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December 2011 Mapping class factorization via fatgraph Nielsen reduction
Alex James Bene
Osaka J. Math. 48(4): 1047-1061 (December 2011).

Abstract

The mapping class group of a genus $g$ surface $\Sigma_{g,1}$ with one boundary component is known to have a simple yet infinite presentation with generators given by elementary Whitehead moves on marked bordered fatgraphs. In this paper, we introduce an algorithm called fatgraph Nielsen reduction which, from the action of a mapping class $\varphi \in MC_{g,1}$ of $\Sigma_{g,1}$ on the fundamental group $\pi_{1}(\Sigma_{g,1})$ of $\Sigma_{g,1}$, determines a sequence of Whitehead moves representing $\varphi$ beginning at any choice of marked bordered fatgraph. The algorithm utilizes a reduction of bordered fatgraphs to linear chord diagrams, where the desired sequence is given in terms of elementary chord slide moves which continuously decrease some energy function. As a consequence, this leads to an algorithm which factors any mapping class given by its action on $\pi(\Sigma_{g,1})$ in terms of any convenient generating set for $MC_{g,1}$.

Citation

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Alex James Bene. "Mapping class factorization via fatgraph Nielsen reduction." Osaka J. Math. 48 (4) 1047 - 1061, December 2011.

Information

Published: December 2011
First available in Project Euclid: 11 January 2012

zbMATH: 1242.57012
MathSciNet: MR2871293

Subjects:
Primary: 05C25, 20F38
Secondary: 20F05, 20F06, 20F34, 20F99, 32G15, 57M99

Rights: Copyright © 2011 Osaka University and Osaka City University, Departments of Mathematics

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Vol.48 • No. 4 • December 2011
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