Homomorphisms between mapping class groups

TL;DR

This paper proves that non-trivial homomorphisms between mapping class groups of certain surfaces are always induced by embeddings, leading to a classification of holomorphic maps between their moduli spaces as forgetful maps.

Contribution

It establishes that all non-trivial homomorphisms between mapping class groups under specified conditions are induced by embeddings, and classifies holomorphic maps between moduli spaces.

Findings

Non-trivial homomorphisms are induced by embeddings.

Endomorphisms of mapping class groups are isomorphisms.

Holomorphic maps between moduli spaces are forgetful maps.

Abstract

Suppose that $X$ and $Y$ are surfaces of finite topological type, where $X$ has genus $g\geq 6$ 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 $X$.