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Mapping class group

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Infinite group

The Michigan Mathematical Journal

We prove that for any infinite-type orientable surface S, there exists a collection of essential curves Γ in S such that any homeomorphism that preserves the isotopy classes of the elements of Γ is isotopic to the identity. The collection Γ is countable and has an infinite complement in C(S), the curve complex of S. As a consequence, we obtain that the natural action of the extended mapping class group of S on C(S) is faithful.

The Alexander Method for Infinite-Type Surfaces