(Co)homologies and K-theory of Bianchi groups using computational geometric models

Abstract : This thesis consists of the study of the geometry of a certain class of arithmetic groups, by means of a proper action on a contractible space. We will explicitly compute their group homology, and their equivariant K-homology. More precisely, consider an imaginary quadratic number field, and its ring of integers R. The Bianchi groups are the groups SL_2(R) and PSL_2(R). These groups act in a natural way on hyperbolic three-space. The Bianchi groups are a key to the study of a larger class of groups, the Kleinian groups, which dates back to works of Poincaré. In fact, each non-cocompact arithmetic Kleinian group is commensurable with some Bianchi group. The author has implemented the computation of a fundamental domain for the Bianchi groups. By computing the stabilisers and identifications on this fundamental domain, we obtain an explicit orbifold structure. We use it to study different aspects of the geometry of our groups. Firstly, we compute group homology with integer coefficients, using the equivariant Leray/Serre spectral sequence. Secondly, we compute the Bredon homology of the Bianchi groups, from which we deduce their equivariant K-homology. By the Baum/Connes conjecture, which is verified by the Bianchi groups, we obtain the K-theory of the reduced C*-algebras of the Bianchi groups, as isomorphic images. Finally, we complexify our orbifolds, by complexifying the real hyperbolic three-space. We obtain orbifolds given by the induced action of the Bianchi groups on complex hyperbolic three-space. Then we compute the Chen/Ruan orbifold cohomology for these complex orbifolds. This is one side of Ruan's cohomological crepant resolution conjecture.
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Mathematics [math]. Université Joseph-Fourier - Grenoble I, 2010. English


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Submitted on : Monday, December 19, 2011 - 9:49:16 PM
Last modification on : Tuesday, December 20, 2011 - 10:14:57 AM

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Alexander Rahm. (Co)homologies and K-theory of Bianchi groups using computational geometric models. Mathematics [math]. Université Joseph-Fourier - Grenoble I, 2010. English. <tel-00526976v5>

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