Principal Investigator: Ilya Kapovich
The outer automorphism group of a free group is a more complicated and much less well understood cousin of the modular group of a Riemann surface. The project aims to study the outer automorphism group of a free group via analyzing the interaction between the Culler-Vogtmann Outer space (the traditionally considered free group analog of the Teichmuller space) and the space of geodesic currents on a free group. A geodesic current is a measure-theoretic generalization of the notion of a closed curve or a conjugacy class. The space of geodesic currents turns out to be a natural companion of the Outer space and studying them together should produce substantial new information of the dynamics and geometry of the outer automorphism group of a free group. A key tool to be used for such study is the so-called ``geometric intersection form'' introduced in the earlier work of the proposer. Specific questions to be studied include: studying the geometry of various free group analogs of the curve complex, constructing domains of discontinuity and developing a theory of convex co-compactness for subgroups of the outer automorphism group of a free group, developing a theory of generalized (or ``non-commutative'') currents, and others. In combinatorics and computer science there is an active search for theories of words based on domains other than an integer segment (e.g. the integer lattice of the Cayley graph of a free group). The theory of generalized currents, introduced in the proposal, provides a good model of this kind that should be useful in the study of words based on non-linear domains. The proposal aims to develop this theory further and to investigate in more detail the interactions between geometric group theory and computer science that arise in this context.
The goal of the project is to study dynamics and geometry of the outer automorphism group of a free group, one of the most mysterious, interesting and least understood mathematical objects. Among the key ideas is the use of a new tool, called a ``geodesic current'', which allows one to bring substantial new machinery and techniques from analysis and ergodic theory to the subject. The project will also explore, via the notion of generalized geodesic currents, interactions between geometric group theory and computer science, aimed at understanding new types of data structures - namely words that, unlike standard words, are not written on straight line segments. A significant component of the project includes REU (``Research Experience for Undergraduates'') activities, that will involve U.S. undergraduate students in advanced mathematical research and help prepare them for future careers in science and technology. The proposed research has served as a basis for a project, funded for two years, under the umbrella of the UIUC-CNRS institutional collaborative agreement, for collaborative research in geometric group theory between researchers from UIUC and Marseille. The NSF grant will be used to expand and enhance that project, to involve additional research groups and to further build up institutional international research ties between UIUC and CNRS.
