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通过将离散群体视为几何对象来促进对它们的理解。这是Gromov研究群的拟等距性质及其与代数性质的关系的程序的一部分。该提案特别强调作用在非正曲线空间上的群，如双曲群和相对双曲群、小消去群、CAT(0)群等。该建议有三个一般目的：*1.理解在有限呈现子群下闭合的非正曲群的类及其与维度的关系。特别地，为了确定群G的子群结构的性质所蕴含的G-复形的组合条件，我们建议使用代数拓扑学中的技巧，这些技巧在凝聚性和局部拟凸性的研究中没有得到充分的探索。具体地说，L^2 Betti数在相干性和局部拟凸性研究中的应用，以及在轨道范畴上的Bredon模与同调等周不等式的关系。这些都是首席研究员(及其合作者)最近引入的工具，用于研究非正曲群的子群。这些工具的使用在该领域是新颖的，仍有几个方面有待探索。我们期望我们的研究能够为剩余有限性这一领域的悬而未决的问题提供一些启示。*2.推进定义准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展示了这些不变量的代数方法，并利用它得到了关于某些离散群的子群结构的结果。结合离散群的子群的研究，有几个方向可以进一步发展这些不变量的研究。我们期望我们的填充函数的代数方法将揭示同调几何群不变量和粗几何群不变量之间的新的联系。*3.经典的图上的组合博弈可以产生无限图的准等距不变量，从而产生有限生成群的不变量(通过Cayley图)。这些准等距不变量与离散群理论之间的关系大多是未被探索的。我们计划调查这些关系。目前正在进行的工作表明，双曲群的新特征；群分裂与遏制博弈之间的关系；以及顺从性的某些方面似乎与特定的博弈有关。这些研究将在图论博弈论和几何群论家之间架起桥梁。