Dimensions of locally and asymptotically self-similar spaces

Two results are obtained, in a sense dual to each other. First, the capacity dimension of every compact, locally self-similar metric space coincides with the topological dimension, and second, a metric space asymptotically similar to its compact subspace has asymptotic dimension equal to the topological dimension of the subspace. As an application of the first result, the following Gromov conjecture is proved: the asymptotic dimension of every hyperbolic group $G$ equals the topological dimension of its boundary at infinity plus 1, $\operatorname{asdim}G=\dim\partial_{\infty}G+1$. As an application of the second result, we construct Pontryagin surfaces for the asymptotic dimension; in particular, these surfaces are examples of metric spaces $X$, $Y$ with $\operatorname{asdim}(X\times Y)<\operatorname{asdim}X+\operatorname{asdim}Y$. Other applications are also given.