Aspects of the coarse geometry of discrete groups

离散群的粗略几何的各个方面

• 批准号: RGPIN-2018-06841
• 负责人: MartinezPedroza, Eduardo
• 金额: $ 1.46万
• 依托单位: Memorial University of Newfoundland
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=714500
• 关键词: Aspects; coarse; geometry; discrete; groups

The proposed research program aims to advance the understanding of discrete groups by regarding them as geometric objects. This is part of Gromov's program of studying quasi-isometry properties of groups and their relations to algebraic properties. The proposal has particular emphasis on groups acting on non-positively curved spaces as hyperbolic and relatively hyperbolic groups, small cancellation groups, CAT(0) groups, among others. The proposal has three general objectives: 1. To understand the classes of nonpositively curved groups that are closed under taking finitely presented subgroups and the relation to dimension. In particular, to identify combinatorial conditions on G-complexes implying properties of the subgroup structure of the group G. We propose to use techniques from algebraic topology that have not been fully explored in the study of coherence and local quasiconvexity. Specifically, the use of L^2 Betti numbers in the study of coherence and local quasiconvexity, and the use of Bredon modules over the orbit category in relation with homological isoperimetric inequalities. These are tools that the Principal Investigator (and its collaborators) have recently introduced to the study of subgroups of non-positively curved groups. The use of these tools is novel in the area and several aspects remain to be explored. We expect our investigations to shed some light into outstanding questions in the area as residual finiteness. 2. To advance the study of homological approaches to define quasi-isometry invariants. The emphasis here is to continue to develop the theory of homological higher dimensional Dehn functions. Recently, Hanlon and the PI exhibited an algebraic approach to these invariants and used it to obtain results on the subgroup structure of certain classes of discrete groups. There are several directions to further develop the study of these invariants in connection with the study of subgroups of discrete groups. We expect that our algebraic approach to filling functions will reveal new connections between homological and coarse geometric group invariants. 3. Classical combinatorial games on graphs have versions that yield quasi-isometry invariants of infinite graphs, and hence invariants of finitely generated groups (via Cayley graphs). The relation between these quasi-isometry invariants and the theory of discrete groups is mostly unexplored. We plan to investigate these relations. Current work in progress suggests new characterizations of hyperbolic groups; relations between splittings of groups and containment games; and certain aspects of amenability seem to be related to particular games. These investigations will create bridges between the community in game theory on graphs, and geometric group theorists.

拟议的研究计划旨在通过将它们作为几何对象来提高对离散群体的理解。这是Gromov研究群体的准等级特性及其与代数特性的关系的一部分。该提案特别强调了作用于非物质弯曲空间的组，例如双曲和相对双曲线组，小取消组，CAT（0）组等。该提案有三个一般目标： 1。了解在有限呈现的亚组和维度的关系下关闭的非阳性弯曲组的类别。特别是，为了确定G型复合物的组合条件，暗示了G组的亚组结构的特性。我们建议使用来自代数拓扑的技术，这些技术在相干性和局部Quasiconvexity研究中尚未得到充分探索。具体而言，在相干性和局部准蔬菜的研究中使用l^2 betti数，以及与轨道类别相对于同源等等的不等式而言，在轨道类别上使用了布雷登模块。这些是主要研究人员（及其合作者）最近引入非弯曲组亚组的工具。这些工具的使用在该区域是新颖的，还有几个方面待探索。我们希望我们的调查能够以剩余的有限态度阐明该地区的出色​​问题。 2。促进对定义准偶像分析不变的同源方法的研究。 这里的重点是继续发展同源较高的DEHN功能的理论。最近，Hanlon和PI对这些不变式表现出了代数方法，并将其用于获得某些离散组的亚组结构的结果。有几个方向可以进一步开发与离散组亚组的研究有关的这些不变的研究。我们预计我们的代数方法填充功能将揭示同源和粗几何组不变的新联系。 3。图上的经典组合游戏具有无限图的准代码不变的版本，因此产生了有限生成的组的不变式（通过Cayley图）。这些准偶然的不变性与离散组理论之间的关系大多未开发。我们计划调查这些关系。当前正在进行的工作表明双曲线群的新特征。团体分裂与遏制游戏之间的关系；不舒适性的某些方面似乎与特定游戏有关。这些调查将在游戏理论上的社区与几何群体理论家之间建立桥梁。