Lower Bound for the Simplicial Volume of Closed Manifolds Covered by ${\mathbb{H}}^2\times {\mathbb{H}}^2\times {\mathbb{H}}^2$

Abstract

We estimate the upper bound for the ${\ell }^{\infty }$-norm of the volume form on ${\mathbb{H}}^2\times {\mathbb{H}}^2\times {\mathbb{H}}^2$ seen as a class in $H_c^6({\mathrm{PSL}}_2\mathbb{R}\times {\mathrm{PSL}}_2\mathbb{R}\times {\mathrm{PSL}}_2\mathbb{R};\mathbb{R})$. This gives the lower bound for the simplicial volume of closed Riemennian manifolds covered by ${\mathbb{H}}^2\times {\mathbb{H}}^2\times {\mathbb{H}}^2$. The proof of these facts yields an algorithm to compute the lower bound of closed Riemannian manifolds covered by $({\mathbb{H}}^2{)}^n$.