Geodesic currents on free groups

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The main theme of this proposal is to study the dynamics andgeometry of automorphisms of free groups via geodesic currents. Inthe free group case the Teichmuller space of a surface is replacedby the so-called ``outer space" of a free group, introduced byCuller and Vogtmann. Most of the existing work in this areainvolved topologically inspired methods. A ``geodesic current" isa measure-theoretic analogue of a closed curve. The geodesiccurrents approach, that is being developed by the proposer in hisrecent work, transfers the investigation into the setting ofmeasure-theoretic and symbolic dynamics, which is ``native'' for afree group. This setting allows the use of the powerful machineryof ergodic theory, measure theory and probability theory. Theproject will concentrate on several specific topics where theproposer has already made progress. These include: theintersection form, generic stretching factors of automorphisms,extremal entropy problems, the new compactification of the outerspace obtained via the Patterson-Sullivan embedding, random lengthspectrum rigidity, ideal version of Whitehead's algorithm, andothers.The outer automorphism group of a finitely generated free group isone of the most mysterious and least understood objects inGeometric Group Theory. This group is a (more difficult to study)``cousin'' of the modular group of a compact surface, that plays afundamental role in many branches of Mathematics, such as theTeichmuller theory, low-dimensional topology, hyperbolic geometry,complex analysis, and so on. The proposal aims to investigate theouter automorphism group of a free group by exploiting new ideasfrom other branches of Mathematics, particularly the tools ofmeasure-theoretic dynamics. The subject of the proposal is related to thestudy of patterns in infinite words, which is important in combinatorics,and the action of substitution-like maps on such words. As the proposer'swork shows, there is also a connection, particularly viathe study of Whitehead's algorithm, with the complexity theory andthe study of practical (as opposed to ``worst-case" or theoretical)behavior of various algorithms.