The genus of curve, pants and flip graphs

TL;DR

This paper investigates the genus of the curve, pants, and flip graphs in surface theory, revealing their infinite genus in most cases and providing growth rates for their quotients, with results derived using probabilistic methods.

Contribution

It identifies the genus of quotient graphs and their growth rates, extending understanding of these graphs in surface topology.

Findings

Most graphs have infinite genus.

The genus of quotient graphs matches that of complete graphs.

Growth rates are precisely characterized for the other two graphs.

Abstract

This article is about the graph genus of certain well studied graphs in surface theory: the curve, pants and flip graphs. We study both the genus of these graphs and the genus of their quotients by the mapping class group. The full graphs, except for in some low complexity cases, all have infinite genus. The curve graph once quotiented by the mapping class group has the genus of a complete graph so its genus is well known by a theorem of Ringel and Youngs. For the other two graphs we are able to identify the precise growth rate of the graph genus in terms of the genus of the underlying surface. The lower bounds are shown using probabilistic methods.