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

Given a discrete group $Γ$ acting on a compact space $X$, and a stabilizer subgroup $Λ\leq Γ$ of $X$, we use the Rieffel induction of covariant $(Λ, X)$-representations to study classes of representations induced by certain characters on $Λ$ and the $\text{C}^*$-algebras they generate. When $X$ is a