The Selberg–Weil–Kobayashi Local Rigidity Theorem for Exponential Lie Groups

A local rigidity Theorem proved by Selberg and Weil for Riemannian symmetric spaces and generalized by T. Kobayashi for a non-Riemannian homogeneous space G/H, asserts that there are no continuous deformations of a cocompact discontinuous subgroup Γ for G/H in the setting of a linear noncompact semi-simple Lie group G except some few cases: G is not locally isomorphic to ${\mathrm SL}_2(\mathbb R)$ for H compact or G is not locally isomorphic to SO(n,1) or SU(n,1) for G×G and H=ΔG. When in large contrast G is assumed to be exponential solvable and H⊂G a maximal subgroup, we prove an analog of such a Theorem stating that the local rigidity holds on the parameter space $\mathcal R(\Gamma,\ G,\ H)$ if and only if G is isomorphic to group ax+b of affine transformations of the real line. Remarkably, we do also drop the assumption on Γ to be uniform for G/H. This distinguished phenomenon comes out as a specific instance from the general theory of exponential Lie groups where any abelian discrete subgroup uniquely admits a syndetic hull. Unlike the setting of nilpotent Lie groups, the parameter space may fail to be open even when the Clifford–Klein form in question is compact.