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Analytic problems in Coarse Geometry and Geometric Group Theory

Final Report Summary - COARSE ANALYSIS (Analytic problems in Coarse Geometry and Geometric Group Theory)

The overall goal of the project was 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 aims to address questions relating to geometric group theory, Roe algebras and variants of the Baum-Connes conjectures.

The envisioned impact is primarily scientific, pushing the state-of-the-art further in the field and stimulating new research.

The work on this project has allowed the researcher to continue existing collaborations (R. Willett, B. Nica) and establish new ones: locally with J. Zhang and N. Wright, internationally within EU with M. Finn-Sell, K. Li and P. Nowak, and world-wide with A. Tikuisis and E. Guentner.

We take on the concrete objectives one by one: the first objective was to compute the nuclear dimension of Roe algebras, with the expectation that it equals the asymptotic dimension of the underlying space. This task has seen progress, but it is not yet complete, as the problem appears substantially more difficult than anticipated. The work with A. Tikuisis (U Toronto, Canada), and subsequently with J. Zhang (Southampton) provides a new tool, useful for this goal: a new way of deciding whether an operator belongs to the Roe algebra – the so-called ‘quasi-locality’. To the best of our knowledge, this is state-of-the-art about this question.
Further investigation (jointly with K. Li and P. Nowak (both IMPAN, Warsaw) and J. Zhang), we have related quasi-locality of certain operators to new coarse-geometric shape: asymptotic expanders, and their weighted versions. This work has resulted in 4 papers (2 published, 1 submitted, 1 in preparation) in leading journals.

The second objective was to prove the Baum-Connes conjecture for certain limits of hyperbolic groups. Here, the collaboration with M. Finn-Sell (Vienna) continues. We have made progress in the right direction: We are
in process of working out an explicit formula for the assembly map for
hyperbolic groups. Obtaining the correct bounds on the quantities involved
will enable us to utilize the quantitative K-theory framework of Oyono-Oyono and Yu to obtain the result (this part has been already worked out).
Within this theme, we have started a new project with E. Guentner and B. Nica on uniformly bounded representations on boundaries of groups acting on CAT(0) cubical complexes (partially building on the work with B. Nica on strong hyperbolicity). This work resulted in 1 published paper so far.

The third objective of the project was to produce concrete examples of non-exact groups. While this question was resolved in 2014 by another researcher (D Osajda, Wroclaw), the technique proposed in this project has proved fruitful for further research: in collaboration with N. Wright (Southampton), we have proved a result about coarse median structures of B. Bowditch. This is a successful example of applying coarse geometric techniques and expertise in geometric group theory. This work has resulted in 1 published paper.

The project website is at http://www.personal.soton.ac.uk/js2m12/cig-coarse-analysis/