Applications of $C^*$-Entropy to Representation Growth for Discrete Groups

Abstract

The topological $1$-bounded entropy $h^\text{top}(C^*(Γ))$ of the full group $C^*$-algebra of a finitely generated group $Γ$ is given a representation-theoretic interpretation. The implications of Hilbert-Schmidt and matricial stability in this setting are investigated, recovering and strengthening a representation growth result of Hadwin and Shulman. The entropy framework is used to recover upper bounds on representation growth for superrigid groups and property (T) groups. This recovers a result of De La Harpe, Robinson, and Valette on the growth rate of the irreducible unitary representations of a property (T) group within the framework of $1$-bounded entropy. A soft proof that $SL_n(\mathbb{Z})$ admits subexponential representation growth is recorded.