The geometry of flip graphs and mapping class groups

TL;DR

This paper explores the geometric structure of flip graphs related to surface triangulations, revealing convexity properties and analyzing the growth of associated moduli spaces, thereby providing insights into the mapping class group's geometry.

Contribution

It demonstrates strong convexity of certain strata in flip graphs and characterizes the diameter growth of the quotient moduli spaces in relation to surface complexity.

Findings

Strata with the same multiarc are strongly convex in flip graphs.

The diameter of the quotient moduli space grows with surface complexity.

Insights into the geometric properties of the mapping class group.

Abstract

The space of topological decompositions into triangulations of a surface has a natural graph structure where two triangulations share an edge if they are related by a so-called flip. This space is a sort of combinatorial Teichm\"uller space and is quasi-isometric to the underlying mapping class group. We study this space in two main directions. We first show that strata corresponding to triangulations containing a same multiarc are strongly convex within the whole space and use this result to deduce properties about the mapping class group. We then focus on the quotient of this space by the mapping class group to obtain a type of combinatorial moduli space. In particular, we are able to identity how the diameters of the resulting spaces grow in terms of the complexity of the underlying surfaces.