Inner C^-Selfless Groups

We introduce the notion of inner C^-selfless groups and study their structural properties. We prove a Fourier coefficient extraction theorem for invariant subalgebras of crossed products by such groups under a compatibility assumption on the action. Consequently, every inner C^-selfless group has the relative C^-invariant subalgebra rigidity property, and every finite product of such groups has the C^-invariant subalgebra rigidity property. We also introduce another group-theoretic notion, which we call cyclic Powers property. This implies inner C^-selflessness, complete C^-selflessness, and the Powers property, among others. We establish it for groups admitting a topologically free extreme boundary action on a compact Hausdorff space with more than two points, and for Zariski-dense discrete subgroups of PSL(d, K) containing a biproximal element, where K is a local field and d≥2. In particular, we obtain the Powers property for PSL(d, Z), d≥3, answering a question posed more than twenty years ago by Bridson-de la Harpe. Finally, we show that every non-trivial mixed identity-free discrete group has the relative invariant subalgebra rigidity property for its group von Neumann algebra. This shows that the Baumslag-Solitar groups BS(m,n) with |m|,|n|≥2 and |m|≠|n| have the relative ISR property, thereby answering a question of Dudko-Jiang.