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 [3] we show that the volume of a representation rho :pi _1(M)rightarrow mathrm {Isom}^+({{mathbb {H}}}^n), properly normalized, takes integer values if n is even and ge 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.