Deformations of complex structures on \(\Gamma\backslash SL_ 2(\mathbb{C})\)
type: Item
instance of: scholarly article
title: Deformations of complex structures on \(\Gamma\backslash SL_ 2(\mathbb{C})\)
zbMATH Open document ID: 0837.11034
DOI: 10.1007/BF02863419
author: C. S. Rajan
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 1994-08-14T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 11F75
Mathematics Subject Classification ID: 22E40
zbMATH DE Number: 618132
zbMATH Keywords: compact hyperbolic manifold
zbMATH Keywords: torsionfree cocompact lattice
zbMATH Keywords: nontrivial deformations
zbMATH Keywords: complex structure
zbMATH Keywords: first Betti number
reviewed by: Joachim Schwermer
MaRDI profile type: Publication
cites work: One-dimensional cohomologies of discrete subgroups
cites work: Q3885948
cites work: Q3704179
cites work: On liftings and cusp cohomology of arithmetic groups
cites work: On the first Betti number of a constant negatively curved manifold
cites work: Geometric construction of cohomology for arithmetic groups. I
cites work: Vanishing theorems for cohomology groups associated to discrete subgroups of semisimple Lie groups
full work available at URL: https://doi.org/10.1007/bf02863419
OpenAlex ID: W2015485004
description: scientific article; zbMATH DE number 618132