A cofinite universal space for proper actions for mapping class groups

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