Arithmetic structure of fundamental groups and actions of semisimple Lie groups (Q5950616)

Let \(G\) be a connected Lie group with \(\mathbb{R}\)-rank \(\geq 2\) acting continuously on a compact space \(M\) equipped with a \(G\)-invariant ergodic measure \(\mu\). The action is called engaging if, for every finite covering \(M'\to M\), the action of \(\widetilde G\) on \(M'\) is ergodic. The action is called totally engaging if there is no \(\widetilde G\)-equivariant measurable section of the covering for any nontrivial covering space. Let, moreover, \(\pi_1(M)\to GL (M)\) be a finite-dimensional linear representation over \(\mathbb{C}\), let \(\Gamma\) be the image and assume \(\Gamma\) to be finite. A: If the action of \(G\) on \(M\) is totally engaging, then \(\Gamma\) is an arithmetic group. B: If the action of \(G\) on \(M\) is engaging, then \(\Gamma\) is \(s\)-arithmetic. C: Let \(G,M,\Gamma\) be as in \(B\), and let \(\Gamma_\infty\) be the arithmetic subgroup of \(\Gamma\), then \(M\) has a virtual arithmetic quotient of the form \(K\setminus H/ \Gamma_\infty\). In fact, the results \(A,B,C\) are proved in sharper form and examples showing the necessity of the hypothesis are presented.