Mapping Class Factorization via Fatgraph Nielsen Reduction

TL;DR

This paper introduces an algorithm called fatgraph Nielsen reduction that determines a sequence of Whitehead moves representing a mapping class from its action on the fundamental group of a genus g surface, enabling explicit factorization.

Contribution

It presents a novel algorithm for factorizing mapping classes via fatgraph Nielsen reduction based on their action on the fundamental group.

Findings

Algorithm effectively finds Whitehead move sequences for mapping classes.

Enables explicit factorization of mapping classes from their fundamental group action.

Provides a practical method for understanding the structure of the mapping class group.

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 $\varphi\in MC_{g,1}$ 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 $MC_{g,1}$.