Integrality of Volumes of Representations

Michelle Bucher, Marc Burger, Alessandra Iozzi

Published 2014-07-02, updated 2020-03-01Version 2

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.