Topological deformation rigidity of higher rank lattice actions
Let \(\Gamma\) be an irreducible higher rank lattice (i.e., the \(\mathbb{R}\)- rank of \(G\) is at least 2) in a connected semisimple Lie group \(G\) with finite center and without compact factor. Let \(\rho_0\) be a \(C^1\)- action of \(\Gamma\) on a compact smooth manifold \(M\). Let \(\gamma_1, \dots, \gamma_s \in \Gamma\) be such that \(\rho_0 (\gamma_i)\) is 1-normally hyperbolic at a \(C^1\)-foliation \(\mathcal C\), the intersection \(\bigcap T_x {\mathcal C} (\rho_0 (\gamma_i))\) is empty for every \(x \in M\), and periodic points of \(\rho_0\) are dense in \(M\). Then the action \(\rho_0\) is deformation rigid. (The proof employs that periodical points persist under deformations \(\rho_t\) which extend to a conjugacy.) As a further result, the rigidity of certain linear actions follows, e.g., this is the case for a subgroup of finite index in \(\text{SL} (n, \mathbb{Z})\) \((n \geq 3)\) or \(\text{Sp} (2n, \mathbb{Z})\) \((n \geq 2)\) with action on a torus or nilmanifold induced by a finite-dimensional continuous representation.
finite-dimensional continuous representation
scientific article; zbMATH DE number 817713