Proportionality principle for the simplicial volume of families of Q-rank 1 locally symmetric spaces

TL;DR

This paper proves a proportionality principle linking Riemannian volume and simplicial volume for certain Q-rank 1 locally symmetric spaces, including new cases with non-amenable cusp groups, and provides a simplified proof for known cases.

Contribution

It establishes the proportionality principle for a broader class of Q-rank 1 locally symmetric spaces, including those with non-amenable cusp groups, and offers a simplified proof for existing results.

Findings

Proportionality principle holds for spaces with non-amenable cusp groups.

Simplified proof of the proportionality principle for spaces with amenable cusp groups.

First examples of manifolds with non-amenable cusp groups satisfying the principle.

Abstract

We establish the proportionality principle between the Riemannian volume and locally finite simplicial volume for Q-rank 1 locally symmetric spaces covered by products of hyperbolic spaces, giving the first examples for manifolds whose cusp groups are not necessarily amenable. Also, we give a simple direct proof of the proportionality principle for the locally finite simplicial volume and the relative simplicial volume of Q-rank 1 locally symmetric spaces with amenable cusp groups established by L\"oh and Sauer.