Research in Geometry and Topology

University Of Utah, Salt Lake City UT

Investigators

Abstract

Abstract Award: DMS 1607236, Principal Investigator: Mladen Bestvina 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 proposal 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, Lobacevski, Poincare and others, most recently to Gromov and Thurston. The goal of the project is to better understand this phenomenon. The study of the large-scale geometry of the outer automorphism group of a free group on n generators, usually denoted Out(F_n) has made great strides in recent years, but several fundamental questions are still open. The principal investigator has proposed a strategy for proving the Novikov and Farrell-Jones conjectures for this group. The strategy is modeled on the proofs of corresponding conjectures for mapping class groups, taking into account the extra complications present in Out(F_n). The steps in the strategy provide concrete questions the PI plans to attack. More specifically, a goal is to prove that Out(F_n) has finite asymptotic dimension, to better understand the boundary of the complex of free splittings, and to study the extent to which the orbit map from Out(F_n) to a suitable product of projection complexes fails to be a quasi-isometric embedding. Additional questions include characterization of convex cocompactness in mapping class groups, uniform hyperbolicity of standard Out(F_n)-complexes, and cubing the pants complex.