Unicorn paths and hyperfiniteness for the mapping class group

TL;DR

This paper demonstrates the hyperfiniteness of orbit equivalence relations from the mapping class group's actions on the Gromov boundaries of the arc and curve graphs of a surface, using infinite unicorn paths.

Contribution

It introduces a new approach utilizing Pho-On's infinite unicorn paths to establish hyperfiniteness for these group actions, strengthening previous results on amenability and exactness.

Findings

Proves hyperfiniteness of orbit equivalence relations for the mapping class group actions.

Strengthens previous results on the group's amenability and exactness.

Utilizes infinite unicorn paths to analyze the Gromov boundaries.

Abstract

Let S be an orientable surface of finite type. Using Pho-On's infinite unicorn paths, we prove the hyperfiniteness of orbit equivalence relations induced by the actions of the mapping class group of S on the Gromov boundaries of the arc graph and the curve graph of S. In the curve graph case, this strengthens the results of Hamenst\"adt and Kida that this action is universally amenable and that the mapping class group of S is exact.