Boundary $C^*$-algebras for acylindrical groups

TL;DR

This paper studies boundary C*-algebras associated with acylindrical groups acting on infinite trees, showing they are simple Cuntz-Krieger algebras with explicitly computed K-theory, advancing understanding of their structure.

Contribution

It establishes that boundary algebras for acylindrical uniform lattices on trees are simple Cuntz-Krieger algebras with explicit K-theory calculations.

Findings

Boundary algebra is a simple Cuntz-Krieger algebra.

K-theory of the algebra is explicitly determined.

Results apply to groups acting on infinite trees with more than two ends.

Abstract

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