A combinatorial algorithm to compute presentations of mapping-class groups of orientable surfaces with one boundary component

TL;DR

This paper introduces a combinatorial algorithm to compute presentations of the mapping-class group of orientable surfaces with one boundary component, by relating it to a subgroup of automorphisms of a free group and defining a variation of Auter space.

Contribution

It provides a novel combinatorial algorithm for deriving presentations of these mapping-class groups, connecting geometric topology with algebraic automorphism groups.

Findings

Algorithm computes presentations for the subgroup M_{g,1,p}

Establishes isomorphism with the mapping-class group of a surface

Defines a variation of Auter space for this purpose

Abstract

We give an algorithm which computes a presentation for a subgroup, denoted $\AM_{g,1,p}$, of the automorphism group of a free group. It is known that $\AM_{g,1,p}$ is isomorphic to the mapping-class group of an orientable genus-$g$ surface with one boundary component and $p$ punctures. We define a variation of Auter space.