Boundary $C^{\ast }$-algebras for acylindrical groups

Let $\mathrm{\Delta }$ be an infinite, locally finite tree with more than two ends. Let $\mathrm{\Gamma }<\text{aut}(\mathrm{\Delta })$ be an acylindrical uniform lattice. Then the boundary algebra $\text{cl}{A}_{\mathrm{\Gamma }}=C(\mathrm{\partial }\mathrm{\Delta })⋊\mathrm{\Gamma }$ is a simple Cuntz-Krieger algebra whose K-theory is determined explicitly.