Hierarchy Theory for Automorphism and Outer Automorphism Groups of Free Groups

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The study of symmetry dates to ancient times - classic theorems on Platonic solids are found in Euclid's "Elements." Medieval artisans decorated the walls of the Alhambra palace with beautiful depictions of planar symmetries also known, in modern language, as 2-dimensional crystallographic groups. There is great power in abstraction, and the study of symmetry leaped forward in the mid 19th century when mathematicians formulated the abstract concept of a group. These advances enabled a flowering of applications to science, such as the use of 3-dimensional crystallographic groups to explore the atomic structure of crystalline arrays. Modern geometric group theory is a vast abstraction of the study of crystallographic groups. The research accomplished under this award will advance our understanding of geometric group theory by focusing on a class of groups at the current frontier of knowledge, namely the automorphism and outer automorphism groups of free groups. These groups encode symmetries and other relations amongst networks which mathematicians call graphs. Work under this award will involve construction of new models of relations and symmetries amongst graphs, leading to the discovery and proof of new theorems in geometric group theory. Some of this work will be carried out by graduate students under the advisement of the principal investigator, and mathematical monographs aimed at beginning graduate students will be produced to aid in their training and for use by the broader mathematical community.Research activity under this award (expected to be joint work with Michael Handel of CUNY) will focus on studying the large scale geometry of the groups Aut(F) and Out(F), the automorphism and outer automorphism groups of a free group F of finite rank. In large scale geometry one studies a group by its actions on various geometries, focusing not on small scale features of those geometries, but instead on differences and commonalities of large scale features. Under this award Aut(F) and Out(F) will be studied by developing hierarchy theories for these groups, in analogy to hierarchy theories used so successfully to study the large scale geometry of mapping class groups of surfaces. Constructing a hierarchy theory requires, at a minimum, actions of the group and various of its subgroups on hyperbolic geodesic metric spaces. Some things are known already about a hierarchy for Out(F), including actions representing the top of a hierarchy. But there remain many large gaps in our knowledge of this topic. To plug gaps in the middle of an Out(F) hierarchy, one strategy to be pursued under this award will be constructing actions representing the top of Aut(A) hierarchies for free factors A of F. Another strategy will be constructing actions that encode Dehn twist behavior, needed to plug gaps at the bottom of an Out(F) hierarchy. Medium term applications are expected to reveal geometric properties of word and conjugacy problems for Out(F). Longer term applications may include shedding light on deep conjectures about finite asymptotic dimension and quasi-isometric rigidity of Out(F). Graduate work done under this award is expected to extend hierarchy methods beyond outer automorphism groups of free groups to include broader categories of outer automorphism groups.