Higher rank hyperbolicity and homological isoperimetric inequalities

Poincaré的经典定理Poincaré对偶定理指出，对于闭的、可定向的n维流形M，对所有k，存在k阶上同调与由具有基本类的盖积给出的n-k同调之间的同构.根据群上同调的定义，对于非球面的、闭的、定向的n-流形的基本群也是如此。定义：群G是n-Poincaré对偶群，如果G的系数在整数上的群环中的n次上同调是给定的具有平凡G-模结构的整数，且对于所有G-模N，系数在N中的k次上同调与系数在N中的n-k同调之间存在同构。无论有限表示的Poincaré对偶群是非球面的，闭的，对于n大于或等于3的n-流形，定向n-流形是一个未解决的问题。我们将研究有限表示的Poincaré对偶群。在最近的一篇工作[KL]中，Kleiner和Lang引入了高阶双曲性的概念，它是由同调等周不等式定义的。在Kielak和Kropholler对Poincaré对偶群的研究中，自然出现了略有不同的不同版本[KK]。Kielak和Kropholler证明了任何2-Poincaré对偶群都是曲面的基本群。我们的目的是证明同调等周不等式的Kleiner-Lang定义可以用来代替Kielak-Kropholler的版本。一旦完成，下一步将是研究Poincaré对偶群在Kleiner-Lang定义的边界上的作用，使用他们的高阶双曲性概念，以模仿n-Poincaré对偶群的Kielak-Kropholler证明。这个项目属于EPSRC几何和拓扑研究领域。书目[KL]Bruce Kleiner，Urs Lang高阶双曲性.发明。数学课。221,597-664(2020)。Https://doi.org/10.1007/s00222-020-00955-w[KK]Dawid Kielak，Peter Kropholler(2021年)Poincaré对偶群的等周不等式。美国数学学会论文集，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).