Higher rank hyperbolicity and homological isoperimetric inequalities

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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).