The Cohomology of Lattices in ${\SL}(2,\mathbb{C})$

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 $E_n = \Sym^n\otimes \overline{\Sym}{}^n$, $n\in \N\cup \{0\}$, 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.