Inner $\mathrm{C}^{\ast}$-Selfless Groups

We introduce the notion of inner $\mathrm{C}^{\ast}$-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 $\mathrm{C}^{\ast}$-selfless group has the relative $\mathrm{C}^{\ast}$-invariant subalgebra rigidity property, and every finite product of such groups has the $\mathrm{C}^{\ast}$-invariant subalgebra rigidity property. We also introduce another group-theoretic notion, which we call cyclic Powers property. This implies inner $\mathrm{C}^{\ast}$-selflessness, complete $\mathrm{C}^{\ast}$-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 $\mathrm{PSL}(d,\mathbb K)$ containing a biproximal element, where $\mathbb K$ is a local field and $d\geq2$. In particular, we obtain the Powers property for $\mathrm{PSL}(d,\mathbb Z)$, $d\geq3$, 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 $\mathrm{BS}(m,n)$ with $|m|,|n|\geq2$ and $|m|\ne|n|$ have the relative ISR property, thereby answering a question of Dudko-Jiang.