The Baum-Connes assembly map and Langlands duality (Joint work with Nick Wright and Roger Plymen) We show that the left hand side of the Baum Connes assembly map for the extended affine Weyl group of a compact semi-simple Lie group is naturally isomorphic to the left hand side for its Langland’s dual. The proof generalise to show that for a finite extension of a free abelian group the Baum-Connes isomorphism can be decomposed as a composition of Poincaré, T- and Fourier dualities.
Topological superrigidity (Joint work with Aditi Kar) Following our earlier work on Bass-Serre splittings for Poincaré duality groups, we discovered a new topological splitting theorem for high dimensional manifolds in the spirit of the classical torus theorem. In the following statement a (T)-manifold is any closed manifold such that the universal cover is an irreducible globally symmetric Riemannian manifold of non-compact type, and which is one of the following: of real rank at least 2; quaternionic hyperbolic space; the Cayley hyperbolic plane.
The topological superrigidity theorem: Let N be a closed (T)-manifold of dimension 2k>5. Given any aspherical topological manifold M of dimension 2k+1 and any π1-injective, continuous function j from N to M, there is a manifold M’ homotopy equivalent to M and an embedded, 2-sided, π1-injective codimension-1 submanifold N' in M’ such that up to homotopy, the map j factors through a finite degree covering p of N'.
When M, N satisfy the Borel conjecture, for example, following the recent work of Bartels and Leuck if M, N are CAT(0) manifolds (or more generally have (CAT(0) or hyperbolic fundamental groups), then the conclusion may be strengthened to deduce that M’=M.The relationship with Waldhausen’s torus theorem is as follows: starting with a π1-injective map one deduces the existence of a π1-injective embedding. However the conclusion in the topological superrigidity theorem is considerably stronger: in the torus theorem the embedded submanifold may be topologically unrelated to the original one whereas here the resulting embedding is homotopy equivalent to the original one. Whether or not M satisfies the Borel conjecture we obtain the following corollary: If M is a 2k+1 manifold with trivial first Betti number and N has a non-trivial Pontryagin number (for example it is quaternionic hyperbolic) then there are no π1--injective maps from N to M. This generalises the known fact that there are no such immersions. When the target manifold M is CAT(0) the hypotheses of the topological superrigidity theorem closely parallel those of the geometric superrigidity theorem and together they give strong constraints on the existence of non-trivial codimension-1 maps of T-manifolds into aspherical manifolds.
Amenability, cohomology and Yu’s Property A (Joint work with Nick Wright, Piotr Nowak 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 Følner'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 have answered this question in several different ways. In the context of discrete metric spaces I, together with Wright and Brodzki, introduced an analogue of bounded cohomology which characterises Yu’s property A in terms of vanishing, and provides the notion of an asymptotically invariant mean for a space. In the context of topological actions we introduced, with Nowak, a class of Banach modules associated to the action such that the vanishing of group cohomology over these coefficients characterises topological amenability of the action. Specialising to the action of a group on its Stone Cech compactification we obtain a characterisaction of C* exactness for the group generalising Johnson’s celebrated characterisation of amenability in terms of vanishing bounded cohomology. An alternative homological view of amenability was provided by Block and Weinberger who showed that for a metric space amenability could be characterised by non vanishing uniformly finite homology. In recent work we dualised the cohomology theory described above to provide a notion of the uf homology of a topological action providing a characterisation of topological amenability in terms of the non vanishing of the first uf homology. This dualisation arose from an earlier study by Brodzki, Niblo and Wright of the relationship between classical bounded cohomology and uniformly finite homology for 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.
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.