Asymptotic dimension and asymptotic property C.

摘要

This thesis will be concerned with the study of some "large-scale" properties of metric spaces. This area evolved from the study of geometric group theory.;Chapter 1 lays out some of the fundamental notions of geometric group theory including information about word metrics, Cayley graphs, quasi-isometries, and ends of groups and graphs.;Chapter 2 introduces the idea of "large-scale" or "asymptotic" properties of metric spaces along the lines proposed by Gromov in [Gro93]. After looking at some elementary asymptotic versions of common topological notions, such as connectedness, we focus on asymptotic dimension, the large-scale analog of ordinary covering dimension.;In the final chapter, we focus on Dranishnikov's asymptotic version of Haver's property C; see [Dra00]. We provide some basic results on metric spaces with asymptotic property C, studying subspaces and unions. We also prove a result involving the product of metric spaces with asymptotic property C and exhibit a metric space with asymptotic property C and infinite asymptotic dimension. In addition, we study the relationships between asymptotic property C and some of our previously introduced concepts such as quasi-isometries and asymptotic dimension. *Please refer to dissertation for references.