On volumes of quasi-arithmetic hyperbolic lattices

TL;DR

This paper proves that the covolume of quasi-arithmetic hyperbolic lattices is a rational multiple of that of arithmetic lattices, providing insights into the volume structure of hyperbolic manifolds in higher dimensions.

Contribution

It establishes a fundamental relationship between quasi-arithmetic and arithmetic hyperbolic lattices' covolumes, extending understanding of their volume spectra.

Findings

Covolume of quasi-arithmetic lattices is a rational multiple of arithmetic ones.

Provides a description of volume shapes for most known hyperbolic n-manifolds with n>3.

Enhances understanding of volume distribution in hyperbolic geometry.

Abstract

We prove that the covolume of any quasi-arithmetic hyperbolic lattice (a notion that generalizes the definition of arithmetic subgroups) is a rational multiple of the covolume of an arithmetic subgroup. As a corollary, we obtain a good description for the shape of the volumes of most of the known hyperbolic n-manifolds with n>3.