An extreme boundary of acylindrically hyperbolic groups

TL;DR

The paper proves that every non-elementary acylindrically hyperbolic group G admits a minimal, extremely proximal action on a compact metrizable space, yielding a topologically free action when G has no nontrivial finite normal subgroups. The construction leverages Bestvina–Bromberg–Fujiwara projection complexes and the horofunction boundary, together with Myrberg limit points, to produce an extreme boundary ∂G with north–south dynamics. This boundary action provides new C*-algebra consequences, including C*-selflessness and, under mild hypotheses, C*-simplicity, removing prior rapid decay requirements. The authors illustrate the approach in curve graphs and coned-off Cayley graphs, highlighting the method’s broad applicability across hyperbolic-like settings and sharp boundary-dynamics results that drive operator-algebraic implications.

Abstract

We prove that every acylindrically hyperbolic group admits a minimal and extremely proximal action on a compact metrizable space. If there are no nontrivial finite normal subgroups, then the action is topologically free. This answers positively a question of Ozawa and the applications to $C^\ast$-algebras are discussed.

Theorems & Definitions (84)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.7
• Lemma 2.1
• Definition 2.2
• Lemma 2.3
• proof