Doctoral Dissertations

Date of Award

5-2022

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematics

Major Professor

Jerzy Dydak

Committee Members

Nikolay Brodskiy, Mathew Langford, Michael Berry.

Abstract

The main purpose of this work is to present a coarse counterpart to the Freudenthal compactification and its corona (the space of ends) that generalizes the Freudenthal compactification of a Freudenthal topological space X (connected, locally connected, locally compact and σ-compact) and its corona; then applying it to groups as coarse space to obtain generalizations to many well-known results in the theory of ends of groups. To this end, we present two constructions:

1. The Coarse Freudenthal compactification of a proper metric space which is a coarse compactification that coincides with the Freudenthal compactification when the metric space is geodesic.

2. The Freudenthal large scale compactification of a Hausdorff topological coarse space X that coincides with the Coarse Freudenthal compactification when X is a proper metric space equipped with the coarse structure induced by its metric.

Coarse spaces of special interest are coarse groups equipped with coarse structures induced by a generating family, a class of groups that includes: finitely generated groups, countable, metrizable groups, and topological groups.

Considering the space of coarse ends to coarse groups, we present generalization, under the framework of coarse geometry, to classical results in the theory of ends of finitely generated groups and the theory of ends of locally compact, metrizable connected groups; such as the result that the number of ends is either is infinite of at most 2, and the result that, in such groups, groups of two ends are characterized as having a copy of Z of either finite index (in the case of finitely generated groups) or of compact index in the sense that coset space is compact (in the case of locally compact, metrizable connected groups).

We give the necessary and sufficient conditions for metrizability of the space of coarse ends of a coarse group that is coarsely geodesic; and prove that if the space of coarse ends is infinite and metrizable, then it is a Cantor space; which generalizes the fact that the space of ends of a finitely generated group that has infinitely many ends is a Cantor space.

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