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The stamping deformation was defined by Apanasov as the first example of a deformation of the flat conformal structure on a hyperbolic 3-orbifold distinct from bending. We show that in fact the stamping cocycle is equal to the sum of three bending cocycles. We also obtain a more general result, showing that derivatives of geodesic lengths vanish at the base representation under deformations of the flat conformal structure of a finite-volume hyperbolic 3-orbifold.

Abstract. Let Γ ⊂ SO(3, 1) be a lattice. The well known bending deformations, introduced by Thurston and Apanasov, can be used to construct non-trivial curves of representations of Γ into SO(4, 1) when Γ\H 3 contains an embedded totally geodesic surface. A tangent vector to such a curve is given by a non-zero group cohomology class in H 1 (Γ, R 4 1 ). Our main result generalizes this construction of cohomology to the context of "branched" totally geodesic surfaces. We also consider a natural generalization of the famous cuspidal cohomology problem for the Bianchi groups (to coefficients in non-trivial representations), and perform calculations in a finite range. These calculations lead directly to an interesting example of a link complement in S 3 which is not infinitesimally rigid in SO(4, 1). The first order deformations of this link complement are supported on a piecewise totally geodesic 2-complex.

Given a Bianchi Group Γ d = P SL 2 (O d ), and a Hyperbolic manifold M , where π 1 (M ) is of finite index in Γ d , we show that all boundary slopes are realized as the boundary slope of an immersed totally geodesic surface and hence are virtually embedded boundary slopes.