Metric dimension of $C^{\ast}$-algebras of cocycle twisted transformation groupoids: Growth and dynamical complexity

Abstract

We consider a natural CQMS structure on a twisted transformation groupoid $C^{\ast}$-algebra coming from stratified $_{\text {c}}$Lip-norm introduced by Austad. We obtain upper bounds of metric dimension of reduced $C^{\ast}$-algebra of a transformation groupoid $Γ\rtimes X$ and its cocycle twist for a suitably chosen CQMS structure, provided $(X,d)$ is a compact metric space of finite Kolmogorov dimension and $Γ$ is a discrete group of polynomial growth. When $Γ$ has exponential growth, we prove that the dimension is generically $+\infty$ proving that the dichotomy between polynomial growth and exponential growth of groups survive even after considering cocycle twists of transformation groupoids.