Simplicity and pure infiniteness for $\text{C}^*$-algebras associated with stabilizers of boundary actions
Given a discrete group $\Gamma$ acting on a compact space $X$, and a stabilizer subgroup $\Lambda\leq \Gamma$ of $X$, we use the Rieffel induction of covariant $(\Lambda, X)$-representations to study classes of representations induced by certain characters on $\Lambda$ and the $\text{C}^*$-algebras they generate. When $X$ is a $\Gamma$-boundary, we obtain new classes of simple, traceless group $\text{C}^*$-algebras. When the $\Gamma$-boundary $X$ is an extreme boundary, we show that the associated group $\text{C}^*$-algebras are also purely infinite, answering part of a question posed by Kalantar and Scarparo on $\text{C}^*$-algebras associated with Thompson's groups. The latter $\text{C}^*$-algebras are shown to be selfless in the sense of Robert.