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

## Table of Contents

## 1 The grant has ended, this page is for archival purposes only

### 1.1 Duration: October 2015 - October 2019

I am partially supported by this grant, which includes funds for **3
years of PhD studentship** to work on one of the objectives below -
please contact me about this if you are interested!

### 1.2 Current status of research (last updated Oct 2019)

For the research background, please see the official abstract below.

Quick summaries of the status of the three objectives follow here:

#### 1.2.1 Objective 1: Roe C*-algebras

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).

#### 1.2.2 Objective 2: Concrete non-exact groups

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).

#### 1.2.3 Objective 3: The Baum–Connes conjecture for limits of hyperbolic groups

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.

#### 1.2.4 Contingency objective: Uniformly bounded representations of hyperbolic groups

### 1.3 Dissemination

#### 1.3.1 In preparation

*Weighted asymptotic expanders*, with Kang Li and Jiawen Zhang

#### 1.3.2 Preprints

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

#### 1.3.3 Papers

*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

#### 1.3.4 Conference talks

- 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*

#### 1.3.5 Other talks

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

## 2 Official Abstract

The overall goal of this proposal is a systematic study of C*-algebras related to coarse structures of metric spaces and discrete groups. The background theme is the interplay between analysis and coarse geometry. It addresses questions relating to exactness of discrete groups and spaces, Roe algebras and the Baum–Connes conjectures.

The interplay between coarse and analytic properties is exemplified by the first objective: computing the nuclear dimension of Roe algebras in terms of asymptotic dimension of the underlying space. Nuclear dimension of C/-algebras is a recent notion that plays a tremendous role in Elliott’s Classification Program of C/-algebras. The motivation for this objective is to systematically study the parallels between C*-algebraic methods of the Classification Program and topological and K-theoretic methods used for Novikov-type conjectures.

The second objective is to expand the techniques from Geometric Group Theory to produce a concrete example of a non-exact group. So far the only such examples are shown to exist by probabilistic methods, after an outline by M. Gromov. As non-exactness is highly relevant for (the potential failure of) the Baum-Connes conjecture, having concrete examples to study would be paradigm-shifting. The idea for such a construction is to generalize the small cancellation theory to a coarse setting.

The last objective is to prove the Baum-Connes conjecture for certain limits of hyperbolic groups, using the quantitative K- theory of Oyono-Oyono and Yu. Since most of the examples of discrete groups with unusual properties (e.g. non-exact) are constructed as such limits, showing that some of them do satisfy the conjecture is desirable.