On invertible elements in reduced $C^*$-algebras of acylindrically hyperbolic groups

TL;DR

This paper proves that for acylindrically hyperbolic groups with no non-trivial finite normal subgroups, the invertible elements are dense in their reduced $C^*$-algebras, extending to finite products of such groups.

Contribution

It establishes the density of invertible elements in reduced $C^*$-algebras for a broad class of acylindrically hyperbolic groups, including their finite products.

Findings

Invertible elements are dense in the reduced $C^*$-algebra of certain acylindrically hyperbolic groups.

The result applies to groups with no non-trivial finite normal subgroups.

Extension of the density result to finite direct products of these groups.

Abstract

Let $G$ be an acylindrically hyperbolic group. We prove that if $G$ has no non-trivial finite normal subgroups, then the set of invertible elements is dense in the reduced $C^\ast$-algebra of $G$. The same result is obtained for finite direct products of acylindrically hyperbolic groups.