The cohomology of lattices in sl(2, c)

Abstract

This paper contains both theoretical results and experimental data on the behavior of the dimensions of the cohomology spaces H1(Γ, En), where Γ is a lattice in SL(2,ℂ) and (Formula Presented), n ∈ ℕ ∪ {0}, is one of the standard selfdual modules. In the case Γ = 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(n2/log n) for any fixed lattice Γ in the general case. We discuss a number of new questions and conjectures suggested by our results and our experimental data. © A K Peters, Ltd.