Geodesic currents on free groups

这一提议的主要主题是通过测地线电流研究自由群自同构的动力学和几何。在自由群的情况下，曲面的Teichmuller空间被由Culler和Vogtmann提出的自由群的所谓‘外层空间’所代替。这一领域的现有工作大多涉及拓扑学的启发方法。‘测地线流’是闭合曲线的测度论模拟。这位提出者在他最近的工作中提出的大地流方法，将研究转移到测度论和符号动力学的背景下，这对一组人来说是“原生的”。这一设置允许使用遍历理论、测度论和概率论的强大机制。该项目将集中在提出者已经取得进展的几个具体主题上。这些问题包括：交形式，自同构的一般伸缩因子，极值熵问题，通过Patterson-Sullivan嵌入得到的外空间的新紧化，随机长谱刚性，理想形式的Whitehead算法等等。有限生成自由群的外自同构群是几何群论中最神秘和最不被理解的对象之一。这个群是紧曲面的模群的一个(更难研究的)“表亲”，它在许多数学分支中扮演着基本的角色，如TeichMuller理论、低维拓扑学、双曲几何、复分析等。这个提议旨在通过利用其他数学分支的新概念，特别是测度论动力学的工具来研究自由群的外自同构群。该提案的主题涉及对无限单词中模式的研究，这在组合学中是重要的，以及在这些单词上的类代换映射的作用。正如提出者的工作所表明的，还有一种联系，特别是通过对怀特黑德算法的研究，与复杂性理论和对各种算法的实际(而不是“最坏情况”或理论上的)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.