Algebraic linearity for an automorphism of a surface group
type: Item
instance of: scholarly article
title: Algebraic linearity for an automorphism of a surface group
zbMATH Open document ID: 0646.57007
DOI: 10.1016/0022-4049(88)90094-1
author: Joan Birman
author: Caroline Series
published in: Journal of Pure and Applied Algebra
publication date: 1988-01-01T00:00:00Z
review text: Let W be a finite set of (cyclic) words in a free group. When studying the question whether W can be extended to a basis of the free group, Whitehead introduced a labelled graph G(W) now known as the coinitial graph associated to W. Now let M be a compact surface with boundary (so it has free fundamental group) and S(M) be the set of isotopy classes of families of pairwise disjoint simple loops on M. Then each \(\Lambda\in S(M)\) is represented by a set of cyclic words \(W_{\Lambda}\) in the fundamental group. ``Our first result is to show that the set of coinitial graphs of the \(W_{\Lambda}\), as \(\Lambda\) ranges over S(M), may be reinterpreted as a finite set of canonical `train tracks' (in the sense of Thurston) where the labels of the graph correspond to weights on these tracks. The second and main result of this paper is a linearity theorem which asserts that if an automorphism of the fundamental group of the surface is induced by a diffeomorphism and maps a set of weights supported to one track to a set of weights supported on the same or any other track, then, with some appropriate algebraic restrictions, the action is linear on the positive linear span of the weights (this is related to Thurston's theorem about the piecewise integral linearity of the action of pseudo-Anosov maps on S(M)).'' Then similar results are discussed for the case of closed surfaces, with some stronger restrictions on the automorphism.
Mathematics Subject Classification ID: 57N05
Mathematics Subject Classification ID: 20F34
Mathematics Subject Classification ID: 57M05
zbMATH DE Number: 4054613
zbMATH Keywords: action on the positive linear span of weights
zbMATH Keywords: automorphism of the fundamental group of a surface
zbMATH Keywords: compact surface with boundary
zbMATH Keywords: isotopy classes of families of pairwise disjoint simple loops
zbMATH Keywords: coinitial graphs
zbMATH Keywords: train tracks
reviewed by: Bruno Zimmermann
MaRDI profile type: Publication
full work available at URL: 0022-4049(88)90094-1
OpenAlex ID: W2150948712
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description: scientific article; zbMATH DE number 4054613