Constructible groups and ramifications of JSJ theory

JSJ 理论的可构造群和分支

• 批准号: RGPIN-2019-04319
• 负责人: MathesonTouikan, Nicholas
• 金额: $ 1.75万
• 依托单位: University of New Brunswick
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758885
• 关键词: Constructible; groups; ramifications; JSJ; theory

The discovery by Poincaré of fundamental groups showed that groups could capture aspects of topological spaces. This phenomenon goes both ways: topological spaces can also describe groups. This perspective lead to group presentations, a description given by generators and relations, making a group into a set of strings of symbols. This formalism enabled Dehn in the 1920s to pose the three fundamental algorithmic problems of group theory: the word problem (are two elements equal?), the conjugacy problem (are two elements conjugate?), and the isomorphism problem (are two groups isomorphic?). In the 1950s, Novikov proved that the word problem in general is undecidable. As a consequence, a systematic study of all group considering only presentations is not feasible. However, further taking into account the metric on a group induced by a presentation, as well as the topology of presentation complexes, has been a spectacularly successful approach for certain classes of groups, yielding solutions to Dehn's problems. This is geometric group theory, the study of groups as geometric objects. My research program consists of several interconnected projects in geometric group theory aimed at describing algebraic structure and finding algorithms, unified by the themes of constructible groups and JSJ theory. Constructible groups are groups that can be obtained from the trivial group by successive gluings, or amalgamations, along 2-ended subgroups. JSJ theory (named after Jaco-Shalen and Johansson) describes the ways in which a group can be decomposed as an amalgam of groups. Constructible groups are relatively well-understood. This research program aims to describe quasi-isometry and commensurability classes of constructible groups. This description is one of the few remaining open questions about these groups. The proposed method to pursue this investigation is a novel application of JSJ theory to quasi-isometric rigidity. Until now, the techniques used to prove quasi-isometric rigidity results applied to different kinds of groups. Fast algorithms, that will be implemented on computers, will also be developed for constructible groups. I will also continue building on my work on the isomorphism problem for relatively hyperbolic groups to solve the conjugacy problem in Out(Fn),  an important open problem in my field that has withstood decades of attacks. An important component of the proposed approach is to exploit the fact that constructible groups actually play a key role in this problem. JSJ theory is powerful, but inaccessible. I will explore how the topic of the collapse of CAT(0) cube complexes can be used to unify and simplify many of the basic theorems of JSJ theory, as well as related results, while simultaneously  providing generalizations of JSJ theory. This research program will therefore provide radically new perspectives on well-established concepts of geometric group theory, yielding unexpected and far-reaching applications.

Poincaré对基本群的发现表明，群可以捕捉到拓扑空间的各个方面。这种现象是双向的：拓扑空间也可以描述群。这种视角导致了组表示，这是由生成器和关系给出的描述，使组成为一组符号字符串。这种形式主义使Dehn在20世纪20年代提出了群论的三个基本算法问题：字问题(两个元素是否相等？)、共轭问题(两个元素是否共轭？)和同构问题(两个群是否同构？)。在20世纪50年代，诺维科夫证明了问题这个词总体上是不可判定的。因此，对所有群体进行仅考虑陈述的系统研究是不可行的。然而，进一步考虑由表示诱导的群上的度量以及表示复形的拓扑，对于某些类型的群来说一直是一个非常成功的方法，产生了Dehn问题的解决方案。这就是几何群论，把群作为几何对象来研究。我的研究计划包括几个相互关联的几何群论项目，旨在描述代数结构和寻找算法，以可构群和JSJ理论为主题统一起来。可构群是指由平凡群沿2端子群连续粘合或合并而成的群。JSJ理论(以Jaco-Shalen和Johansson命名)描述了将一个群分解为群的混合体的方法。可构造性群体相对容易理解。本研究的目的是刻划可构群的拟等距类和可公度类。这个描述是关于这些群体的为数不多的悬而未决的问题之一。提出的研究方法是JSJ理论在准等距刚性中的一种新的应用。到目前为止，用于证明准等距刚性结果的技术适用于不同类型的群。将在计算机上实现的快速算法也将被开发用于可构造群。我还将继续在我关于相对双曲群的同构问题的工作的基础上，解决Out(Fn)中的共轭问题，这是我所在领域中的一个重要的公开问题，经受住了数十年的攻击。提出的方法的一个重要组成部分是利用可构建群体实际上在这个问题中发挥关键作用这一事实。JSJ理论很强大，但很难理解。我将探索如何使用CAT(0)立方体复形的折叠这一主题来统一和简化JSJ理论的许多基本定理以及相关结果，同时提供JSJ理论的推广。因此，这一研究计划将为几何群论的成熟概念提供全新的视角，产生意想不到的和深远的应用。