Topics in geometric group theory

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Title Topics in geometric group theory
Publication Type dissertation
School or College College of Science
Department Mathematics
Author Malone, William
Date 2010-05
Description This document contains results in a couple of nonrelated areas of geometric group theory. What follows are abstracts for each part. Let Mi and Ni be path-connected locally uniquely geodesic metric spaces that are not points and f : ?mi=1Mi ? ?ni=1 Ni be an isometry where ?ni=1 Ni and ?mi=1Mi are given the sup metric. Then m = n and after reindexing Mi is isometric to Ni for all i. Moreover f is a composition of an isometry that reindexes the factor spaces and an isometry that is a product of isometries fi : Mi ? Ni. Given a geometric amalgamation of free groups G and the associated simple thick two-dimensional hyperbolic piecewise manifold M, the visual boundary ?M is a complete quasi-isometry invariant. This invariant can be effciently computed for any G using an adaptation of Leighton's Theorem. Let G and G0 be geometric amalgamation of free groups with a single Z vertex. If the associated simple thick two-dimensional hyperbolic piecewise manifolds M and M0 have the same Euler characteristic, then G is commensurable to G0 if and only if M and M0 are homeomorphic. The proof is then extended to the case where G and G0 have more than a single Z vertex, but more conditions have to be placed on G and G0. With these results an elementary example of two geometric amalgamations of free groups that are quasi-isometric but not commensurable can be given.
Type Text
Publisher University of Utah
Subject Commensurable; Quasi-isometry; Reducible; Sup metric
Subject LCSH Geometric group theory
Dissertation Institution University of Utah
Dissertation Name PhD
Language eng
Rights Management ©William Malone
Format application/pdf
Format Medium application/pdf
Format Extent 484,261 bytes
Identifier us-etd2,149205
Source Original in Marriott Library Special Collections, QA3.5 2010 .M35
ARK ark:/87278/s6s75wz8
Setname ir_etd
ID 193679
Reference URL https://collections.lib.utah.edu/ark:/87278/s6s75wz8
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