Regular combings, nonpositive curvature and the quasiconvexity of Abelian subgroups
type: Item
instance of: scholarly article
title: Regular combings, nonpositive curvature and the quasiconvexity of Abelian subgroups
zbMATH Open document ID: 0802.20033
DOI: 10.1016/0022-4049(93)90009-I
author: Martin R. Bridson
published in: Journal of Pure and Applied Algebra
publication date: 1993-11-03T00:00:00Z
review text: The author studies existence and properties of abelian subgroups in (bi- )automatic groups and in groups which act properly and cocompactly on simply-connected spaces of nonpositive curvature. The main technical result is the following: let \(\mathcal A\) be a finite set and \(\mu: {\mathcal A}^* \to \Gamma\) a monoid epimorphism to a group \(\Gamma\). Suppose that \(H \subset \Gamma\) is an abelian subgroup for which there exists a regular language \({\mathcal L} \subset {\mathcal A}^*\) such that \(\mu({\mathcal L}) = H\). Then, there exist only finitely many commensurability classes of subgroups \(K'\) in \(H\) such that each class is represented by a subgroup \(K \in K'\), \(\mu^{-1}(K) \cap {\mathcal L}\) is a regular language. As an application, the author gives a criterion (in terms of regular languages) for a finitely generated group to contain a subgroup isomorphic to \(Z \times Z\). Another result is the following: Let \(X\) be a simply-connected geodesic metric space of nonpositive curvature and \(\Gamma\) a finitely generated group which acts properly and cocompactly by isometries on \(X\). Then the author gives a sufficient condition for every abelian subgroup of \(\Gamma\) to be virtually cyclic in terms of the combing associated to the action.
reviewed by: Athanase Papadopoulos
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20M35
zbMATH DE Number: 427654
zbMATH Keywords: abelian subgroups
zbMATH Keywords: (bi-)automatic groups
zbMATH Keywords: simply-connected spaces
zbMATH Keywords: monoid epimorphism
zbMATH Keywords: regular language
zbMATH Keywords: commensurability classes of subgroups
zbMATH Keywords: finitely generated group
zbMATH Keywords: simply-connected geodesic metric space
zbMATH Keywords: nonpositive curvature
zbMATH Keywords: combing
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 427654