Representations and Rigidity

William Marsh Rice University, Houston TX

Investigators

Abstract

A three-dimensional manifold is locally modelled on the space in which we live, but globally it can be quite different. Such manifolds and their higher-dimensional variants have proved to be astonishingly important, arising in many branches of mathematics and physics. Recently, study of these manifolds has seen a period of remarkable transformation, with progress made by understanding symmetries of the manifolds and the related so-called "covering spaces." These are succinctly encoded by a purely algebraic object known as the fundamental group. The interplay between the fundamental group and the manifold is at the center of this research project. The project aims to better understand more general groups in addition to those arising from manifolds. Recent work has focused on the longstanding question of how one might recognize various classes of groups from "local data" and on trying to reconstitute the group as global data. This project is focused on representations, in various settings, of groups arising in low-dimensional geometry and topology (e.g. free groups, surface groups, and Kleinian groups). The project addresses questions of distinguishing such groups in the setting of finitely-generated residually finite groups by their profinite completions. Remarkably, the case of the free group is still open and continues to provide focus for work in this direction. The project also addresses connections with number theory, algebraic geometry, mapping class groups, and the topology of higher dimensional hyperbolic manifolds. Among other objectives is a better understanding of the arithmetic of canonical components of character varieties of knot groups. 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.