Volume rigidity at ideal points of the character variety of hyperbolic 3-manifolds

Given the fundamental group Γ of a finite-volume complete hyperbolic 3-manifold M, it is possible to associate to any representation ρ:Γ→Isom(H3) a numerical invariant called volume. This invariant is bounded by the hyperbolic volume of M and satisfies a rigidity condition: if the volume of ρ is maximal, then ρ must be conjugated to the holonomy of the hyperbolic structure of M. This paper generalizes this rigidity result by showing that if a sequence of representations of Γ into Isom(H3) satisfies limn→∞Vol(ρn)=Vol(M), then there must exist a sequence of elements gn∈Isom(H3) such that the representations gn∘ρn∘g−1n converge to the holonomy of M. In particular if the sequence ρn converges to an ideal point of the character variety, then the sequence of volumes must stay away from the maximum. We conclude by generalizing the result to the case of k-manifolds and representations in Isom(Hm), where m≥k.