Homomorphisms between mapping class groups
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Aramayona, Javier; Souto, Juan (2012). Homomorphisms between mapping class groups. Geometry & Topology 16 (4), 2285-2341
Suppose that X and Y are surfaces of finite topological type, where X has genus g &gt;= 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 nontrivial homomorphism Map(X ) -&gt; Map(Y) is induced by an embedding, ie a combination of forgetting punctures, deleting boundary components and subsurface embeddings. In particular, if X has no boundary then every nontrivial endomorphism Map(X) -&gt; Map(X) is in fact an isomorphism.