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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=658919
• 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推进离散群体的理解，把他们作为几何对象。这是格罗莫夫的计划的一部分，研究拟等距性质的团体和他们的关系，代数性质。该建议特别强调作用于非正弯曲空间的群，如双曲群和相对双曲群、小消去群、CAT（0）群等。该提案有三个总目标：**1。了解非正曲群的类在取群表示下是闭的，以及与维数的关系。特别是，确定G-复形上的组合条件，这意味着群G的子群结构的性质。我们建议使用的技术，代数拓扑尚未充分探讨在研究的连贯性和局部拟凸。具体来说，在相干性和局部拟凸性的研究中使用L^2贝蒂数，以及在同调等周不等式中使用轨道范畴上的Bredon模。这些是主要研究者（及其合作者）最近引入到非正弯曲组的亚组研究中的工具。这些工具的使用在该领域是新颖的，有几个方面仍有待探索。我们希望我们的研究能为剩余有限性这一领域中悬而未决的问题提供一些线索。2.推进定义拟等距不变量的同调方法的研究。 这里的重点是继续发展同调高维Dehn函数理论。最近，Hanlon和PI展示了一种代数方法来处理这些不变量，并使用它来获得某些离散群类的子群结构结果。有几个方向可以进一步发展这些不变量的研究， 离散群的子群。我们期望我们的填充函数的代数方法将揭示同调和粗几何群不变量之间的新联系。3.图上的经典组合游戏有产生无限图的准等距不变量的版本，因此也有产生无限生成群的不变量（通过凯莱图）。这些准等距不变量和离散群理论之间的关系大多是未探索的。我们计划调查这些关系。目前正在进行的工作表明，新的双曲群的特征;分裂的群体和遏制游戏之间的关系;和某些方面的顺从性似乎与特定的游戏。这些调查将创建社区之间的桥梁博弈论的图形，几何群理论。