Abstract
A local rigidity theorem was proved by Selberg and Weil for Riemannian symmetric spaces and generalized by Kobayashi for a non-Riemannian homogeneous space G/H, determining explicitly which homogeneous spaces G/H allow nontrivial continuous deformations of co-compact discontinuous groups. When G is assumed to be exponential solvable and H subset of G is a maximal subgroup, an analog of such a theorem states that the local rigidity holds if and only if G is isomorphic to the group Aff(R) of affine transformations of the real line (cf. [L. Abdelmoula, A. Baklouti and I. Kedim, The Selberg-Weil-Kobayashi rigidity theorem for exponential Lie groups, Int. Math. Res. Not. 17 (2012) 4062-4084.]). The present paper deals with the more general context, when G is a connected solvable Lie group and H a maximal nonnormal subgroup of G. We prove that any discontinuous group G for a homogeneous space G/H is abelian and at most of rank 2. Then we discuss an analog of the Selberg-Weil-Kobayashi local rigidity theorem in this solvable setting. In contrast to the semi-simple setting, the G-action on G/H is not always effective, and thus the space of group theoretic deformations (formal deformations) I (Gamma, G; G/H) could be larger than geometric deformation spaces. We determine I (Gamma, G; G/H) and also its quotient modulo uneffective parts when the rank Gamma = 1. Unlike the context of exponential solvable case, we prove the existence of formal colored discontinuous groups. That is, the parameter space admits a mixture of locally rigid and formally nonrigid deformations.