Asymptotic dimension and asymptotic property
- UNCG Author/Contributor (non-UNCG co-authors, if there are any, appear on document)
- Lauren Danielle Sher (Creator)
- Institution
- The University of North Carolina at Greensboro (UNCG )
- Web Site: http://library.uncg.edu/
- Advisor
- 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.
Asymptotic dimension and asymptotic property
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Created on 5/1/2011
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Additional Information
- Publication
- Thesis
- Language: English
- Date: 2011
- Keywords
- Asymptotic Dimension, Asymptotic Property C
- Subjects
- Geometric group theory