A Polygonal Perspective of Nielsen Reduction and the Chord Slide Groupoid

TL;DR

This paper explores a geometric approach to Nielsen reduction for surface automorphisms, using polygonal decompositions and chord slide groupoids to provide a topological interpretation.

Contribution

It introduces fatgraph Nielsen reduction and describes the chord slide groupoid with generators and relations, linking algebraic and topological methods.

Findings

Fatgraph Nielsen reduction decomposes surface automorphisms into elementary moves.

Chord slide groupoid is generated by specific elementary rearrangements.

Provides a presentation of the groupoid with explicit generators and relations.

Abstract

Nielsen reduction is an algorithm which decomposes any automorphism of a free group into a product of elementary Nielsen transformations. While this may be applied to a mapping class of a surface $S_{g,1}$ with one boundary component, the resulting decomposition in general will not have a topological interpretation. In this survey, we discuss a variation called fatgraph Nielsen reduction which decomposes such a mapping class into elementary Nielsen transformations interpreted as rearrangements of polygon domains for $S_{g,1}$ described by systems of arcs in $S_{g,1}$. These elementary moves generate the chord slide groupoid of $S_{g,1}$, which we survey and describe in terms of generators and relations.