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The Generalized Cuspidal Cohomology Problem

 Printed: Aug 2006
  • Anneke Bart
  • Kevin P. Scannell
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Let $\Gamma \subset \SO(3,1)$ be a lattice. The well known \emph{bending deformations}, introduced by \linebreak Thurston and Apanasov, can be used to construct non-trivial curves of representations of $\Gamma$ into $\SO(4,1)$ when $\Gamma \backslash \hype{3}$ contains an embedded totally geodesic surface. A tangent vector to such a curve is given by a non-zero group cohomology class in $\H^1(\Gamma, \mink{4})$. 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.
MSC Classifications: 57M50, 22E40 show english descriptions Geometric structures on low-dimensional manifolds
Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx]
57M50 - Geometric structures on low-dimensional manifolds
22E40 - Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx]

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