Mapping Class Factorization via Fatgraph Nielsen Reduction

Details Mapping Class Factorization via Fatgraph Nielsen Reduction Mapping Class Factorization via Fatgraph Nielsen Reduction

2009-04-27

arxiv.org/abs/0904.4067v1

Mathematics

Geometric Topology

16 pages, 4 figures

Scientific paper

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 moves called Whitehead moves on so-called 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. 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 a certain generating set for $MC_{g,1}$.

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