Kangasniemi, Ilmari
(2016)
Coarse structures are an abstract construction describing the behavior of a space at a large distance. In this thesis, a variety of existing results on coarse structures are presented, with the main focus being coarse embeddability into Hilbert spaces. The end goal is to present a hierarchy of three coarse invariants, namely coarse embeddability into a Hilbert space, a property of metric spaces known as Property A, and a finite-valued asymptotic dimension.
After outlining the necessary prerequisites and notation, the first main part of the thesis is an introduction to the basics of coarse geometry. Coarse structures are defined, and it is shown how a metric induces a coarse structure. Coarse maps, equivalences and embeddings are defined, and some of their basic properties are presented. Alongside this, comparisons are made to both topology and uniform topology, and results related to metrizability of coarse spaces are outlined.
Once the basics of coarse structures have been presented, the focus shifts to coarse embeddability into Hilbert spaces, which has become a point of interest due to its applications to several unsolved conjectures. Two concepts are presented related to coarse embeddability into Hilbert spaces, the first one being Property A. It is shown that Property A implies coarse embeddability into a Hilbert space, and that it is a coarse invariant.
The second main concept related to coarse embeddability is asymptotic dimension. Asymptotic dimension is a coarse counterpart to the Lebesgue dimension of topological spaces. Various definitions of asymptotic dimension are given and shown equivalent. The coarse invariance of asymptotic dimension is shown, and the dimensions of several example spaces are derived. Finally, it is shown that a finite asymptotic dimension implies coarse embeddability into a Hilbert space, and in the case of spaces with bounded geometry it also implies Property A.