Homomorphisms between mapping class groups

Abstract

Suppose that $X$ and $Y$ are surfaces of finite topological type, where $X$ has genus $g\ge 6$ and $Y$ has genus at most $2g-1$; in addition, suppose that $Y$ is not closed if it has genus $2g-1$. Our main result asserts that every non-trivial homomorphism $\Map(X) \to \Map(Y)$ is induced by an {\em embedding}, i.e. a combination of forgetting punctures, deleting boundary components and subsurface embeddings. In particular, if $X$ has no boundary then every non-trivial endomorphism $\Map(X)\to\Map(X)$ is in fact an isomorphism. As an application of our main theorem we obtain that, under the same hypotheses on genus, if $X$ and $Y$ have finite analytic type then every non-constant holomorphic map $\CM(X)\to\CM(Y)$ between the corresponding moduli spaces is a forgetful map. In particular, there are no such holomorphic maps unless $X$ and $Y$ have the same genus and $Y$ has at most as many marked points as $X$.

Cited by in corpus (10)

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• The first integral cohomology of pure mapping class groups
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• Low dimensional linear representations of the mapping class group of a nonorientable surface
• The universal $n$-pointed surface bundle only has $n$ sections
• Lower central series, surface braid groups, surjections and permutations
• Homomorphisms between pure mapping class groups
• Non-planar ends are continuously unforgettable