The scale function for locally compact groups acting on non-positively curved spaces

Let $G$ be a totally disconnected, locally compact (t.d.l.c.) group. The scale $s_G(g)$ of $g\in G$ in the sense of Willis is given by the minimum value of the index $\mathrm{\mid }gU{g}^{-1}:U\cap gU{g}^{-1}\mathrm{\mid }$ as $U$ ranges over the compact open subgroups; the theory associated to the scale has been very successful in describing general dynamical features of automorphisms of t.d.l.c. groups. We focus on the case where $G$ acts properly and continuously by isometries on a geodesic space $X$, where $X$ is complete CAT(0) or proper and Gromov-hyperbolic, and $g\in G$ is hyperbolic. In this context, we find geometric descriptions of the parabolic and contraction groups, tidy subgroups, and structures in the $G$-action that encode the scale, including criteria for $g$ to have scale $1$.