PROPERTIES OF HYPERBOLIC GROUPS: FREE NORMAL SUBGROUPS, QUASICONVEX SUBGROUPS AND ACTIONS OF MAXIMAL GROWTH
Hyperbolic groups are defined using the analogy between algebraic objects groups and hyperbolic metric spaces and manifolds. Our research involves the study and use of two very different, yet very natural, classes of subgroups in a hyperbolic group G: normal subgroups and quasiconvex subgroups. Norm...
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ndltd-VANDERBILT-oai-VANDERBILTETD-etd-06212012-1720482013-01-08T17:16:58Z PROPERTIES OF HYPERBOLIC GROUPS: FREE NORMAL SUBGROUPS, QUASICONVEX SUBGROUPS AND ACTIONS OF MAXIMAL GROWTH Chaynikov, Vladimir Vladimirovich Mathematics Hyperbolic groups are defined using the analogy between algebraic objects groups and hyperbolic metric spaces and manifolds. Our research involves the study and use of two very different, yet very natural, classes of subgroups in a hyperbolic group G: normal subgroups and quasiconvex subgroups. Normal subgroups are embedded nicely in G in the classical group theoretic sense, while the quasiconvex subgroups are embedded hyperbolically in G as geometric objects. We prove that if R is a (not necessarily finite) set of words satisfying certain small cancellation condition in a hyperbolic group G then the normal closure of R is free. This result was first presented (for finite set R) by T. Delzant but the proof seems to require some additional argument. New applications are provided, the connection between different small cancellation techniques is studied. One of our main results concerns the existence of highly transitive actions of non-elementary hyperbolic groups (i.e. the actions which are ktransitive for every natural k) on infinite countable sets. The construction of such examples involves limits of quasiconvex subgroups and some quantitative estimates on maximal growth of action. The main corollary is that almost every group admits a highly transitive action with finite kernel on a countable set. As a side-product of our approach we prove that for a non-elementary hyperbolic group G and a quasiconvex subgroup of infinite index H in G there exists g in G such that <H,g> is quasiconvex of infinite index and is isomorphic to the free product of H and <g> if and only if H and E(G) intersect trivially, where E(G) is the maximal finite normal subgroup of G. Alexander Olshanskiy Mark Sapir Mike Mihalik John Ratcliffe Thomas W. Kephart VANDERBILT 2012-06-22 text application/pdf http://etd.library.vanderbilt.edu/available/etd-06212012-172048/ http://etd.library.vanderbilt.edu/available/etd-06212012-172048/ 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 Chaynikov, Vladimir Vladimirovich PROPERTIES OF HYPERBOLIC GROUPS: FREE NORMAL SUBGROUPS, QUASICONVEX SUBGROUPS AND ACTIONS OF MAXIMAL GROWTH |
description |
Hyperbolic groups are defined using the analogy between algebraic objects groups and hyperbolic metric spaces and manifolds. Our research involves the study and use of two very different, yet very natural, classes of subgroups in a hyperbolic group G: normal subgroups and quasiconvex subgroups. Normal subgroups are embedded nicely in G in the classical group theoretic sense, while the quasiconvex subgroups
are embedded hyperbolically in G as geometric objects.
We prove that if R is a (not necessarily finite) set of words satisfying certain small cancellation condition in a hyperbolic group G then the normal closure of R is free. This result was first presented (for finite set R) by T. Delzant but the proof seems to require some additional argument. New applications are provided, the connection between different small cancellation techniques is studied.
One of our main results concerns the existence of highly transitive actions of non-elementary hyperbolic groups (i.e. the actions which are ktransitive for every natural k) on infinite countable sets. The construction of such examples involves limits of quasiconvex subgroups and some quantitative estimates on maximal growth
of action. The main corollary is that almost every group admits a highly transitive action with finite kernel on a countable set. As a side-product of our approach we prove that for a non-elementary hyperbolic group G and a quasiconvex subgroup of infinite index H in G there exists g in G such that <H,g> is quasiconvex of infinite index and is isomorphic to the free product of H and <g> if and only if H and E(G) intersect trivially, where E(G) is the maximal finite normal subgroup of G. |
author2 |
Alexander Olshanskiy |
author_facet |
Alexander Olshanskiy Chaynikov, Vladimir Vladimirovich |
author |
Chaynikov, Vladimir Vladimirovich |
author_sort |
Chaynikov, Vladimir Vladimirovich |
title |
PROPERTIES OF HYPERBOLIC GROUPS: FREE NORMAL SUBGROUPS, QUASICONVEX SUBGROUPS AND ACTIONS OF MAXIMAL GROWTH |
title_short |
PROPERTIES OF HYPERBOLIC GROUPS: FREE NORMAL SUBGROUPS, QUASICONVEX SUBGROUPS AND ACTIONS OF MAXIMAL GROWTH |
title_full |
PROPERTIES OF HYPERBOLIC GROUPS: FREE NORMAL SUBGROUPS, QUASICONVEX SUBGROUPS AND ACTIONS OF MAXIMAL GROWTH |
title_fullStr |
PROPERTIES OF HYPERBOLIC GROUPS: FREE NORMAL SUBGROUPS, QUASICONVEX SUBGROUPS AND ACTIONS OF MAXIMAL GROWTH |
title_full_unstemmed |
PROPERTIES OF HYPERBOLIC GROUPS: FREE NORMAL SUBGROUPS, QUASICONVEX SUBGROUPS AND ACTIONS OF MAXIMAL GROWTH |
title_sort |
properties of hyperbolic groups: free normal subgroups, quasiconvex subgroups and actions of maximal growth |
publisher |
VANDERBILT |
publishDate |
2012 |
url |
http://etd.library.vanderbilt.edu/available/etd-06212012-172048/ |
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AT chaynikovvladimirvladimirovich propertiesofhyperbolicgroupsfreenormalsubgroupsquasiconvexsubgroupsandactionsofmaximalgrowth |
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1716570550692741120 |