Higher rank hyperbolicity and homological isoperimetric inequalities

poincar<e:1>对偶定理是poincar<e:1>的一个经典定理，它指出对于一个闭合的、可定向的n流形M，对于所有k，在第k个上同构和由与基本类的帽积给出的n-k个同构之间存在同构。通过群上同调的定义，非球、封闭、取向n流形的基群也成立。由这个结果产生的群的一个自然类别是庞卡罗对偶群。定义:如果群环上系数G的第N次上同调是用平凡G模结构给出的整数，并且对于所有G模N，在N上系数的第k次上同调与N上系数的第N -k次上同调之间存在同构，那么群G就是N - poincar<e:1>对偶群。有向n流形对于n大于等于3是一个开放的问题。我们将学习有限表示的poincarcars对偶群。在最近的一篇论文[KL]中，Kleiner和Lang引入了一个用同调等周不等式定义的高阶双曲的概念。在Kielak和Kropholler [KK]对poincar<s:1>对偶群的研究中，自然出现了一个略有不同的不平等版本。Kielak和Kropholler证明了任何2- poincar<s:1>对偶群都是曲面的基本群。我们的目的是证明Kleiner-Lang对同调等周不等式的定义可以用来代替Kielak-Kropholler的定义。一旦完成，下一步将是研究一个庞卡罗莱对偶群在Kleiner-Lang定义的边界上的作用，使用他们的高阶双曲的概念，以模仿Kielak-Kropholler对n-庞卡罗莱对偶群的证明。该项目属于EPSRC几何和拓扑研究领域。参考文献[l] Bruce Kleiner， Urs Lang。发明。数学。21, 597-664（2020）。https://doi.org/10.1007/s00222-020-00955-w[KK] david Kielak, Peter Kropholler (2021) poincar<s:1>对偶群的等周不等式。数学学报，149(11),4685-4698。(doi: 10.1090 / proc / 15596)。

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).