Geodesic currents on free groups

本文的主要目的是通过测地流来研究自由群自同构的动力学和几何学。在自由群的情况下，表面的Teichmuller空间被自由群的所谓“外层空间”所包围，由Culler和Vogtmann引入。现有的大多数工作在这方面involved拓扑启发的方法。“测地电流”伊萨封闭曲线的测量理论模拟。 在最近的工作中，提出者正在发展的大地电流方法，将研究转移到测量理论和符号动力学的背景下，这对自由群来说是“本土的”。这种设置允许使用强大的机器遍历理论，测度理论和概率论。该项目将集中在几个具体的主题，其中投标人已经取得了进展。其中包括：交形式、自同构的一般伸缩因子、极值熵问题、通过Patterson-Sullivan嵌入得到的外层空间的新紧化、随机长度谱刚性、Whitehead算法的理想版本等等。该集团紧曲面的模群的“外自同构群”（较难研究），它在Teichmuller理论、低维拓扑、双曲几何、复分析等数学分支中起着基础性的作用.该建议旨在利用数学其他分支的新思想，研究自由群的外自同构群，特别是测量理论动力学的工具。该提案的主题涉及无限单词中模式的研究，这在组合学中很重要，以及替代类映射对这些单词的作用。正如提出者的工作所表明的那样，还有一种联系，特别是对怀特海算法的生动研究，与复杂性理论和对各种算法的实际（而不是“最坏情况”或理论）行为的研究。

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.