LEAPS-MPS: Commensurations and Automorphisms of Free Groups via Topological Models

A group is a mathematical structure that encodes the symmetry of an object or system. Groups appear in a broad range applied contexts including particle physics, encryption systems, crystallography, linguistics, and phylogenetics. As a mathematical object, a group itself is highly symmetric, and the symmetry group of a group is an important and well-studied part of group theory. Further insight into a group can be obtained from its partial symmetries; the set of partial symmetries also form a group. Presently, little is known about the partial symmetry groups outside of some specific contexts. In applied contexts, it is common to start with an object and determine its symmetry group; in contrast, mathematicians understand an abstract group by investigating the various objects that the group is the symmetries of. The PI has introduced a new object whose symmetries are the partial symmetries of a given group. This project will investigate the partial symmetries of some particular groups of broad mathematical interest using the PI's new model object. In addition, this project will use a separate new model due to the PI to address a long-standing open question about the symmetry group of a free group---free groups are basic building blocks of groups. The project will incorporate undergraduate and graduate trainees, providing concentrated STEM workforce development and support a regional seminar connecting students and researchers from local institutions including San José State University, UC Berkeley, and UC Santa Cruz. Automorphism groups are central examples in geometric group theory. The abstract commensurator of a group is a natural relaxation and appears in the study of rigidity phenomena of groups (such as Margulis’ arithmeticity theorem). The first aim of this project develops the theory of full solenoids as a topological model of an abstract commensurator of the group, previously introduced by the PI. In collaboration with Studenmund, the PI will construct a moduli space of full solenoids over graphs and use this space to analyze the abstract commensurator of a free group (by analogy with the space of conformal structures on the disk). The PI will demonstrate the effectiveness of full solenoids by computing the abstract commensurators of several families of group using a common proof method, in collaboration with Pengitore. The second aim of this project will approach the long-standing open question of whether all finite sets of elements the outer automorphism group of a free group have uniformly short independent words. Specifically, the PI will prove that sets of polynomially-growing outer automorphisms have this property. These efforts will be complemented by undergraduate and graduate student projects exploring many of the finitary structures that appear in the course of the investigation. 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.