Asymptotic dimension of a hyperbolic space and capacity dimension of its boundary at infinity

A quasisymmetry invariant of a metric space $Z$ (called the capacity dimension, $\operatorname{cdim} Z$) is introduced. The main result says that the asymptotic dimension of a visual Gromov hyperbolic space $X$ is at most the capacity dimension of its boundary at infinity plus 1, $\operatorname{asdim} X \le \operatorname{cdim} \partial_\infty X+1$.