My main research interests lie in the area of analytic, topological and geometric  methods in group theory with applications in geometry. I have published several  results concerning the splitting theory of discrete groups with related results  concerning the structure of 3-manifolds. My work with Lawrence Reeves laid the  foundations for the study of group actions on CAT(0) cube complexes which has been applied by others working on the Baum-Connes conjecture and has come to  prominence partly due to the recent results of Haglund and Wise showing that many  important groups act on such spaces. Current and recent projects are outlined below.

Amenability, cohomology and Yu’s Property A (Joint work with Nick Wright and Jacek Brodzki) Amenability appears as one of the fundamental concepts bridging the  

worlds of functional analysis and geometric group theory. A group is  

said to be amenable if it admits an invariant mean on the space of  

bounded functions on the group. While the definition can be extended  

to an abstract metric space using Folner's criterion instead, in the  

absence of a group action the notion is not sufficiently powerful to  

encode the coarse geometry of the space and this is not a particularly  

fruitful approach. In his work on the Novikov conjecture Yu introduced  

an alternative non-equivariant generalisation of amenability, Yu's  

Property A, in which equivariance is replaced by a controlled support  

condition which captures more of the geometry. Spaces satisfying Yu's  

condition also satisfy the Coarse Baum Connes conjecture. There are  

several well known homological characterisations of amenability and  

Higson asked if there are analogous characterisations of property A.  

We will consider coarse generalisations of bounded cohomology and  

Block & Weinberger's uniformly finite homology  which provide a  

positive answer to Higson's question and illuminate the extent to  

which property A provides asymptotic means on a group.

Asymptotic Compression (Joint work with Dr. Sarah Campbell and Rob Francis) In  our paper “Exactness and Hilbert space compression for discrete groups” we showed  that groups acting properly and co-compactly on CAT(0) cube complexes admit a family  of embeddings into Hilbert space such that the asymptotic compression tends to 1. We  are currently investigating generalisations of this result to affine buildings and to study  the compression of embeddings in Lp spaces. Given the result of Arzhantseva and Sapir  that the Hilbert space compression of Thompson’s group is exactly ½ we are also  interested in understanding the Lp compression of this group. Of related interest is the  question of which groups admit a proper (co-compact) action on a product of locally  finite trees. This surprisingly delicate question is being studied by my student Gemma  Holloway. In particular it appears to be unknown whether or not the fundamental group  of a hyperbolic surface admits such an action despite the fact that proper actions on  locally infinite trees are abundant for such groups, as are proper co-compact actions on  CAT(0) cubical complexes. This work is also connected with work of Stalder and  Valette in which they considered the Hilbert space compression for wreath products. 

Embeddings of spaces and groups (Joint work with Dr. Jacek Brodzki and Dr. Nick  Wright) Ozawa proved that for a discrete group G, the uniform Roe algebra is nuclear  if and only if G has Yu’s property A. In this project we are studying the extent to  which this result can be generalized to discrete metric spaces. An important ingredient  of this work is the introduction of a partial translation structure for a space, which  plays the role of the left/right actions of a group on itself. We introduce an invariant  for a countable discrete metric space, which measures how group-like the partial  translation structure is, and have generalized Ozawa’s result to spaces that are  sufficiently group-like. Our invariant takes the value 1 when the space admits an  injective, uniform embedding into a group, so it provides an obstruction to the  existence of such an embedding. To date we have been unable to find any spaces  which do not embed in this way and we continue to investigate whether or not such  embeddings always exist. In passing we also give a short direct proof that the uniform  Roe algebra is an invariant up to Morita equivalence.    

Characterisations of hyperbolicity (Joint work with Prof. Indira Chatterji) We have  introduced a new characterisation of hyperbolicity for geodesic metric spaces in terms  of the geometry of balls. This work, which is connected with Papasoglu’s  characterisation of hyperbolicity in terms of bigons is currently being generalised by  my student Rob Francis who is investigating hyperbolicity for length spaces. One  consequence is that R-trees may be characterised as those geodesic spaces in which  the intersection of any two metric balls is a metric ball. Further questions concerning  characterisations of non-positive curvature in similar terms are being investigated.   

CAT(0) Dimension of Artin Groups (Joint work with Dr. Paul Hanham) Crisp and  Brady showed that there are Artin groups which admit 2-dimensional Eilenberg  MacLean spaces but which do not act properly and co-compactly on any 2-  dimensional CAT(0) complexes. In this project we show that most rank 3 Artin  groups do have 2-dimensional CAT(0) Eilenberg MacLean spaces by constructing  them explicitly. Further work in this area is aimed at settling the question of rigidity  for such actions.   

Measured Wall Spaces and CAT(0) cube complexes Measured wall spaces  simultaneously generalise the notion of an R-tree and that of a CAT(0) cube complex.  They are abstract objects which appear classically in the guise of non-discrete median  algebras and there are several foundational questions to be answered, for example,  what is the correct notion of the geometric realisation of such a space. Valette et al  have shown that for many interesting classes of groups (including countable discrete  groups) the Haagerup property is equivalent to the existence of a proper action of the  group on a measured wall space. It is a long-term ambition to try to understand and  classify such actions. A recent result of Dunwoody on the existence of a JSJ-type  decomposition for group actions on R-trees may be of use here but this work is highly  speculative.

The modified de Bruijn graph for DNA sequencing (Joint work with Dr. Jim  Anderson and Prof. Keith Fox) The de Bruijn graph is used extensively in gene  sequencing. In recent work we introduced a modified graph (technically the quotient  of the de Bruijn graph by the involution arising from the symmetry between the two  strands of a DNA segment) and used it to answer questions about the existence of  universal footprinting templates for use in protein assay.