EPSRC Reference: 
EP/V002899/1 
Title: 
Boundary representations of nonpositively curved groups 
Principal Investigator: 
Spakula, Dr J 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Sch of Mathematical Sciences 
Organisation: 
University of Southampton 
Scheme: 
Standard Research 
Starts: 
01 July 2021 
Ends: 
18 December 2024 
Value (£): 
367,941

EPSRC Research Topic Classifications: 
Algebra & Geometry 
Mathematical Analysis 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
The proposed research generally fits into the framework of Noncommutative Geometry, in particular research related to the BaumConnes conjecture, analysis on groups, and representation theory. The BaumConnes conjecture connects geometry, topology and algebra. From one point of view, it proposes a way to understand the algebraic topology (Ktheory) of (a part of) the representation space of a group. While it is possible to effectively describe all the representations of (semisimple) Lie groups, this task is impossible for discrete groups in general. Here we propose to construct explicit families of representations for large classes of discrete groups, using geometry (nonpositive curvature) and boundaries. They directly address important questions (Shalom's conjecture), relate to existing approaches to the BaumConnes conjecture, and harmonic analysis on discrete groups. The proposed pathway combines ideas from analytic and geometric group theory, representation theory of Lie groups and random walks.
The philosophy of this project is to capitalise on, and further develop, connections between Geometric Group Theory and Analysis/Noncommutative Geometry. We propose to construct a "compact picture" for (uniformly bounded) representations of prominent classes of nonpositively curved groups.
First, we deal with the case where one can do ``combinatorial harmonic analysis'', i.e. the case of groups acting properly on (finite dimensional) CAT(0) cube complexes.
Second, we distill the main features of the construction and perform it with hyperbolic groups, thus establishing Shalom's conjecture.

Key Findings 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Potential use in nonacademic contexts 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Impacts 
Description 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk 
Summary 

Date Materialised 


Sectors submitted by the Researcher 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Project URL: 

Further Information: 

Organisation Website: 
http://www.soton.ac.uk 