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The Generalized Cuspidal Cohomology Problem

Published online by Cambridge University Press:  20 November 2018

Anneke Bart
Department of Mathematics and Computer Science, Saint Louis University, St. Louis, MI 63103 e-mail:
Kevin P. Scannell
Department of Mathematics and Computer Science, Saint Louis University, St. Louis, MI 63103 e-mail:
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Let $\Gamma \subset SO(3,1)$ be a lattice. The well known bending deformations, introduced by Thurston and Apanasov, can be used to construct non-trivial curves of representations of $\Gamma $ into $SO(4,1)$ when $\Gamma \backslash {{\mathbb{H}}^{3}}$ contains an embedded totally geodesic surface. A tangent vector to such a curve is given by a non-zero group cohomology class in ${{\text{H}}^{1}}\left( \Gamma ,\mathbb{R}_{1}^{4} \right)$. 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.


Research Article
Copyright © Canadian Mathematical Society 2006


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