Dimensions of locally and asymptotically self-similar spaces

Abstract

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.

Asymptotic dimension

“…In fact, this applies to more general hyperbolic spaces, see, for example [10,15,85]. We note that in the case of hyperbolic groups asdim Γ = dim ∂ ∞ Γ + 1 [27,26].…”

“…The inequality in this theorem can be strict [27]. It can be strict even when one of the factors is the reals R. In [37] an example of a metric space (uniform simplicial complex) X is constructed with the properties asdim X = 2 and asdim(X × R) = 2.…”

On asymptotic dimension of amalgamated products and right-angled Coxeter groups

“…For polycyclic groups, the asymptotic dimension equals the Hirsch length: asdim D h./ (see Bell and Dranishnikov [4] for the inequality in one direction and Dranishnikov and Smith [16] for the other direction). The asymptotic dimension of a finitely generated hyperbolic group equals the covering dimension of its boundary plus one: asdim D dim @ 1 C 1 by Buyalo [7] and Buyalo and Lebedeva [8]. In view of Bestvina-Mess' formula [6] we have asdim D vcd./ for those hyperbolic groups for which the virtual cohomological dimension is defined (eg for residually finite hyperbolic groups).…”

Remarks on coarse triviality of asymptotic Assouad–Nagata dimension

“…Due to the latter one can immediately construct examples realising a strict inequality in the logarithmic law: [7, section 5]). Earlier examples with such properties can be found in [3] and [7].…”