Deformations of complex structures on Γ\SL₂(C)

Abstract

Let G be a connected complex semisimple Lie group. Let Γ be a cocompact lattice in G. In this paper, we show that when G is SL₂(C), nontrivial deformations of the canonical complex structure on X exist if and only if the first Betti number of the lattice Γ is non-zero. It may be remarked that for a wide class of arithmetic groups Γ, one can find a subgroup Γ' of finite index in Γ, such that Γ'/[Γ',Γ'] is finite (it is a conjecture of Thurston that this is true for all cocompact lattices in SL(2, C)). We also show thatG acts trivially on the coherent cohomology groups Hⁱ(Γ\G,O) for any i≥0.