Relative hyperbolicity and asymptotic invariants of groups

本课题主要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。第一部分讨论相对双曲群的小抵消理论。也就是说，我们打算使用PI所阐述的方法来构造具有“奇异”性质的新群，以解决经典群理论中一些长期存在的问题。第二部分以虚拟哈肯猜想为动力，基于Thurston的双曲Dehn手术理论在相对双曲群背景下的推广。第三部分是关于群体嵌入的新研究成果。这里的最终目标是获得希格曼嵌入定理的一个相对双曲的版本。我们解决这些问题的方法是基于van kampen图的渐近和组合分析的一种新方法。相对双曲群的概念最初是由gromov在1987年提出的，用于推广代数和几何性质的许多例子。从那时起，这一理论从不同的角度得到了阐述，现在它是群论中最突出的部分之一。相对双曲群包含了许多有趣的例子，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。这个项目不仅仅是将双曲群的理论推广到相关的情况，而是旨在研究在普通双曲的背景下看不见的新现象，有时是意想不到的现象。相对双曲群理论与几何、低维拓扑、动力系统理论之间的密切互动，为我们的项目提供了丰富的问题来源和动力。

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.