Research in Geometry and Topology

University Of Utah, Salt Lake City UT

Investigators

Abstract

The interplay between algebra and geometry is one of the classical themes in mathematics. Traditionally, one studies geometric objects via their symmetries. In geometric group theory the situation is reversed: one starts with a group of symmetries of an algebraic object (for example, another group) and constructs a geometric object with the same symmetries. This research project focuses on the study of symmetries of a free group. Surprisingly, its geometry is to a large degree governed by hyperbolic geometry that goes back to Gauss, Lobachevski, Poincare and others, most recently to Gromov and Thurston. The goal of the project is to better understand this phenomenon. The award includes funds for graduate student support. The study of the large-scale geometry of the outer automorphism group Out(Fn) of the free group Fn has made great strides in recent years, but several fundamental questions are still open. The PI has proposed a strategy for proving the Farrell-Jones conjecture for this group. The strategy is modelled on the proofs of corresponding conjectures for mapping class groups, taking into account the extra complications present in Out(Fn). The steps in the strategy provide concrete questions the PI plans to attack. More specifically, a goal is to prove that Out(Fn) has finite asymptotic dimension, to better understand the boundary of the complex of free factors and its local connectivity, and to isolate the part of large scale geometry of Out(Fn) not governed by hyperbolicity by proving a distance formula for a natural electrification of Out(Fn). In a different direction, the project attempts to construct infinitely many quasi-isometry types of hyperbolic free by cyclic groups with fully irreducible monodromies, providing a phenomenon opposite from the situation for hyperbolic 3-manifolds that fiber over the circle with pseudo-Anosov monodromies. 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.