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: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=737554
• 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.

庞加莱发现的基本群体表明，群体可以捕捉方面的拓扑空间。这种现象是双向的：拓扑空间也可以描述群。这种观点导致了群体呈现，即由生成元和关系给出的描述，使群体成为一组符号串。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：单词问题（两个元素相等吗？），共轭问题（两个元素是共轭的吗？），同构问题（两个群同构吗？）在1950年代，诺维科夫证明了一般的文字问题是不可判定的。因此，对所有群体的系统研究只考虑陈述是不可行的。然而，进一步考虑到度量的一组诱导的演示，以及拓扑结构的演示复合体，已成为一个非常成功的方法，为某些类别的群体，产生解决方案德恩的问题。这是几何群论，研究作为几何对象的群。我的研究计划包括几何群论中的几个相互关联的项目，旨在描述代数结构和寻找算法，由可构造群和JSJ理论的主题统一。可构造群是可以通过沿沿着2-末端子群的连续胶合或合并从平凡群获得的群。JSJ理论（以Johansson和Shalen的名字命名）描述了一个群可以被分解为一个群的混合体的方式。可构造的群相对来说是很容易理解的。本研究计划旨在描述可构造群的拟等距类和可构造类。这一描述是关于这些群体的少数几个悬而未决的问题之一。所提出的方法进行这项调查是一个新的应用JSJ理论准等距刚度。到目前为止，用于证明拟等距刚性结果的技术适用于不同类型的群体。将在计算机上实现的快速算法也将为可构造的组开发。我也将继续建设我的工作同构问题相对双曲群解决共轭问题出（Fn），一个重要的开放问题，在我的领域，经受住了几十年的攻击。所提出的方法的一个重要组成部分是利用的事实，建设组实际上在这个问题中发挥了关键作用。JSJ理论是强大的，但无法实现。我将探讨CAT（0）立方体复形的坍缩如何被用来统一和简化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，这项研究计划将为几何群论的既定概念提供全新的视角，产生意想不到的和深远的应用。