On the Teichmüller stack of compact quotients of SL2(C)

This article aims to pursue and generalize, by using the global point of view offered by the stacks, the local study made by Ghys(J f & uuml;r die reine und angewandte Mathematik468:113-138, 1995) concerning the deformations of complex structures of compact quotients of SL2(C). In his article, Ghys showed that the analytic germ of the representation variety R(Gamma ) := Hom(Gamma, SL2(C)) of Gamma in SL2(C), pointed at the trivial morphism, determines the Kuranishi space of SL2(C)/Gamma. In this note, we show that the tautological family above a Zariski analytic open subset V in R(Gamma ) remains complete. Moreover, the computation of the isotropy group of a complex structure in Teichm & uuml;ller space, allows us to affirm that the quotient stack [V/SL2(C)]is an open substack of the Teichm & uuml;ller stack of SL2(C)/Gamma.