A cofinite universal space for proper actions for mapping class groups

TL;DR

This paper constructs a cofinite universal space for proper actions of mapping class groups of surfaces, facilitating the study of their cohomology and related conjectures in K-theory.

Contribution

It introduces a truncated Teichmueller space as a deformation retract, providing a new geometric model for the classifying space of proper actions of mapping class groups.

Findings

Existence of a cofinite universal space for proper actions of mapping class groups.

Finiteness of conjugacy classes of finite subgroups in these groups.

Verification of the rational Novikov conjecture for mapping class groups.

Abstract

We prove that the mapping class group $\Gamma_{g,n}$ for surfaces of negative Euler characteristic has a cofinite universal space $\E$ for proper actions (the resulting quotient is a finite $CW$-complex). The approach is to construct a truncated Teichmueller space $\T_{g,n}(\epsilon)$ by introducing a lower bound for the length of shortest closed geodesics and showing that $\T_{g,n}(\epsilon)$ is a $\Gamma_{g,n}$ equivariant deformation retract of the Teichmueller space $\T_{g, n}$. The existence of such a cofinite universal space is important in the study of the cohomology of the group $\gag$. As an application, we note that there are only finitely many conjugacy classes of finite subgroups of $\Gamma_{g,n}$. Another application is that the rational Novikov conjecture in K-theory holds for $\Gamma_{g,n}$.