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 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 $\phi \in M{C}_{g,1}$ of ${\Sigma }_{g,1}$ on the fundamental group ${\pi }_1({\Sigma }_{g,1})$ of ${\Sigma }_{g,1}$, determines a sequence of Whitehead moves representing $\phi$ 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 $M{C}_{g,1}$.

16 pages, 4 figures