TOPOLOGICAL PROPERTIES OF ASYMPTOTIC CONES
Gromov asked whether an asymptotic cone of a finitely generated group was always simply connected or had uncountable fundamental group. We prove that Gromov's dichotomy holds for asymptotic cones with cut points as well as HNN extensions and amalgamated products where the associated subgroups a...
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ndltd-VANDERBILT-oai-VANDERBILTETD-etd-03202013-1047182013-04-10T04:43:09Z TOPOLOGICAL PROPERTIES OF ASYMPTOTIC CONES Kent, Curtis Andrew Mathematics Gromov asked whether an asymptotic cone of a finitely generated group was always simply connected or had uncountable fundamental group. We prove that Gromov's dichotomy holds for asymptotic cones with cut points as well as HNN extensions and amalgamated products where the associated subgroups are nicely embedded. We also show a slightly weaker dichotomy for multiple HNN extensions of free groups. We define an analogue to Gromov's loop division property which we use to give a sufficient condition for an asymptotic cone of a complete geodesic metric space to have uncountable fundamental group. This is used to understand the asymptotic cones of many groups currently in the literature. As a corollary, we show that an infinite group is virtually cyclic if and only if an asymptotic cone of the group has exactly two-ends. As well we show that in every asymptotic cone of a finitely generated group which contains a cut-point, the maximal transversal trees are universal R-trees with continuum branching at every point. Mark Sapir Michael Mihalik Bruce Hughes Denis Osin Will Johns VANDERBILT 2013-04-09 text application/pdf http://etd.library.vanderbilt.edu/available/etd-03202013-104718/ http://etd.library.vanderbilt.edu/available/etd-03202013-104718/ en unrestricted I hereby certify that, if appropriate, I have obtained and attached hereto a written permission statement from the owner(s) of each third party copyrighted matter to be included in my thesis, dissertation, or project report, allowing distribution as specified below. I certify that the version I submitted is the same as that approved by my advisory committee. I hereby grant to Vanderbilt University or its agents the non-exclusive license to archive and make accessible, under the conditions specified below, my thesis, dissertation, or project report in whole or in part in all forms of media, now or hereafter known. I retain all other ownership rights to the copyright of the thesis, dissertation or project report. I also retain the right to use in future works (such as articles or books) all or part of this thesis, dissertation, or project report. |
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Mathematics Kent, Curtis Andrew TOPOLOGICAL PROPERTIES OF ASYMPTOTIC CONES |
description |
Gromov asked whether an asymptotic cone of a finitely generated group was always simply connected or had uncountable fundamental group. We prove that Gromov's dichotomy holds for asymptotic cones with cut points as well as HNN extensions and amalgamated products where the associated subgroups are nicely embedded. We also show a slightly weaker dichotomy for multiple HNN extensions of free groups.
We define an analogue to Gromov's loop division property which we use to give a sufficient condition for an asymptotic cone of a complete geodesic metric space to have uncountable fundamental group. This is used to understand the asymptotic cones of many groups currently in the literature.
As a corollary, we show that an infinite group is virtually cyclic if and only if an asymptotic cone of the group has exactly two-ends. As well we show that in every asymptotic cone of a finitely generated group which contains a cut-point, the maximal transversal trees are universal R-trees with continuum branching at every point. |
author2 |
Mark Sapir |
author_facet |
Mark Sapir Kent, Curtis Andrew |
author |
Kent, Curtis Andrew |
author_sort |
Kent, Curtis Andrew |
title |
TOPOLOGICAL PROPERTIES OF ASYMPTOTIC CONES |
title_short |
TOPOLOGICAL PROPERTIES OF ASYMPTOTIC CONES |
title_full |
TOPOLOGICAL PROPERTIES OF ASYMPTOTIC CONES |
title_fullStr |
TOPOLOGICAL PROPERTIES OF ASYMPTOTIC CONES |
title_full_unstemmed |
TOPOLOGICAL PROPERTIES OF ASYMPTOTIC CONES |
title_sort |
topological properties of asymptotic cones |
publisher |
VANDERBILT |
publishDate |
2013 |
url |
http://etd.library.vanderbilt.edu/available/etd-03202013-104718/ |
work_keys_str_mv |
AT kentcurtisandrew topologicalpropertiesofasymptoticcones |
_version_ |
1716580014352236544 |