A cofinite universal space for proper actions 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 $\Gamma_{g,n}$.