Unicorn paths and hyperfiniteness for the mapping class group

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 Hamenstädt and Kida that this action is universally amenable and that the mapping class group of S is exact.

Abstract licence: CC BY-ND

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