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The University of Southampton
Mathematical Sciences

Research project: Coarse geometry of non-positively curved spaces

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Non positively curved spaces play a central role in geometry, topology and geometric group theory. Natural examples arise from the study of right angled Artin groups, Coxeter groups, knot groups and limit groups. The work of Connes, Higson, Yu and others has shown that asymptotic properties, as encoded by coarse geometry, play a surprising role in determining their structure, linking the unitary representation theory of their fundamental groups with the topology of the spaces themselves.

In a series of papers we studied the asymptotic Hilbert space compression of CAT(0) cube complexes and showed that they satisfy Yu’s generalized amenability condition of property A. As an interesting side effect we showed that if a group acts properly on a finite dimensional CAT(0) cube complex then stabilisers at infinity are amenable. More recently we have developed a combinatorial theory of coarse median algebras base on Bowditch’s concept of a coarse median space, which opens new possibilities for application of these ideas.

Related research groups

Pure Mathematics
Geometric Group Theory
Noncommutative Geometry

Key Publications

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