A note on the integrality of volumes of representations

TL;DR

This paper provides an elementary, combinatorial proof that the volume of certain geometric representations is an integer in higher dimensions, confirming a conjecture for cases where n ≥ 2.

Contribution

It offers a new, elementary proof of the integrality of representation volumes in higher-dimensional hyperbolic geometry, simplifying previous complex arguments.

Findings

Volumes are integer-valued for n ≥ 2

Elementary proof simplifies understanding of volume integrality

Confirms a conjecture by Bucher, Burger, and Iozzi

Abstract

Let $\Gamma$ be a torsion-free, non-uniform lattice in $\mathrm{SO}(2n,1)$. We present an elementary, combinatorial-geometrical proof of a theorem of Bucher, Burger, and Iozzi which states that the volume of a representation $\rho:\Gamma\to\mathrm{SO}(2n,1)$, properly normalized, is an integer if $n$ is greater than or equal to $2$.