Rigidity of discontinuous actions on diamond homogeneous spaces (Q2814326)

Let \(G\) be a locally compact group, \(H\) a closed subgroup of \(G\) and \(\Gamma\) a discontinuous group for the homogeneous space \(G/H\), which is a discrete subgroup of \(G\) acting properly and pointwise freely on \(G/H\). The parameter space of the action of \(\Gamma\) on \(G/H\) is defined as \(R(\Gamma,G,H)=\{\varphi\in \mathrm{Hom}(\Gamma,G): \varphi(\Gamma)\text{ is a discontinuous group for }G/H\}\), where \(\mathrm{Hom}(\Gamma,G)\) designates the set of all homomorphism from \(\Gamma\) to \(G\).NEWLINENEWLINE A point \(\varphi\in R(\Gamma,G,H)\) is said to be locally rigid if its \(G\)-orbit through the inner conjugation is open in \(R(\Gamma,G,H)\) and strongly locally rigid, if this \(G\)-orbit is open in \(\mathrm{Hom}(\Gamma,G)\).NEWLINENEWLINE The author shows that in the case where \(G\) is the diamond group; i.e \(G=\mathbb{R}^n \ltimes\mathbb{H}_{2n+1}\), the semi-direct group of \(\mathbb{R}^n\) and the Heisenberg group \(\mathbb{H}_{2n+1}\), the property of strong local rigidity fails to hold. When \(H\) is dilation invariant local rigidity also fails to hold.