Superrigidity of lattices in solvable Lie groups
Let \(\Gamma\) be a closed cocompact subgroup of a simply connected solvable Lie group \(G\), such that \(Adj_G \Gamma\) has the same Zariski closure as \(Ad G\). Main results: Any finite-dimensional representation \(\alpha : \Gamma \to GL_n (\mathbb{R})\) can be virtually extended to a representation of \(G\). If \(\Gamma\) is isomorphic to a closed cocompact subgroup \(\Gamma'\) of another simply connected cocompact Lie group \(G'\), then any isomorphism from \(\Gamma\) to \(\Gamma'\) extends to a crossed isomorphism from \(G\) to \(G'\). -- The author uses two technical tools -- the nilshadow map and the existence of syndetic hulls. The main superrigidity theorem is described for solvable groups and for lattices in nonsolvable groups.
scientific article; zbMATH DE number 861961
scientific article