Curriculum vitae:
My CV can be downloaded here.
Current work and research interests:
In 2005 I was awarded an EPSRC Postdoctoral Fellowship and have joined the group at Southampton to work with Professor Graham Niblo, Dr Jacek Brodzki and Dr Nick Wright. I am also working on a slightly separate subject in collaboration with Prof Robertson from the University of Newcastle upon Tyne.
My research is centred around Yu's Property A and related properties such as exactness, Hilbert space compression and uniform embeddability. Yu’s notion of Property A for a discrete metric space has been shown to hold for hyperbolic groups, amenable groups, free groups, and groups acting on CAT(0) cube complexes. Property A links together many complex geometric and analytic properties. For example it implies that a space is uniformly embeddable in Hilbert space but is also equivalent to more analytic properties such as the group admitting an amenable action on some compact space. Remarkably, it has also been shown to imply both the Novikov and the Coarse Baum Connes conjecture making this property of considerable use and international importance.
As my own research has shown, alternative formulations of Property A can be used to extend many known results to much wider classes of groups and spaces. These formulations include measure theoretic properties, amenable actions, a characterization of weak amenability by Ozawa and Hilbert space compression. I am currently studying some large classes of groups in order to show that these satisfy Property A by both generalising previous results and using the afore mentioned equivalent properties to Property A. In particular, I am concentrating on groups which have either strong geometric or growth properties. These include for instance groups acting on buildings. Another particularly interesting group I am looking at is Thompson's group F which is the focus of some very exciting international research.
[3] S.J.Campbell, Exactness for free and amenable groups by construction of Ozawa kernels, Bull. London Math. Soc.38 (2006) 441-446.
Download here.
[4] S.J.Campbell, G.A.Niblo, Hilbert space compression and exactness of discrete groups, Journal of functional analysis 222, 2005, pp 292-305.
Download here.
Thesis:
I worked as a PhD student under the supervision of Dr Graham Niblo, studying the geometry of group actions on CAT(0) cube complexes. The main result of the thesis was to prove that groups acting properly and cocompactly on CAT(0) cube complexes are exact. Other results included using a new more geometric method to show that groups such as amenable and free groups are exact.
I successfully defended my thesis entitled "A study of exactness for discrete groups" on 16 September 2005.
Both the Abstract and the full Thesis are available in pdf format.
Talks and conferences:
11/04/06 Newcastle BMC 2006, group theory splinter talk: Yu's Property A and CAT(0) cube complexes. Slides
17/03/06 Southampton Pure Maths Colloquium: Some constructions of Ozawa kernels. Slides
23/05/05 Southampton Pure Maths Colloquium: Hilbert space compression and CAT(0) cube complexes. Slides
24/05/04 Invited speaker at Oxford K-Theory day: Ozawa kernels and amenable groups. Slides
27/03/04 National Postgraduate Conference at Warwick: Hilbert space compression and free groups. Slides
22/09/03 Pure group seminar, Southampton University: Exactness and the integers.
18/01/03 Postgraduate pure group seminar, Southampton University: Kazhdan's property T.
Last modified February 2008.