Geodesic currents on free groups and the Outer space

摘要奖：DMS-0904200主要研究人员：伊利亚·卡波维奇自由群的外自同构群是黎曼曲面的模群的一个更复杂、更不为人所知的表亲。该项目旨在通过分析库勒-沃格特曼外层空间(传统上被认为是TeichMuller空间的类似自由群)与自由群上测地线电流空间之间的相互作用来研究自由群的外自同构群。测地线流是闭曲线或共轭类概念在测度论上的推广。测地线电流空间变成了外太空的天然伴侣，对它们的共同研究将产生关于自由群的外自同构群的动力学和几何的实质性新信息。用于这类研究的一个关键工具是在提出者的早期工作中介绍的所谓的“几何交集形式”。要研究的具体问题包括：研究曲线复形的各种自由群类似的几何，构造不连续域，发展自由群的外自同构群的子群的凸余紧性理论，发展广义(或‘非交换’)流的理论，等等。在组合学和计算机科学中，有一种基于域而不是整数段(例如，自由群的Cayley图的整数格)的词理论的积极搜索。建议中引入的广义电流理论提供了一种很好的模型，在研究基于非线性域的字时应该是有用的。该提案旨在进一步发展这一理论，并更详细地研究几何群论与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，自由群是最神秘、最有趣、最不被理解的数学对象之一。其中的关键思想是使用一种名为“测地线流”的新工具，它允许人们将大量新的机械和技术从分析和遍历理论带到主题中。该项目还将通过广义测地线电流的概念，探索几何群论和计算机科学之间的相互作用，目的是了解新型数据结构--即与标准词不同的词不是写在直线段上的词。该项目的一个重要组成部分包括REU活动，该活动将让美国本科生参与高级数学研究，并帮助他们为未来的科学和技术职业生涯做好准备。拟议的研究是一个项目的基础，该项目在UIUC-CNRS机构合作协议的框架下资助了两年，用于UIUC和马赛的研究人员在几何群论方面的合作研究。NSF补助金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之间的机构国际研究联系。

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.