Applications of $C^{\ast }$-Entropy to Representation Growth for Discrete Groups

The topological $1$-bounded entropy $h^{\text{top}}(C^{\ast }(\mathrm{\Gamma }))$ of the full group $C^{\ast }$-algebra of a finitely generated group $\mathrm{\Gamma }$ is given a representation-theoretic interpretation. The implications of Hilbert-Schmidt stability and matricial stability are investigated within this setting, recovering and strengthening a representation growth result of Hadwin and Shulman. The entropy framework is used to prove 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 $S{L}_n(\mathbb{Z})$ admits subexponential representation growth is recorded.