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Details of Grant 

EPSRC Reference: EP/P010245/1
Title: Optimal geometric structures for hyperbolic groups
Principal Investigator: Mackay, Dr J M
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department: Mathematics
Organisation: University of Bristol
Scheme: First Grant - Revised 2009
Starts: 16 November 2016 Ends: 15 September 2018 Value (£): 98,324
EPSRC Research Topic Classifications:
Algebra & Geometry
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
07 Sep 2016 EPSRC Mathematical Sciences Prioritisation Panel September 2016 Announced
Summary on Grant Application Form
This project studies what are known as Gromov hyperbolic groups. A 'group' is a collection of symmetries of some geometric object, and properties of this geometric object correspond to algebraic properties of the group. One of the most important properties of a geometric object is its curvature: is it flat like a plane, positively curved like a sphere, or negatively curved. This last case, called hyperbolic geometry, is less familiar, but is very important as it is the one which arises most often. In the 1980s, Mikhail Gromov introduced what are now known as Gromov hyperbolic groups as a very large and flexible class of negatively curved groups; in some sense almost every group is Gromov hyperbolic.

Rigidity phenomena are of interest throughout mathematics, where some weak resemblance can be upgraded to a strong similarity. A deeply influential example of this is Mostow's Rigidity Theorem, which says that hyperbolic geometry in dimensions three and higher is rigid: if one finite volume hyperbolic object can be deformed into another such object, then they must have had the exact same shape to begin with. The aim of this project is to understand the geometric structure of Gromov hyperbolic groups and explore surprising rigidity properties which they have.

One goal concerns Gromov hyperbolic groups which are close to being one-dimensional, that is, they are groups of symmetries of spaces which grow at rates arbitrarily close to that of a hyperbolic plane. Sometimes these actually are groups of symmetries of the hyperbolic plane: they have an optimal geometric structure as follows from a deep theorem of Casson-Jungreis and Gabai. This part of the project is concerned with the near-misses which do not have the optimal structure: it aims to show that such groups must be of a very specific algebraic form.

The project will also study how hyperbolic groups can act in a volume-respecting way as is of fundamental importance in ergodic theory and dynamical systems. In this context, I will look at hyperbolic groups which have Kazhdan's property (T). This property, introduced by Kazhdan in the 1960s, is a rigidity property in dynamics which is extremely useful in algebra and geometry, and has applications in computer science through the construction of expander graphs. Hyperbolic groups with property (T) are interesting for their connections with probability: Zuk showed in 2003 that certain random groups almost surely have these properties, and recently such groups have arisen as fundamental groups of random topological objects. The project aims to show that hyperbolic groups with property (T) are particularly well-behaved and have an optimal geometric structure.

Approaching these goals will require the use of a mix of ideas and expertise from algebra, analysis and the geometry of group actions.

Key Findings
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Potential use in non-academic contexts
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Organisation Website: http://www.bris.ac.uk