Constraining mapping class group homomorphisms using finite subgroups

Abstract

We classify homomorphisms from mapping class groups by using finite subgroups. First, we give a new proof of a result of Aramayona-Souto that homomorphisms between mapping class groups of closed surfaces are trivial for a range of genera. Second, we show that only finitely many mapping class groups of closed surfaces have non-trivial homomorphisms into Homeo(S n ) for any n. We also prove that every homomorphism from Mod(Sg) to Homeo(S 2 ) or Homeo(S 3 ) is trivial if g ≥ 3, extending a result of Franks-Handel.

Normal generators for mapping class groups are abundant

“…(2) Chen and the first author proved that for g 3 and h < 2g 1 with h ¤ g, any homomorphism Mod.S g / ! Mod.S h / is trivial [8]. (3) Chen, Kordek, and the second author showed that any homomorphism from the braid group B n to the braid group B 2n is either cyclic or is equivalent to one of the standard inclusions [7].…”