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Author

Date

2017Type

- Doctoral Thesis

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yes
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Abstract

The Nagata dimension of a metric space X is the least integer n, such that for all r>0 there is a cover of X with sets whose diameter is bounded by cr and every set of diameter bounded by r does intersect at most n+1 of the sets of the cover. Here, c>0 is a constant independent of r. This notion of dimension can be viewed as a local and global metric variant of the topological dimension and has notable applications in Lipschitz extension theory. In this thesis, we consider CAT(0) spaces, those are non-positively curved metric spaces.
A useful way to construct a cover of a metric space is to start from covers of its spheres. So we show that for a CAT(0) space X with the property that all of its spheres centered at a base point z in X have Nagata dimension n (finite) with the same constant c, we have that the Nagata dimension of X is bounded above by n+1. If complete, such spaces are thus absolute Lipschitz retracts.
For a geodesically complete CAT(0) space X, one can define the boundary of X as a set of equivalence classes of geodesic rays. There are several ways to define a metric on the boundary of X and we will introduce a new way to do so. We will discuss several properties and dimension results for this metric. For another metric on the boundary that was introduced by M. Gromov, we will answer two previously open questions. Show more

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https://doi.org/10.3929/ethz-b-000256908Publication status

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Publisher

ETH ZurichSubject

Metric geometryOrganisational unit

03500 - Lang, Urs / Lang, Urs
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