Higher rank hyperbolicity and homological isoperimetric inequalities

高阶双曲性和同调等周不等式

• 批准号: 2785744
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2785744
• 关键词: Higher; rank; hyperbolicity; homological; isoperimetric

The Poincaré duality theorem, a classical theorem by Poincaré, states that for a closed, orientable, n-manifold M, for all k there are isomorphisms between the k-th cohomology and the n-k homology given by the cap product with the fundamental class. By definition of group cohomology, the same holds for the fundamental group of an aspherical, closed, oriented n-manifold. A natural class of groups that arise from this result is the class of Poincaré duality groups.Definition: A group G is an n-Poincaré duality groups if the n-th cohomology of G with coefficients in the group ring over the integers is the integers given with the trivial G-module structure, and for all G-modules N, there are isomorphisms between the k-th cohomology with coefficients in N and the n-k homology with coefficients in N.Whether finitely presented Poincaré duality groups are exactly the fundamental groups of aspherical, closed, oriented n-manifolds is an open question for n greater or equal to 3. We will study finitely presented Poincaré duality groups.In a recent work [KL], Kleiner and Lang introduced a notion of higher rank hyperbolicity defined in terms of homological isoperimetric inequalities. A slightly different version of the inequality appears naturally in the study of Poincaré duality groups by Kielak and Kropholler [KK]. Kielak and Kropholler showed that any 2-Poincaré duality group is the fundamental group of a surface. We aim to show that the Kleiner-Lang definition of homological isoperimetric inequalities can be used in place of Kielak-Kropholler's version. Once this is done, the next step would be to investigate the action of a Poincaré duality group on the boundary that Kleiner-Lang defined, using their notion of higher rank hyperbolicity, in order to mimic the proof of Kielak-Kropholler for n-Poincaré duality groups.This project falls within the EPSRC Geometry and Topology research area. Bibliography[KL] Bruce Kleiner, Urs Lang Higher rank hyperbolicity. Invent. math. 221, 597-664 (2020). https://doi.org/10.1007/s00222-020-00955-w[KK] Dawid Kielak, Peter Kropholler (2021) Isoperimetric inequalities for Poincaré duality groups. Proceedings of the American Mathematical Society, 149 (11), 4685-4698. (doi:10.1090/proc/15596).

Poincaré的古典定理Poincaré二元定理指出，对于封闭的，定向的，N-Manifold M，对于所有K，K-TH的同一个同源物与CAP产品提供的N-K同源性之间都是同构的。根据组的定义，对于非球形，封闭，定向的N-manifold的基本群体来说，相同的群体也是如此。 A natural class of groups that arise from this result is the class of Poincaré duality groups.Definition: A group G is an n-Poincaré duality groups if the n-th cohomology of G with coefficients in the group ring over the integers is the integers given with the trivial G-module structure, and for all G-modules N, there are isomorphisms between the k-th cohomology with coefficients在N和N-K同源物中具有系数的N.最终呈现的庞加莱二重性组正是非球体，封闭，定向的N-manifolds的基本组，这是一个更大或等于3的悬而未决的问题。我们将最终研究的poincaréDualityGroups的研究最终提出的poincaré二元组。等距不平等。在Kielak和Kropholler [KK]的庞加莱二元组研究中，自然而然地出现了略有不同的不平等版本。 Kielak和Kropholler表明，任何2-Poincaré二元组都是表面的基本组。我们的目的是表明，可以使用kleiner-lang的同源等含量不平等的定义，以代替Kielak-Kropholler的版本。完成此操作后，下一步将是调查庞加莱二元组在Kleiner-lang定义的边界上的作用，使用其较高等级双曲线的概念，以模仿N-PoincaréDiality小组的Kielak-Krophollers的证明。参考书目[KL]布鲁斯·克莱纳（Bruce Kleiner），urs lang级别的双曲线。发明。数学。 221，597-664（2020）。 https://doi.org/10.1007/s00222-020-00955-w [kk] Dawid Kielak，Peter Kropholler（2021）PoincaréTualityduality群体的等距不平等。美国数学学会论文集，149（11），4685-4698。 （doi：10.1090/proc/15596）。