Let $Gamma subset SO(3,1)$ be a lattice.The well known emph{bending deformations}, introduced bylinebreak Thurstonand Apanasov, can be usedto construct non-trivial curves of representations of $Gamma$into $SO(4,1)$ when $Gamma ackslash hype{3}$ containsan embedded totally geodesic surface. A tangent vector to such acurve is given by a non-zero group cohomology classin $H^1(Gamma, mink{4})$. Our main result generalizes thisconstruction of cohomology to the context of ``branched''totally geodesic surfaces.We also consider a natural generalization of the famouscuspidal cohomology problem for the Bianchi groups(to coefficients in non-trivial representations), andperform calculations in a finite range.These calculations lead directly to an interesting example of alink complement in $S^3$which is not infinitesimally rigid in $SO(4,1)$.The first order deformations of this link complement are supportedon a piecewise totally geodesic $2$-complex.
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