Geodesic currents on free groups

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

The main theme of this proposal is to study the dynamics and geometry of automorphisms of free groups via geodesic currents. In the free group case the Teichmuller space of a surface is replaced by the so-called ``outer space" of a free group, introduced by Culler and Vogtmann. Most of the existing work in this area involved topologically inspired methods. A ``geodesic current" is a measure-theoretic analogue of a closed curve. The geodesic currents approach, that is being developed by the proposer in his recent work, transfers the investigation into the setting of measure-theoretic and symbolic dynamics, which is ``native'' for a free group. This setting allows the use of the powerful machinery of ergodic theory, measure theory and probability theory. The project will concentrate on several specific topics where the proposer has already made progress. These include: the intersection form, generic stretching factors of automorphisms, extremal entropy problems, the new compactification of the outer space obtained via the Patterson-Sullivan embedding, random length spectrum rigidity, ideal version of Whitehead's algorithm, and others. The outer automorphism group of a finitely generated free group is one of the most mysterious and least understood objects in Geometric Group Theory. This group is a (more difficult to study) ``cousin'' of the modular group of a compact surface, that plays a fundamental role in many branches of Mathematics, such as the Teichmuller theory, low-dimensional topology, hyperbolic geometry, complex analysis, and so on. The proposal aims to investigate the outer automorphism group of a free group by exploiting new ideas from other branches of Mathematics, particularly the tools of measure-theoretic dynamics. The subject of the proposal is related to the study of patterns in infinite words, which is important in combinatorics, and the action of substitution-like maps on such words. As the proposer's work shows, there is also a connection, particularly via the study of Whitehead's algorithm, with the complexity theory and the study of practical (as opposed to ``worst-case" or theoretical) behavior of various algorithms.