Constraining mapping class group homomorphisms using finite subgroups

2021/12/15 by Chen, Lei, Lanier, Justin
#Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2112.07843

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(\mathbbSⁿ) for any n. We also prove that every homomorphism from Mod(S_g) to Homeo(\mathbbS²) or Homeo(\mathbbS³) is trivial if g≥ 3, extending a result of Franks--Handel.

Related

• Homomorphisms from mapping class groups
• The mapping class group cannot be realized by homeomorphisms
• Low-dimensional linear representations of mapping class groups
• Homomorphisms into mapping class groups. An addendum
• Injective homomorphisms of mapping class groups of non-orientable surfaces