The word problem for geometrically finite groups
type: Item
instance of: scholarly article
title: The word problem for geometrically finite groups
zbMATH Open document ID: 0595.20032
DOI: 10.1007/BF00164400
published in: Geometriae Dedicata
publication date: 1986-01-01T00:00:00Z
review text: Let \(<X:R>\) be a presentation of a group G, with \(G=F/N\), where F is a free group with basis X and N is the normal closure of R in F. For \(n\in {\mathbb{N}}\) let R(n) be the set of all words in X which are conjugate to some \(r\in R\) by some reduced word of length at most n. Let F(n) be the set of initial segments of words in R(n). On F(n) there is an equivalence relation \(\sim_ n\) where \(w_ 1w_ 2\sim_ nw_ 1w_ 0w_ 2\) whenever both are in F(n) and \(w_ 0\) is in R(n). Let f be any function \(f: {\mathbb{N}}\to {\mathbb{N}}\). The presentation \(<X:R>\) is said to be f-decidable if given any element in F represented by a word w of length m then \(w\in N\) iff \(w\sim_ n1\) for \(n=f(m)\). The Todd- Coxeter algorithm shows that any presentation is f-decidable for some f. But in general f needs not to be recursive. The word problem for G is decidable iff G possesses a presentation such that this presentation is f-decidable for some recursive f. In the paper under review the authors prove the following: (i) Let G be a cocompact group of isometries of hyperbolic or euclidean space of dimension \(d\geq 2\) and \(<X:R>\) be the presentation obtained from a Dirichlet region for G, then there is a natural number s such that \(<X:R>\) is f-decidable for \(f(m)=sm^ 2\). (ii) Let G be a discrete group of isometries of hyperbolic d-space such that either \(d\geq 2\) and H/G has finite volume or \(d=3\) and G is geometrically finite. Let \(<X:R>\) be a presentation as before, then \(<X:R>\) is f-decidable for \(f(m)=[e^{sm}]\) and suitable s.
Mathematics Subject Classification ID: 20F10
Mathematics Subject Classification ID: 51M05
Mathematics Subject Classification ID: 51M10
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20F65
zbMATH DE Number: 3957424
zbMATH Keywords: presentation
zbMATH Keywords: words
zbMATH Keywords: f-decidable
zbMATH Keywords: Todd-Coxeter algorithm
zbMATH Keywords: word problem
zbMATH Keywords: group of isometries
zbMATH Keywords: hyperbolic or euclidean space
zbMATH Keywords: Dirichlet region
zbMATH Keywords: hyperbolic d-space
zbMATH Keywords: geometrically finite
author: William Floyd
author: A. Howard M. Hoare
author: Roger C. Lyndon
reviewed by: Gernot Stroth
MaRDI profile type: Publication
cites work: Q3310101
cites work: The combinatorial structure of cocompact discrete hyperbolic groups
cites work: Group completions and limit sets of Kleinian groups
cites work: Q3877854
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description: scientific article; zbMATH DE number 3957424