Integrality of volumes of representations

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 into the connected component of the isometry group of hyperbolic n-space, properly normalized, takes integer values if n=2m is at least 4. Moreover we prove that the volume is continuous in all dimension and hence, if the dimension of M is even and at least 4, it is constant on connected components of the representation variety.<br/> If M is not compact and 3-dimensional, the volume is not locally constant and we give explicit examples of representations with volume as arbitrary as the volume of hyperbolic manifolds obtained from M via Dehn fillings.