A Polygonal Perspective of Nielsen Reduction and the Chord Slide Groupoid

Details A Polygonal Perspective of Nielsen Reduction and the Chord Slide Groupoid A Polygonal Perspective of Nielsen Reduction and the Chord Slide Groupoid

2010-10-25

arxiv.org/abs/1010.5275v2

Mathematics

Geometric Topology

This survey (minus minor corrections) was written for and published in the abstracts for the 57th Topology Symposium which too

Scientific paper

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.

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