The cohomology of lattices in SL(2,C)

FINIS, TOBIAS; GRUNEWALD, FRITZ; TIRAO, PAULO

EXPERIMENTAL MATHEMATICS

A K PETERS LTD

Año: 2010 vol. 19 p. 29 - 63

1058-6458

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 $E_n = Sym^notimes overline{Sym}{}^n$, $nin Ncup {0}$, is one of the standard self-dual modules. In the case$Gamma = SL(2,O)$ for the ring of integers $O$ in an imaginaryquadratic 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 $G$ in the general case. We discuss a number of new questions and conjectures suggested by our results and our experimental data.