Open Access
2010 The Cohomology of Lattices in ${\SL}(2,\mathbb{C})$
Tobias Finis, Fritz Grunewald, Paulo Tirao
Experiment. Math. 19(1): 29-63 (2010).


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 $E_n = \Sym^n\otimes \overline{\Sym}{}^n$, $n\in \N\cup \{0\}$, 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 present a large amount of experimental data for this case, as well as for some geometrically constructed and mostly nonarithmetic groups. The computations for $\SL(2,\O)$ lead us to discover two instances with nonlifted classes in the cohomology. We also derive an upper bound of size $O(n^2 / \log n)$ for any fixed lattice $\G$ 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. "The Cohomology of Lattices in ${\SL}(2,\mathbb{C})$." Experiment. Math. 19 (1) 29 - 63, 2010.


Published: 2010
First available in Project Euclid: 12 March 2010

zbMATH: 1225.11072
MathSciNet: MR2649984

Primary: 11F75
Secondary: 11F72 , 11Y99 , 30F40 , 32N10

Keywords: automorphic forms , Cohomology of arithmetic groups , Kleinian groups

Rights: Copyright © 2010 A K Peters, Ltd.

Vol.19 • No. 1 • 2010
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