On the local rigidity of discontinuous groups for exponential solvable Lie groups

Let H be an arbitrary closed connected subgroup of an exponential solvable Lie group. Then a deformation of a discontinuous subgroup Γ of G for the homogeneous space G/H may be locally rigid. When G is nilpotent, connected and simply connected, the question whether the Weil-Kobayashi local rigidity fails to hold is posed by Baklouti [Proc. Japan Acad. Ser. A Math. Sci. 87(2011), no. 9, 173-177]. A positive answer is only provided for some very few cases by now. This note aims to positively answer this question for some new settings. Our study goes even farther to exponential groups. In this case, the local rigidity fails to hold if the automorphism group of the Lie algebra of the syndetic hull of Γ is not solvable. In addition, any deformation of an abelian discontinuous subgroup is shown to be continuously deformable outside the setting of the affine group ax + b.