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Analytic problems in Coarse Geometry and Geometric Group Theory

Objective

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.

Field of science

  • /natural sciences/mathematics/pure mathematics/geometry

Call for proposal

FP7-PEOPLE-2013-CIG
See other projects for this call

Funding Scheme

MC-CIG - Support for training and career development of researcher (CIG)

Coordinator

UNIVERSITY OF SOUTHAMPTON
Address
Highfield
SO17 1BJ Southampton
United Kingdom
Activity type
Higher or Secondary Education Establishments
EU contribution
€ 100 000
Administrative Contact
Dewi Tan (Mrs.)