Integrality of Volumes of Representations

TL;DR

This paper proves that the volume of certain geometric representations of hyperbolic manifolds is always an integer in even dimensions at least 4, and provides examples of volume variation in non-compact 3D cases.

Contribution

It establishes the integrality of representation volumes in even dimensions and constructs explicit examples of volume variation in non-compact 3D hyperbolic manifolds.

Findings

Volumes are integer-valued for even dimensions ≥ 4.

Explicit examples show volume can vary in non-compact 3D cases.

Volume is not locally constant in non-compact 3D hyperbolic manifolds.

Abstract

Let M be an oriented complete hyperbolic n-manifold of finite volume. Using the definition of volume of a representation previously given by the authors in [BucherBurgerIozzi2013] we show that the volume of a representation of the fundamental group of M into the connected component of the isometry group of hyperbolic n-space, properly normalized, takes integer values if n=2m is at least 4. If M is not compact and 3-dimensional, it is known that the volume is not locally constant. In this case we give explicit examples of representations with volume as arbitrary as the volume of hyperbolic manifolds obtained from M via Dehn fillings.