Dynamic Asymptotic Dimension of Translation Actions on Compact Lie Groups

We develop a method to bound the dynamic asymptotic dimension of isometric group actions $\mathrm{\Gamma }\curvearrowright X$ in terms of the asymptotic dimension of a space of graphs similar to a box space of $\mathrm{\Gamma }$ (which also determines finite-dimensionality), and a geometric property of $X$ related to the doubling dimension. We apply this method to describe the DAD of translation actions by finitely generated subgroups of compact Lie groups, characterize finite dimensionality of such actions, and consequently bound the nuclear dimension of $C^{\ast }$-algebras arising from such actions by amenable groups.