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arXiv · 0808.1204

The cohomology of lattices in SL(2,C)

Tobias Finis·Fritz Grunewald·Paulo Tirao
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Abstract

This paper contains both theoretical results and experimental data on the behavior of the dimensions of the cohomology spaces H^1(G,E_n), where Gamma is a lattice in SL(2,C) and E_n is one of the standard self-dual modules. In the case Gamma = SL(2,O) for the ring of integers O in an imaginary quadratic number field, we make the theory of lifting explicit and obtain lower bounds linear in n. We have accumulated a large amount of experimental data in this case, as well as for some geometrically constructed and mostly non-arithmetic groups. The computations for SL(2,O) lead us to discover two instances with non-lifted classes in the cohomology. We also derive an upper bound of size O(n^2 / log n) for any fixed lattice Gamma in the general case. We discuss a number of new questions and conjectures suggested by our results and our experimental data.

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Tobias Finis, Fritz Grunewald, Paulo Tirao. 2008-08-08. The cohomology of lattices in SL(2,C). https://arxiv.org/abs/0808.1204

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