Critical Exponents and Thermodynamic Formalism on Geometrically Infinite Spaces

Groups are a way of describing symmetries of geometric objects and these symmetries may often be viewed in terms of tessellations; for example, the pictures produced by the artist M C Escher. Tessellations of this type describe how surfaces may be obtained from a so-called uniform covering space via appropriate symmetries. Apart from a small number of exceptions, the resulting surfaces admit geometries with negative curvature, in which the area around any given point looks like a saddle. There is a natural dynamical system associated to this geometry called the geodesic flow and the negative curvature makes this system chaotic. Furthermore, this chaotic behaviour parallels behaviour "at infinity" in the universal covering space. The same type of phenomena occur in higher dimensions and in situations where the geometric structure is "coarse" rather than "smooth".

The groups that appear in this theory have various numerical characteristics associated to them, notably the so-called critical exponent. This can be characterised as describing the growth in the universal cover under the group action of the dynamical complexity of the geodesic flow. It is often equal to the fractal dimension of a potentially complicated set that sits inside the boundary of the universal cover. The principle aim of this project is to understand this quantity as one varies the group in specific ways. In particular, one starts with a fixed group and then considers various subgroups. We expect to establish relations with purely algebraic properties of these subgroups. The theory becomes interesting when the subgroups give rise to spaces which are geometrically infinite, since much of the stanard theory does not apply in this case.

To analyse these problems, we shall investigate symbolic dynamical systems that serve as models for geodesic flows. This approach allows quantities such as the critical exponent to be described by a body of theory called thermodynamic formalism. This had its origins in statistical mechanics but has been applied with great success to to understand chaotic dynamical systems. Our second objective will be the development of this theory for infinite group extensions of symbolic dynamical systems.

A very successful tool in the analysis of geodesic flows and other dynamical systems has been the so-called zeta functions of the systems. These are functions of a complex variable obtained by combining local data given by the periodic orbits of the system. They are defined by convergence of an infinite product in a suitable region of the complex place but important information can be obtained if one can extend their analytic domain, and obtaining such extensions is closely related to thermodynamic formalism. For example, it has been possible to establish very precise asymptotic results for these systems. However, these functions are poorly understood, even from the point of view of definition, for geometrically infinite spaces. Our third aim is to develop and analyse a suitable theory for these functions in this case.

The proposal is concerned with the development of a deeper understanding of geometrically infinite geometries via dynamical methods, and thus will allow the transfer of tools from hyperbolic dynamical systems and their infinite group extensions to geometry and geometric group theory. This will allow new methods to be applied to open problems. A result of this transfer of knowledge will be to deepen the current understanding of several related areas of mathematics. More specifically, the proposed research will be of interest to and will have impact upon researchers in geometric group theory and hyperbolic geometry; hyperbolic dynamics; thermodynamic formalism; infinite ergodic theory; number theory and discrete groups; spectral geometry; random dynamical systems; and arithmetic geometry.

We will achieve academic impact through the dissemination of our results. We will do this by publishing our results in leading specialist journals in dynamical systems, geometry, and on the interface between them, and in general mathematics journals. We will make preprintrs of our work available on the arXiv preprint server. We will also present our work at mathematical meetings in both dynamical systems and geometry.

The proposal will also have impact through the training of the PDRA, anticipated to be the currrent PhD student Ms R Dougall, who will be at a very early career stage at the start of the project. She will be trained in the theory of discrete group actions on negatively curved spaces and the associated ergodic theory, and will gain experience both in research and the organization and management of research, such as taking responsibility for organizing the regular Ergodic Theory & Dynamical Systems research seminar. She will also benefit from gaining experience in mentoring graduate students within the PI's research group.

The primary impact of the proposed work will be in academia. The main purpose of the proposed research is to build and develop a bridge between ideas in dynamical systems theory, geometry and geometric group theory. We expect that our results will allow for mathematicians working in the two areas to collaborate and thus to be able to make significant advances. This will have the effect of improving the UK's knowledge base in mathematics.

The PDRA will be involved in outreach projects in the local area, for example school visits with the Warwickshire chapter of ScienceGrrl, an initiative to encourage STEM participation in schools, and with the Further Mathematics Programme; and organizing workshops and meetings for postgraduates and staff as part of the university based Women in Science network.