Mapping class factorization via fatgraph Nielsen reduction

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 $MC_{g,1}$.