Relative hyperbolicity and asymptotic invariants of groups

群的相对双曲性和渐近不变量

• 批准号: 0605093
• 负责人: Denis Osin
• 金额: $ 9.33万
• 依托单位: CUNY City College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0605093&HistoricalAwards=false
• 关键词: Relative; hyperbolicity; asymptotic; invariants; groups

This project is devoted to the study of relatively hyperbolicgroups and their applications in low--dimensionaltopology. It consists of three main parts. First part deals withsmall cancellation theory over relatively hyperbolic groups.Namely we intend to use methods elaborated by the PI to constructnew groups with 'exotic' properties that solve some long--standingproblems in the classical theory of groups. The second part ismotivated by the Virtual Haken Conjecture and is based on ageneralization of Thurston's hyperbolic Dehn surgery theory in thecontext of relatively hyperbolic groups. The third part concernsnew results about group embeddings. The ultimate goal here is toobtain a relatively hyperbolic version of the Higman embeddingtheorem. Our approach to these problems is based on a new methodsrelated to asymptotic and combinatorial analysis of van Kampendiagrams.The notion of a relatively hyperbolic group was first proposed byGromov in 1987 to generalize many examples of algebraic andgeometric nature. Since then the theory has been elaborated fromdifferent points of views and now it is one of the most prominentparts of the group theory. The class of relatively hyperbolicgroups encompasses many examples of interest such as fundamentalgroups of finite volume hyperbolic manifolds, geometrically finiteconvergence groups, mapping class groups, fully residually freegroups, etc. Rather than just generalizing the theory ofhyperbolic groups to the relative case, this project is aimed atthe study of new and sometimes unexpected phenomena which areinvisible in the context of ordinary hyperbolicity. The closeinteraction between the theory of relatively hyperbolic groups andgeometry, low-dimensional topology, the theory of dynamicalsystems, provides a rich source of problems and motivations forour project.

本项目致力于研究相对双曲群及其在低维拓扑中的应用。它由三个主要部分组成。第一部分研究相对双曲群上的小消去理论，即我们打算利用PI所阐述的方法来构造具有“奇异”性质的新群，解决经典群论中的一些长期存在的问题。第二部分以虚Haken猜想为基础，在相对双曲群的背景下推广了Thurston的双曲Dehn手术理论。第三部分是关于群嵌入的新结果。这里的最终目标是得到Higman嵌入定理的一个相对双曲的版本。我们对这些问题的研究是基于货车Kampen图的渐近分析和组合分析的一种新方法，相对双曲群的概念是Gromov在1987年首次提出的，它推广了许多代数和几何性质的例子。从那时起，从不同的角度对群论进行了阐述，现在它是群论中最重要的部分之一.相对双曲群的类包括许多感兴趣的例子，如有限体积双曲流形的基本群，几何有限收敛群，映射类群，完全剩余自由群等，而不仅仅是将双曲群理论推广到相对情况下，这个项目的目的是研究新的，有时意想不到的现象，这些现象在普通双曲性的背景下是不可见的。相对双曲群理论与几何学、低维拓扑学、动力系统理论之间的密切联系为我们的项目提供了丰富的问题来源和动力。