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Asymptotic dimension and asymptotic property

UNCG Author/Contributor (non-UNCG co-authors, if there are any, appear on document)
Lauren Danielle Sher (Creator)
The University of North Carolina at Greensboro (UNCG )
Web Site:
Gregory Bell

Abstract: 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 cite{Gromov}. 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 cite{Dranishnikov}. 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.

Additional Information

Language: English
Date: 2011
Asymptotic Dimension, Asymptotic Property C
Geometric group theory