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We obtain two in a sense dual to each other results: First, that the capacity dimension of every compact, locally self-similar metric space coincides with the topological dimension, and second, that the asymptotic dimension of a metric space, which is asymptotically similar to its compact subspace coincides with the topological dimension of the subspace. As an application of the first result, we prove the Gromov conjecture that the asymptotic dimension of every hyperbolic group G equals the topological dimension of its boundary at infinity plus 1, asdim G = dim ∂ ∞ G + 1. As an application of the second result, we construct Pontryagin surfaces for the asymptotic dimension, in particular, those are first examples of metric spaces X, Y with asdim(X ×Y ) < asdim X +asdim Y . Other applications are also given.

We show that for each n 2 there is a quasi-isometric embedding of the hyperbolic space H n in the product T n = T × · · · × T of n copies of a (simplicial) metric tree T . On the other hand, we prove that there is no quasi-isometric embedding H 2 → T × R m for any metric tree T and any m 0. (2000). 54Exx. Mathematics Subject Classification

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