Intrinsic rigid structure in groups and surfaces

Vanderbilt University, Nashville TN

Investigators

Abstract

Award: DMS 2005368, Principal Investigator: Spencer D. Dowdall This project concerns two foundational branches of mathematics: group theory, and low-dimensional topology/geometry. A group is an algebraic framework that encodes symmetries of objects, such as the rigid motions of space, the configurations of a Rubik's cube, or the symmetries of a molecule. Low-dimensional topology and geometry concern the structure of space itself, like the surface of the earth or the universe we live in, together with inherent geometric features like curvature, distance, and volume. One way in which these topics are linked is through the concept of a parameter space (such as the geometric structures supported on a given object, or the configurations of points in that object) and its symmetry group. This project will focus on two important classes of examples in these contexts: group extensions, which are ways of building new groups out of old, and mapping class groups of surfaces, which are the symmetry groups for parameter spaces of hyperbolic structures on a surface. The purpose of the project is to identify and study key structural features, particularly rigid features that are invariant under perturbation or reparametrization, as well as general features that emerge from random constructions. This project provides research training opportunities for graduate students. Specifically, the project will investigate geometric and dynamical aspects of free group and surface automorphisms and explore how these interact with associated group extensions and moduli spaces of geometric structures. Firstly, the project will study cyclic extensions of surface groups and free groups. This entails the use of veering triangulations to study low-dilatation pseudo-Anosov surface homeomorphims, and the development of a parallel theory of veering triangulations for free-by-cyclic groups that will illuminate features of free group automorphisms. The project will also introduce a new universal attracting tree for free-by-cyclic groups that will be used to encode important algebraic and dynamical invariants. Secondly, the project will study hyperbolicity for extensions of free groups and surface groups. Building on past results, the PI will characterize when free group extensions are hyperbolic, and will study algebraic features of randomly constructed free group extensions. By means of analogy with Kleinian groups, the project will also develop a new theory of geometrically finite subgroups of mapping class groups and connect this to the budding theory of hierarchical hyperbolicity. Thirdly, the project will investigate the moduli space of hyperbolic structures on a surface by quantifying the prevalence of certain types of elements of the mapping class group and by studying shrinking target properties of the geodesic and horocycle flow. Finally, in a view towards more general surfaces, the project will explore aspects of algebraic rigidity in mapping class groups of infinite-type surface and dynamical rigidity for length spectra of billiard-style dynamical systems. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.