Hyperbolic Volume of Representations of Fundamental Groups of Cusped 3-Manifolds

Let W be a compact manifold and let ρ be a representation of its fundamental group into PSL(2,C) Isom(H3). The volume of ρ is defined by taking any ρ-equivariant map from the universal cover W̃ to H and then by integrating the pullback of the hyperbolic volume form on a fundamental domain. This volume does not depend on the choice of the equivariant map because two equivariant maps are always equivariantly homotopic and the cohomology class of the pullback of the volume form is invariant under homotopy. In [3], this definition is extended to the case of a noncompact cusped 3-manifold M (see Definitions 4.1 and 2.5). When M is not compact, some problems of integrability arise if one tries to use the above definition of the volume of a representation. The idea of Dunfield for overcoming these difficulties is to use a particular (and natural) class of equivariant maps, called pseudodeveloping maps (see Definition 2.5), that have a nice behavior on the cusps of M allowing to control their volume. Concerning the welldefinition of the volume, working with noncompact manifolds, two pseudodeveloping maps in general are not equivariantly homotopic and in [3] it is not proved that the volume of a representation does not depend on the chosen pseudodeveloping map. In this paper, we show that the volume of a representation is well defined even in the noncompact case and we generalize to noncompact manifolds some results known in the compact case. We restrict to the orientable case. The paper is structured as follows.