EU Marie Curie Career Integration Grant (FP7-PEOPLE-2013-CIG) #631945

The problem of computing the nuclear dimension of Roe algebras is still out of immediate reach. We have re-focussed onto the problem of recognising operators in Roe algebras, to develop more tools to study the structure of Roe algebras.

This question has appeared in the original work of John Roe and has been open for 20 years. One way to state the question is to ask whether any quasi-local operator (i.e. one with finite ϵ-propagation for all ϵ>0) necessarily belongs to the appropriate Roe algebra. This has been only known to hold for ℤⁿ. In joint work with A. Tikusis (U Toronto) we have developed a new technical trick to approach this problem, and then used a chopping technique to settle this question for proper spaces with Finite Decomposition Complexity. Subsequently, with J. Zhang (U Southampton), we applied the technical trick, together with a new quasi-local version of the Operator Norm Localisation Property, to settle the question for bounded geometry metric spaces with Property A. Please see the list of papers below for the publications; including freely accessible versions on arxiv.

Further work focused on the other side of the above question, namely on the finding the difference between quasi-local operators and Roe algebras. In doing that, we have discovered a coarse-geometric condition on a sequence of graphs, which is strictly weaker than being a sequence of expanders: asymptotic expanders. The beginning of a systematic study of this condition is written up in a preprint jointly with K. Li, P. Nowak (Warsaw) and J. Zhang (Southampton).

This question has been essentially settled by Damian Osajda in 2014, in his work on graphical small cancellation: arxiv link.

Note that Osajda’s method is very different to the approach suggested in this grant. Our attempts to “coarsify” the small cancellation theory led us to consider coarse median spaces of B. Bowditch. In a joint work with N. Wright (U Southampton), we have proved that coarse median spaces of finite rank (+ mild technical assumptions) have Property A (please see the list of papers for the publication).

This is a work in progress, joint with M. Finn-Sell (U Vienna). We have developed some quantitative tools for checking the Baum–Connes conjecture (without coefficients) for some direct limits of groups (in the flavour of lacunary hyperbolic groups). The motivation is naturally that this is the construction used to produce “difficult” groups (Gromov monsters). The current research involves rewriting the assembly map in a sufficiently concrete way to fit the quantitative analysis.

The progress in research in this direction has developed into a project, currently work being done jointly with B. Nica (Montreal) and E. Guentner (Hawaii) {partially building upon our joint work with Nica on strong hyperbolicity, see papers below}. There are no public preprints yet.

• Weighted asymptotic expanders, with Kang Li and Jiawen Zhang

• Quasi-local Algebras and Asymptotic Expanders, with Kang Li, Piotr Nowak and Jiawen Zhang, on arxiv

• Quasi-locality and Property A, with Jiawen Zhang, in Journal of Functional Analysis, 2019, doi arxiv
• Relative commutant pictures of Roe algebras, with Aaron Tikuisis, in Communications in Mathematical Physics, 2019, doi arxiv
• Coarse medians and Property A, with Nick Wright, in Algebraic and Geometric Topology, 2018, doi arxiv
• A metric approach to limit operators, with Rufus Willett, in Transactions of the American Mathematical Society, 2017, doi arxiv
• Strong hyperbolicity, with Bogdan Nica, in Groups, Geometry and Dynamics, 2016, doi arxiv

• At Southampton-Bielefeld Geometric Group Theory Meeting, Southampton, UK (19 Sep 2019): Strong hyperbolicity
• At Special session on recent advances in coarse geometry, AMS Sectional Meeting, Auburn, AL, USA (16 Mar 2019): Quasi-locality and Property A
• At Groups and Geometry in South East, at University College London, UK (28 Oct 2016): Coarse medians and Property A
• At Groups, Dynamics, and Operator Algebras, Queen Mary, University of London, UK (20 Jul 2016): Coarse medians and Property A
• At Workshop on C*-algebras: Geometry and Actions in Münster, Germany (17 Jul 2015): Operator theory and coarse geometry

• At Pure Mathematics Colloquium, University of Sheffield, UK (20 Nov 2019): Quasi-locality and Property A