RUI: The Geometry of Arithmetic Locally Symmetric Spaces

Oberlin College, Oberlin OH

Investigators

Abstract

The study of arithmetic groups has its origins in Gauss' work on quadratic forms and has been an active area of research for well over a century. This project will draw upon ideas and techniques from group theory, number theory, geometry and topology in order to study arithmetic groups and the geometry of their associated locally symmetric spaces. Special attention will be paid to arithmetic hyperbolic reflection groups. Reflection groups are ubiquitous in mathematics and, like arithmetic groups, have been studied since the nineteenth century. Poincare's work on hyperbolic reflection groups in dimension 2 played a prominent role in the work of Klein on discrete groups of isometries of the hyperbolic plane, and analogous results for hyperbolic three-space played an important role in Thurston's work on the geometrization of three-dimensional manifolds. Throughout their history hyperbolic reflection groups have been an important source of motivating examples for those studying more general classes of discrete groups of isometries. This project will employ recent advancements in algebraic and analytic number theory in order to further our knowledge of reflection groups. The Principal Investigator's (PI) work as part of this project will study arithmetic locally symmetric spaces in two contexts: (1) the case of hyperbolic reflection groups, and (2) systolic geometry. Seminal work of Vinberg in the 1980s initiated a program to classify those hyperbolic reflection groups which are arithmetic. By bringing together tools from the spectral theory of hyperbolic manifolds, analytic number theory and the arithmetic theory of quadratic forms the PI and his collaborators will make progress towards the complete classification of congruence arithmetic hyperbolic reflection groups. The PI will also study the systolic geometry of arithmetic hyperbolic manifolds. The systole of a manifold is the least length of a closed geodesic on the manifold. One of the biggest open problems concerning the systolic geometry of arithmetic hyperbolic manifolds is the Short Geodesic Conjecture, which asserts that there is a universal positive lower bound for their systoles. As part of this project the PI will prove that the probability that a commensurability class of arithmetic hyperbolic manifolds contains a representative with systole less than any fixed threshold is zero. 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.