Geodesic currents on free groups and the Outer space

AbstractAward：DMS-0904200首席研究员：Ilya Kapovich自由群的外自同构群是黎曼曲面的模群的一个更复杂和更不容易理解的表亲。 该项目旨在通过分析Culler-Vogtmann外层空间（传统上认为的Teichmuller空间的自由群模拟）与自由群上测地流空间之间的相互作用来研究自由群的外自同构群。测地线电流伊萨闭曲线或共轭类概念的测度论推广。测地流空间是外层空间的天然伙伴，将它们结合起来研究，将为自由群的外自同构群的动力学和几何学提供新的信息。用于这种研究的一个关键工具是所谓的“几何相交形式”在早期的工作中引入的提议者。具体问题要研究的包括：studyingthe几何形状的各种自由群类似物的曲线复杂的，建设领域的不连续性和发展的理论ofconvex co-compactness子群的外部automorphismgroup的自由组，发展理论的广义（或"非交换“）电流，和其他人。 在组合学和计算机科学中，人们积极地寻找基于域而不是整数段的词的理论（例如，自由群的凯莱图的整数格）。在该提案中引入的广义电流理论提供了一个很好的模型，在基于非线性域的单词研究中应该是有用的。该计划旨在进一步发展这一理论，并更详细地研究几何群论和计算机科学之间的相互作用。该项目的目标是研究自由群的外自同构群的动力学和几何学，自由群是最神秘，最有趣和最不了解的天体之一。其中的关键思想是使用一种新的工具，称为“测地线电流”，它允许人们从分析和遍历理论中引入大量的新机器和技术。该项目还将通过广义测地线流的概念探索几何群论和计算机科学之间的相互作用，旨在理解新型的数据结构-即与标准单词不同的单词，它们不是写在直线段上。 该项目的一个重要组成部分包括REU（“本科生研究经验”）活动，这将涉及美国。该研究已被作为一个项目的基础，资助了两年，根据IUC-CNRS institutional collaborative agreement的保护伞下，从UIUC和马赛的研究人员之间的几何群理论的合作研究。国家科学基金会的赠款将用于扩大和加强该项目，使更多的研究小组参与进来，并进一步建立UIUC和CNRS之间的国际研究联系。

AbstractAward: DMS-0904200Principal Investigator: Ilya KapovichThe outer automorphism group of a free group is a morecomplicated and much less well understood cousin of the modulargroup of a Riemann surface. The project aims to study the outerautomorphism group of a free group via analyzing the interactionbetween the Culler-Vogtmann Outer space (the traditionallyconsidered free group analog of the Teichmuller space) and thespace of geodesic currents on a free group. A geodesic current isa measure-theoretic generalization of the notion of a closedcurve or a conjugacy class. The space of geodesic currents turnsout to be a natural companion of the Outer space and studyingthem together should produce substantial new information of thedynamics and geometry of the outer automorphism group of a freegroup. A key tool to be used for such study is the so-called``geometric intersection form'' introduced in the earlier work ofthe proposer. Specific questions to be studied include: studyingthe geometry of various free group analogs of the curve complex,constructing domains of discontinuity and developing a theory ofconvex co-compactness for subgroups of the outer automorphismgroup of a free group, developing a theory of generalized (or``non-commutative'') currents, and others. In combinatorics andcomputer science there is an active search for theories of wordsbased on domains other than an integer segment (e.g. the integerlattice of the Cayley graph of a free group). The theory ofgeneralized currents, introduced in the proposal, provides a goodmodel of this kind that should be useful in the study of wordsbased on non-linear domains. The proposal aims to develop thistheory further and to investigate in more detail the interactionsbetween geometric group theory and computer science that arise inthis context.The goal of the project is to study dynamics and geometry of theouter automorphism group of a free group, one of the mostmysterious, interesting and least understood mathematicalobjects. Among the key ideas is the use of a new tool, called a``geodesic current'', which allows one to bring substantial newmachinery and techniques from analysis and ergodic theory to thesubject. The project will also explore, via the notion ofgeneralized geodesic currents, interactions between geometricgroup theory and computer science, aimed at understanding newtypes of data structures - namely words that, unlike standardwords, are not written on straight line segments. A significantcomponent of the project includes REU (``Research Experience forUndergraduates'') activities, that will involveU.S. undergraduate students in advanced mathematical research andhelp prepare them for future careers in science and technology.The proposed research has served as a basis for a project, fundedfor two years, under the umbrella of the UIUC-CNRS institutionalcollaborative agreement, for collaborative research in geometricgroup theory between researchers from UIUC and Marseille. The NSFgrant will be used to expand and enhance that project, to involveadditional research groups and to further build up institutionalinternational research ties between UIUC and CNRS.