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

Unicorn paths and hyperfiniteness for the mapping class group