Automatic groups: A guided tour
type: Item
instance of: scholarly article
title: Automatic groups: A guided tour
zbMATH Open document ID: 0811.20038
author: Benson Farb
published in: L'Enseignement Mathématique. 2e Série
publication date: 1993-06-29T00:00:00Z
review text: This paper is a useful survey on the theory of automatic groups, which has been developed by Thurston et al. It contains self-contained definitions and a few historical remarks. The author explains how that theory grew out of the desire to understand the structure of a finitely generated group by studying the geometry of its Cayley graph. The paper contains also a list of examples and non-examples, with sometimes ideas of proofs. In particular, there is a detailed proof of a result due to Cannon, which asserts that the fundamental group of a compact manifold with strictly negative curvature is automatic. Finally, the paper contains a list of 12 interesting open problems. The reader should be aware of the fact that problem number 1, conjecturing that mapping class groups of surfaces of negative Euler characteristic are automatic (a conjecture due to Thurston) has been solved a couple of years ago by Lee Mosher.
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 20F10
Mathematics Subject Classification ID: 20F34
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20E06
zbMATH DE Number: 221265
zbMATH Keywords: hyperbolic groups
zbMATH Keywords: braid groups
zbMATH Keywords: automatic groups
zbMATH Keywords: finitely generated group
zbMATH Keywords: Cayley graph
zbMATH Keywords: fundamental group
zbMATH Keywords: compact manifold
zbMATH Keywords: mapping class groups
reviewed by: Athanase Papadopoulos
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 221265