Finiteness properties and CAT(0) groups

In this thesis, we explore several areas of geometric topology. We first prove that all groups G which fit into a short exact sequence F<sub>2</sub> → G → Z, act properly, freely, cellularly and cocompactly on CAT(0) square complexes. This shows, among other things, that their cubical di...

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Main Author: Kropholler, Robert
Other Authors: Bridson, Martin
Published: University of Oxford 2016
Online Access:http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.730390
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spelling ndltd-bl.uk-oai-ethos.bl.uk-7303902018-06-12T03:13:08ZFiniteness properties and CAT(0) groupsKropholler, RobertBridson, Martin2016In this thesis, we explore several areas of geometric topology. We first prove that all groups G which fit into a short exact sequence F<sub>2</sub> → G → Z, act properly, freely, cellularly and cocompactly on CAT(0) square complexes. This shows, among other things, that their cubical dimension is equal to their geometric dimension. In the second part, we consider finiteness properties of subgroups of CAT(0) groups. We construct two infinite families of finitely presented subgroups of hyperbolic groups, that are not themselves hyperbolic. We also construct the first examples of CAT(0) groups that do not contain Z<sup>3</sup> subgroups but have subgroups of type FP<sub>2</sub> that are not finitely presented. We give examples of groups that are of type F<sub>n-1</sub> not of type F<sub>n</sub> and contain no free abelian subgroups of rank &gt; &lceil;<sup>n</sup>&frasl;<sub>3</sub>&rceil;. In the final part of the thesis we examine stable diffeomorphisms of smooth 4-manifolds and place an upper bound on the number of S<sup>2</sup>&times; S<sup>2 summands required to gain a stable diffeomorphism.University of Oxfordhttp://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.730390https://ora.ox.ac.uk/objects/uuid:09c91bf4-cac3-4106-9ae6-5644d40a5934Electronic Thesis or Dissertation
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description In this thesis, we explore several areas of geometric topology. We first prove that all groups G which fit into a short exact sequence F<sub>2</sub> → G → Z, act properly, freely, cellularly and cocompactly on CAT(0) square complexes. This shows, among other things, that their cubical dimension is equal to their geometric dimension. In the second part, we consider finiteness properties of subgroups of CAT(0) groups. We construct two infinite families of finitely presented subgroups of hyperbolic groups, that are not themselves hyperbolic. We also construct the first examples of CAT(0) groups that do not contain Z<sup>3</sup> subgroups but have subgroups of type FP<sub>2</sub> that are not finitely presented. We give examples of groups that are of type F<sub>n-1</sub> not of type F<sub>n</sub> and contain no free abelian subgroups of rank &gt; &lceil;<sup>n</sup>&frasl;<sub>3</sub>&rceil;. In the final part of the thesis we examine stable diffeomorphisms of smooth 4-manifolds and place an upper bound on the number of S<sup>2</sup>&times; S<sup>2 summands required to gain a stable diffeomorphism.
author2 Bridson, Martin
author_facet Bridson, Martin
Kropholler, Robert
author Kropholler, Robert
spellingShingle Kropholler, Robert
Finiteness properties and CAT(0) groups
author_sort Kropholler, Robert
title Finiteness properties and CAT(0) groups
title_short Finiteness properties and CAT(0) groups
title_full Finiteness properties and CAT(0) groups
title_fullStr Finiteness properties and CAT(0) groups
title_full_unstemmed Finiteness properties and CAT(0) groups
title_sort finiteness properties and cat(0) groups
publisher University of Oxford
publishDate 2016
url http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.730390
work_keys_str_mv AT krophollerrobert finitenesspropertiesandcat0groups
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