We prove a volume-rigidity theorem for fuchsian representations of fundamental groups of hyperbolic k-manifolds into Isom(H n ). Namely, we show that if M is a complete hyperbolic k-manifold with finite volume, then the volume of any representation of π 1 (M ) into Isom(H n ), 3 ≤ k ≤ n, is less than the volume of M , and the volume is maximal if and only if the representation is discrete, faithful and "k-fuchsian".

Let M be an orientable, cusped hyperbolic 3-manifold of finite volume. We show that the restriction map r : X 0 → X(∂M ) from a Dehn surgery component in the P SL 2 (C)-character variety of M to the character variety of the boundary of M is a birational isomorphism onto its image. This generalises a result by Nathan Dunfield. A key step in our proof is the exactness of Craig Hodgson's volume differential on the eigenvalue variety.

Let K be a tame knot with irreducible exterior M.K/ in a closed, connected, orientable 3-manifold † such that 1 . †/ is cyclic. If 1 is not a strict boundary slope, then the diameter of the set of strict boundary slopes of K , denoted d K , is a numerical invariant of K . We show that either (i) d K 2 or (ii) K is a generalized iterated torus knot. The proof combines results from Culler and Shalen [3] with a result about the effect of cabling on boundary slopes. 57M15, 57M25; 57M50

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