Deformations of lattices in certain solvable groups
Let \(G\) be a Lie subgroup of a Lie group \(H\). The author calls a subgroup \(\Gamma\) of \(G\) \(H\)-rigid if every small enough deformation of \(\Gamma\) in \(G\) is conjugate to \(\Gamma\) by an element of \(H\). Generalizing a theorem of A. Weil, he conjectures that this is the case for a finitely generated group \(\Gamma\) if the induced homomorphism \(H^1(\Gamma,\text{Ad}_G)\rightarrow H^1(\Gamma,\text{Ad}_H)\) in cohomology is zero. He confirms this for the case that \(\Gamma=\mathbb Z^n\rtimes_A\mathbb Z\), where \(n\geq 2\), \(A\) is a hyperbolic matrix in \(SL(n,\mathbb Z)\), and \(G=\mathbb R^n\rtimes_A\mathbb R\) is considered as a subgroup of \(H=SL(n+1,\mathbb R)\).
scientific article; zbMATH DE number 5294950