Finiteness Properties and CAT(0) groups

Thesis

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 F_n-1 not of type F_n and contain no free abelian subgroups of rank > ⌈ⁿ⁄₃⌉.

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.

Authors: Kropholler, R

• DOI: 10.5287/ora-z5kdg7go9
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford