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Asymptotic dimension.(English)Zbl 1149.54017

This article is a survey of asdim, the theory of asymptotic dimension for metric spaces which is an analog to covering dimension, dim, for topological spaces. A brief explanation is in order in accordance with that given by the authors in their Introduction. In topology there are three important distinct definitions of dimension: covering dimension, dim; large inductive dimension, Ind; and small inductive dimension, ind. They all agree on separable metrizable spaces. But, whereas dim and Ind coincide on arbitrary metrizable spaces, they do not always agree with ind in this class. Hence ind is usually less important, albeit useful in its domain of reference.
For a similar reason the asymptotic version asind of ind does not play an important role in this subject and hence is not treated in this survey. Moreover, the asymptotic analog of Ind, asInd, is only encountered in one application at “this stage of the theory” according to the authors. Thus they concentrate almost exclusively on asymptotic dimension, asdim.
As stated by the authors, “this survey is in no way complete,” as they do not discuss the many extant generalizations of asdim.
The paper breaks into two parts, Part I dealing with the asymptotic dimension of metric spaces and consisting of 11 sections. The first one gives a review of many facts from dimension theory, the second presents definitions and basics for asdim, and the third presents union (sometimes known as sum) theorems. Later topics include the Higson corona (a connection with dim occurs here), the large asymptotic inductive dimension asInd, a Hurewicz-type mapping theorem, coarse embedding, spaces of dimensions 0 and 1, linear control, and the dimension of coarse structures.
Part II comprises 11 sections as well, concentrating on the asymptotic dimension of groups. It begins with metrics on groups and then treats a Hurewicz-type theorem for groups. It later engages groups acting on trees, asdim of a free product, Coxeter groups, hyperbolic groups, relatively hyperbolic groups, buildings, and finally infinite-dimensional groups.
The paper is generously supplied with 95 references.

MSC:

54F45 Dimension theory in general topology
55M10 Dimension theory in algebraic topology
20F69 Asymptotic properties of groups

Cite

References:

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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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