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Title: Hölder dimension with applications to hyperbolic groups
Author: Colvin, Samuel J. M.
ISNI:       0000 0004 9359 3887
Awarding Body: University of Bristol
Current Institution: University of Bristol
Date of Award: 2020
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We introduce the ‘Hölder dimension’ of a metric space as the infimum of Hausdorff dimensions of Hölder equivalent spaces. This definition is a Hölder variant of the definition of conformal dimension, the study of which serves as inspiration for some of our work. We present two main results, along with supporting results and examples. Firstly, in Chapter 4, we show that capacity dimension is an upper bound for the Hölder dimension of any compact, doubling metric space with finite capacity dimension. Consequently, for locally self-similar metric spaces, Hölder dimension is equal to topological dimension. We explore some caveats of this result in Chapter 5 where we give examples of Hölder dimension equalling topological dimension but only as a strict infimum, not a minimum, and an example of a space, which is not locally self-similar that has Hölder dimension strictly greater than its topological dimension. Secondly, in Chapter 6, we explain how to reduce understanding the Hölder dimension of boundaries of hyperbolic groups to understanding the Hölder dimension of the boundaries of hyperbolic groups with connected boundary.
Supervisor: Mackay, John ; Snaith, Nina Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available