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Title: Finiteness properties and CAT(0) groups
Author: Kropholler, Robert
ISNI:       0000 0004 6496 7325
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2016
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In this thesis, we explore several areas of geometric topology. We first prove that all groups G which fit into a short exact sequence F2 → 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 Z3 subgroups but have subgroups of type FP2 that are not finitely presented. We give examples of groups that are of type Fn-1 not of type Fn and contain no free abelian subgroups of rank > ⌈n3⌉. In the final part of the thesis we examine stable diffeomorphisms of smooth 4-manifolds and place an upper bound on the number of S2× S2 summands required to gain a stable diffeomorphism.
Supervisor: Bridson, Martin Sponsor: EPSRC
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available