# New bounds for the simplicial volume of complex hyperbolic surfaces

## TL;DR

This paper provides new estimates for the Gromov norm of certain cohomology classes, leading to explicit bounds on the simplicial volume of complex hyperbolic surfaces, advancing understanding in geometric topology.

## Contribution

It introduces novel bounds for the Gromov norm in complex hyperbolic geometry, resulting in explicit simplicial volume estimates for related manifolds.

## Key findings

- Derived explicit upper bounds for the simplicial volume of complex hyperbolic surfaces.
- Provided estimates for the Gromov norm of top-dimensional cohomology classes.
- Enhanced understanding of the relationship between cohomology norms and geometric invariants.

## Abstract

We give estimates of the Gromov norm of the top dimensional class in $H_c^4(\mathrm{Isom}(\mathbb{H}_{\mathbb{C}}^2);\mathbb{R})$. As a consequence, we obtain an explicit upper bound for the simplicial volume of closed oriented manifolds that are locally isometric to $\mathbb{H}_{\mathbb{C}}^2$.