Geometric finiteness and rationality
type: Item
instance of: scholarly article
title: Geometric finiteness and rationality
zbMATH Open document ID: 0806.57003
DOI: 10.1016/0022-4049(93)90107-5
author: S. Singh
published in: Journal of Pure and Applied Algebra
publication date: 1995-02-16T00:00:00Z
review text: The paper under review answers positively one of the main questions raised in a paper of \textit{S. M. Gersten} and \textit{H. B. Short} [Ann. Math., II. Ser. 134, 125-158 (1991; Zbl 0744.20035)]. To state the result, let us recall briefly (and informally) a few definitions. (The paper is concise, and for background and definitions, the reader is referred to the paper of Gersten and Short.) Let \(A\) be a finite set and \(A^*\) the free monoid generated by \(A\). A subset of \(A^*\) is said to be a regular language if it is the set of words recognized by a finite state automaton. An automatic structure for a group \(G\) is given by a pair \((A,L)\) where \(A\) is a finite set which is a semi-group generator of \(G\), and \(L\) a regular language in \(A^*\), with the property that the question whether two elements of \(L\) representing elements of \(G\) at distance \(\leq 1\) in the Cayley graph can be decided by a finite state automaton. The notion of an automatic group is due to Thurston et al. A subset \(S\subset G\) is said to be \(L\)- rational if the complete inverse image of \(S\) in \(L\) is a regular subset. The paper considers geometrically finite groups, that is, discrete subgroups of isometries of the hyperbolic space \(H^ n\) such that the quotient by \(G\) of the Nielsen convex hull of the action is compact (for this, and other equivalent definitions, see Thurston's Princeton lecture notes). We can state now the main theorem of this paper: Let \(G\) be a geometrically finite group without parabolics, and let \((A,L)\) be any rational structure on \(G\) which is automatic. Then, a subgroup \(H\) of \(G\) is \(L\)-rational if and only if \(H\) is geometrically finite. In his proof of this theorem, the author establishes also the following result, which concerns negatively curved groups (i.e. hyperbolic groups in the sense of Gromov): Let \(G\) be a negatively curved group. A subgroup \(H\) of \(G\) is \(L\)-rational with respect to any automatic structure…
reviewed by: Athanase Papadopoulos
Mathematics Subject Classification ID: 57M60
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20H10
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 20M35
zbMATH DE Number: 179081
zbMATH Keywords: rational subgroup
zbMATH Keywords: regular language
zbMATH Keywords: finite state automaton
zbMATH Keywords: automatic structure
zbMATH Keywords: Cayley graph
zbMATH Keywords: automatic group
zbMATH Keywords: geometrically finite groups
zbMATH Keywords: discrete subgroups of isometries of the hyperbolic space
zbMATH Keywords: negatively curved groups
zbMATH Keywords: hyperbolic groups
MaRDI profile type: Publication
full work available at URL: 0022-4049(93)90107-5
OpenAlex ID: W2029354991
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description: scientific article; zbMATH DE number 179081