Higher rank hyperbolicity and homological isoperimetric inequalities

庞加莱对偶定理是庞加莱的一个经典定理，指出对于一个封闭的、可定向的n-流形M，对于所有k，k阶上同调和n-k同调之间存在同构，这是由与基本类的帽积给出的。根据群上同调的定义，这同样适用于非球面的、封闭的、定向的n-流形的基本群。由这个结果产生的一个自然群类是庞加莱对偶群类。定义：一个群G是一个n-Poincaré对偶群，如果G的系数在整数群环上的n阶上同调是具有平凡G-模结构的整数，且对所有G-模N，系数在N中的k阶上同调与系数在N中的n-k同调之间存在同构。定向n-流形对于n大于或等于3是一个开放问题。在最近的工作[KL]中，Kleiner和Lang引入了用同调等周不等式定义的高阶双曲性的概念。一个稍微不同的版本的不平等自然出现在研究庞加莱对偶群Kielak和Kropholler [KK]。Kielak和Kropholler证明了任何2-Poincaré对偶群都是曲面的基本群。我们的目的是表明，Kleiner-Lang定义的同调等周不等式可以用来代替Kielak-Kropholler的版本。一旦完成了这一步，下一步将是调查的行动庞加莱对偶群的边界上，克莱纳朗定义，使用他们的概念，更高的秩双曲，以模仿证明Kielak-Kropholler的n-Poincaré duality groups.This项目属于福尔斯EPSRC几何和拓扑研究领域。参考书目[KL]布鲁斯克莱纳，Urs Lang Higher rank hyperbolicity.发明。math.221，597-664（2020）。https：//doi.org/10.1007/s00222-020-00955-w [KK] Dawid Kielak，Peter Kropholler（2021）Poincaré对偶群的等周不等式。Proceedings of the American Mathematical Society，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).