groups, and some canonical forms of monomorphic images of balls in…
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 20F10
zbMATH DE Number: 750873
zbMATH Keywords: word problem
zbMATH Keywords: conjugacy problem
zbMATH Keywords: isomorphism problem
zbMATH Keywords: fundamental groups
zbMATH Keywords: hyperbolic groups
zbMATH Keywords: outer automorphism
zbMATH Keywords: algorithm
zbMATH Keywords: homeomorphism problem
zbMATH Keywords: hyperbolic manifolds
Wikidata QID: Q61478899
reviewed by: Bruno Zimmermann
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2118520
OpenAlex ID: W1995131860
author: Zlil Sela
description: scientific article; zbMATH DE number 750873