Simplicity and pure infiniteness for ${\text{C}}^{\ast }$-algebras associated with stabilizers of boundary actions

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