The Selberg-Weil-Kobayashi rigidity theorem: The rank one solvable case (Q2831252)

Let \(G\) be a simply connected solvable Lie group, \(H\) be a maximal connected Lie subgroup in \(G\) and \(\Gamma\) be a discontinuous subgroup in \(G\). There is the natural action of \(\Gamma\) on \(G/H\). If this action is properly discontinuous, then the factor-space \(\Gamma \backslash G/H\) is called the Clifford-Klein form for the homogeneous space \(G/H\). In this paper the problems of deformations of Clifford-Klein forms and the parameter spaces of corresponding deformations of injective homomorphisms \(\Gamma \to G\) are considered.NEWLINENEWLINEWith the aid of a description of all maximal subalgebras in real solvable Lie algebras it is proved that such \(\Gamma\) is isomorphic to \(\mathbb Z\) (it is called ``the rank 1 case'') or to \(\mathbb Z^2\).NEWLINENEWLINEFor \(\Gamma = \mathbb Z\) descriptions of the parameter spaces are obtained and the rigidity in the spaces of homomorphisms is investigated. For the case \(\Gamma = \mathbb Z^2\) the parameter spaces are described.