On endomorphisms of surface mapping class groups (Q5939195)

Let \(M_S\) be the mapping class group, i.e. the group of isotopy classes of orientation-preserving homeomorphisms \(S\to S\) where \(S\) is a compact, connected, orientable surface of genus \(g\) with \(b\) boundary components. A group \(G\) is Hopfian if every homomorphism from \(G\) onto itself is an automorphism. \(M_S\) is Hopfian since it is residually finite and finitely generated. The author and McCarthy have previously shown that every homomorphism \(\psi: M_S\to M_S\) with \(\psi(M_S)\) normal and \(M_S/\psi (M_S)\) cyclic is an automorphism unless \(S\) is an annulus, a sphere with four holes or a torus with two or more holes. In this paper, the author shows that if \(\varphi\) is an endomorphism of \(M_S\) such that \(\varphi (M_S)\) has finite index in \(M_S\) then \(\varphi\) is an automorphism except in the case of \(g=0\) with \(b=2\), 3, or 4 and \(g=1\) with \(b=2\).