Automorphism Groups and Morse Boundaries

Brandeis University, Waltham MA

Investigators

Abstract

Abstract Award: DMS 1607616, Principal Investigator: Ruth Charney Mathematics is used to model physical systems and to analyze data sets. Increasingly, these models take the form of geometric objects. This project focuses on a class of geometric objects that arise as models in a number of contexts, including robotics and genetics. Geometric properties of these models and their symmetry groups are reflected in properties of the physical systems. Questions may involve either local geometry (what happens near a particular point) or large-scale geometry (the structure of the object viewed from a distance). This project concerns large-scale geometry. By introducing a new notion of a "boundary" for the geometries in question, we are able to identify and quantify certain types of behavior, known as hyperbolic behavior. The study of these boundaries, and their implications for the geometry and symmetry of the objects in question, is the main theme of this project. Boundaries of geodesic metric spaces have played an important role in the study of hyperbolic groups, for example in proving rigidity theorems and dynamical properties. Boundaries can also be defined for CAT(0) spaces, however they are not quasi-isometry invariant, hence do not give a well-defined boundary for a CAT(0) group. In recent work, the principal investigator and Sultan introduced a new boundary, called the Morse boundary, for CAT(0) spaces which is quasi-isometry invariant and behaves more like the boundary of a hyperbolic space. Subsequently, Cordes generalized this construction to obtain a Morse boundary for any proper geodesic metric space. This offers a potentially powerful new tool for studying large classes of groups, such as acylindrically hyperbolic groups. The first part of this project will investigate properties of these boundaries and their implications for the groups in question. The second part of the project concerns automorphism groups of right-angled Artin groups (RAAGs). Automorphism groups of free groups, mapping class groups, and linear groups have many properties in common. Automorphism groups of RAAGs give a context in which to study these commonalities. As Teichmuller space has been central to the study of mapping class groups, Culler and Vogtmann's Outer space has been a fundamental tool in the study of automorphism groups of free groups. Similarly, analogues of the curve complex of a surface have been introduced for free groups and shown to be hyperbolic. The second part of this project seeks to find an analogous Outer Space for RAAGs and to study generalizations of the free factor and free splitting complexes for all RAAGs.