Deformations of complex structures on \(\Gamma\backslash SL_ 2(\mathbb{C})\)
Let \(M^n\) be a compact hyperbolic manifold. A conjecture of Thurston asserts that \(M\) has a finite sheeted covering \(N\) with nonzero first Betti number. In some cases one could prove this conjecture where \(\pi_1 (M^n)\) is an arithmetic lattice in the orthogonal group \(PO (n,1)\). For \(n = 3\), the group \(PO (3,1)\) is locally isomorphic to \(SL_2 (\mathbb{C})\). In this note it is shown that, given a torsionfree cocompact lattice in \(SL_2 (\mathbb{C})\), nontrivial deformations of the canonical complex structure on \(\Gamma \backslash SL_2 (\mathbb{C})\) exist if and only if the first Betti number of \(\Gamma\) is nonzero.
scientific article; zbMATH DE number 618132