Growth, Gap, and Geometry

University Of Oklahoma Norman Campus, Norman OK

Investigators

Abstract

Award: DMS 1611758, Principal Investigator: Jing Tao Geometry is a basic area of human research, originating with our visual awareness of the world; it is present implicitly or explicitly in every scientific discipline. The discovery of symmetries and their usefulness is arguably one the greatest achievements of human knowledge. Concretely, one of the most effective ways of understanding and controlling geometric shapes is to exploit their symmetries. The modern approach is to encode this information in something we call the fundamental group. One goal of topologists and geometric group theorists is to understand all possible fundamental groups, classify them, and study their own intrinsic geometric properties. Classically, the most studied geometric objects have been in low dimensions, particularly in dimension two. This is called surface theory and is one of the focus areas of the proposed projects. The principal investigator is also interested in taking the intuitions that one gains from studying the fundamental groups of surfaces and extrapolating them to more complex objects. This area is called geometric group theory, an increasingly active subject within mathematics over the last few decades. The research projects address a broad spectrum of research problems in the general areas of geometric group theory and Teichmuller theory, with interactions with hyperbolic geometry, low-dimensional topology, and dynamics. The specific topics include: (1) Growth tightness of group actions on metric spaces. This notion was first introduced by Grigorchuk and de la Harpe for word metrics and it has connections to the Hopfian property for groups and the Rank Rigidity Conjecture. (2) Stable commutator lengths in right-angled Artin groups and related groups. (3) The geometry of Teichmuller space equipped with the Thurston metric.