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

TL;DR

This paper establishes a lower bound for the simplicial volume of closed manifolds covered by the product of three hyperbolic planes, using bounds on the volume form in bounded cohomology, and provides an algorithm for similar bounds in higher dimensions.

Contribution

It introduces a method to estimate the lower bound of simplicial volume for manifolds covered by ext{H}^2 imes ext{H}^2 imes ext{H}^2, extending to an algorithm for higher products.

Findings

Derived an upper bound for the volume form's norm in bounded cohomology.

Established a lower bound for the simplicial volume of certain closed manifolds.

Provided an algorithm for computing bounds in higher-dimensional cases.

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 $\big(\mathbb{H}^2\big)^n$.