Optimal geometric structures for hyperbolic groups

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.

This work will benefit researchers in the areas of geometric group theory, analysis on metric spaces and dynamics of group actions. It will do so by bringing together ideas from these different areas in order to open up new connections and provide new tools for future research.

The groups studied in this project are of interest both inside and outside mathematics. One of the earliest applications of Kazhdan's property (T) was Margulis' construction of explicit families of expander graphs. These are graphs which are very highly connected, and so serve as good models for a robust computer network, for example. Hyperbolic groups with property (T) arise in probability as generic groups, and also as fundamental groups of random simplicial complexes. Such random geometric objects are of interest in topological data science and statistics. In this area, one looks for topological features of large data sets which persist under varying forms of measurement; to help quantify which features are relevant it is potentially useful to better understand generic features of random objects. This project will show that such fundamental groups admit an optimal geometric structure, and contact will be made with experts in this field to explore potential applications of this phenomenon.

Beyond pure mathematics, the economic contributions of mathematical science research to the UK are enormous. The 2012 Deloitte report for EPSRC indicated that around 16% of UK Gross Value Added, and over 2.8 million jobs are in this area. As this same report acknowledges, in pure mathematics often the economic benefits are seen further down the road, and in unexpected ways, so strengthening the UK's research base in this area is a wise long-term investment. For example, coarse geometric methods have recently been used in topological data analysis. The proposed research will further strengthen the UK's position as a world leader in geometric group theory. Moreover, it will reinforce academic links between the University of Bristol and Universite Paris-Sud, as well as forge new connections with the fast-developing mathematical community in Uruguay.