Lawrence Reeves, Peter Scott, Gadde Swarup
Analogues of JSJ decompositions were developed for Poincaré duality pairs in [19]. These decompositions depend only on the group. Our focus will be on describing the edge splittings of these decompositions more precisely. We use our results to compare these decompositions with two other closely related decompositions.
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A combination of Bestvina--Brady Morse theory and an acyclic reflection group trick produces a torsion-free finitely presented Q-Poincaré duality group which is not the fundamental group of an aspherical closed ANR Q-homology manifold. The acyclic construction suggests asking which Q-Poincaré duality groups act freely on Q-acyclic spaces, i.e., which groups are FH(Q). For example, the orbifold fundamental group Γ of a good orbifold satisfies Q-Poincaré duality, and we show Γ is FH(Q) if the Euler characteristics of certain fixed sets vanish.
We prove the analogue of Johannson's Deformation Theorem for PD3 pairs.
We show that every oriented $n$-dimensional Poincaré duality group over a $*$-ring $R$ is amenable or satisfies a linear homological isoperimetric inequality in dimension $n-1$. As an application, we prove the Tits alternative for such groups when $n=2$. We then deduce a new proof of the fact that when $n=2$ and $R = \mathbb Z$ then the group in question is a surface group.
The authors previously described an algebraic analogue of the JSJ-decomposition of a 3-manifold. This analogue is defined for any finitely presented, one-ended group. We study this analogue in the special case of Poincaré duality pairs.
Property (T) and Poincaré duality in dimension three
本文通过边界扩张理论证明三维庞加莱对偶群不具有性质(T)，并由此得出此类群不可能是Kähler群，同时证明了紧致3流形基本群若具有性质(T)则必为有限群。
Pro-$p$ Poincaré-Duality groups of infinite rank
本文引入了满足Poincaré对偶的无限秩pro-$p$广义Poincaré对偶群类，证明了其基本性质，并揭示了特定正合序列条件下群必为Demushkin群的结论。
If $G$ is a group which admits a manifold model for $\mathrm{B}G$ then $G$ is a Poincaré duality group. We study a generalisation of Poincaré duality groups, introduced initially by Davis and Leary, motivated by groups $G$ with cocompact manifold models $M$ for $\underline{\mathrm{E}}G$ where $M^H$ is a contractible submanifold for all finite subgroups $H$ of $G$. We give several sources of examples and constructions of these Bredon-Poincaré duality groups, including using the equivariant reflection group trick of Davis and Leary to construct examples of Bredon-Poincaré duality groups arising …
Virtual fibring of Poincaré-duality groups
该研究刻画了RFRS庞加莱对偶群存在核为庞加莱对偶群的虚拟满同态的等价条件，建立了代数纤维化群的无穷上同调与纤维的关联，并证明三维及以上非球面流形代数纤维化的纤维具有单端性。
We prove a pro-$p$ version of Sela's theorem stating that a finitely generated group is $k$-acylindrically accessible. This result is then used to prove that $\mathrm{PD}^n$ pro-$p$ groups admit a unique $k$-acylindrical JSJ-decomposition.
We prove that every finite connected simplicial complex is homotopy equivalent to the quotient of a contractible manifold by proper actions of a virtually torsion-free group. As a corollary, we obtain that every finite connected simplicial complex is homotopy equivalent to the classifying space for proper bundles of some virtual Poincaré duality group.
We show that for a wide class of manifold pairs N, M satisfying dim(M) = dim(N) + 1, every π_1-injective map f : N --> M factorises up to homotopy as a finite cover of an embedding. This result, in the spirit of Waldhausen's torus theorem, is derived using Cappell's surgery methods from a new algebraic splitting theorem for Poincare duality groups. As an application we derive a new obstruction to the existence of π_1-injective maps.
Some questions on subgroups of 3-dimensional Poincaré duality groups
本文基于三维流形拓扑学提出关于三维庞加莱对偶群及其子群的若干开放性问题，探讨其结构与性质。
Quasimorphisms and Poincaré duality in dimension 3
研究具有粗连通拟核的无界实值拟同态的PD^3群，证明其要么是环面或克莱因瓶丛的基本群，要么拟等距于黎曼流形(R^3,g)，且拟核粗等价于双曲平面；若群双曲，则在S^1上有忠实拟对称作用。
Let $G$ be a finite group of order $n$ and let $M$ be a $G$-module. We construct groups $H_*^\varkappa(G,M)$ for which $H_k^\varkappa (G,M^{tw}) \cong H^{n-k-1}_λ(G,M),$ where $M^{tw}$ is a twisting of a $G$-module $M$ defined in Section $5$ and $H^{*}_λ(G,M)$ is a variation of the group cohomology introduced by Zarelua, which in many cases is isomorphic to the symmetric cohomology of groups defined by Staic. The groups $H_*^\varkappa(G,M)$ come together with transformations from Tate cohomology. We find conditions under which these transformations are isomorphisms.
We show that if $G$ is a group of type $FP_{n+1}^{\mathbb{Z}_2}$ that is coarsely separated into three essential, coarse disjoint, coarse complementary components by a coarse $PD_n^{\mathbb{Z}_2}$ space $W,$ then $W$ is at finite Hausdorff distance from a subgroup $H$ of $G$; moreover, $G$ splits over a subgroup commensurable to a subgroup of $H$. We use this to deduce that splittings of the form $G=A*_HB$, where $G$ is of type $FP_{n+1}^{\mathbb{Z}_2}$ and $H$ is a coarse $PD_n^{\mathbb{Z}_2}$ group such that both $|\mathrm{Comm}_A(H): H|$ and $|\mathrm{Comm}_B(H): H|$ are greater than two, a…
The aim of these notes, originally intended as an appendix to a book on the foundations of equivariant cohomology, is to set up the formalism of the $G$-equivariant Poincaré duality for oriented $G$-manifolds, for any connected compact Lie group $G$, following the work of J.-L. Brylinski leading to the spectral sequence $$\mathop{\rm Extgr}\nolimits_{H_G}(H_{G,\rm c} (M),H_G)\Rightarrow H_{G}(M)[d_{M}]\,.$$ The equivariant Gysin functor $(\_)_!:=Ω_{G}(\_)\in\mathcal D^{+}(\mathord{\rm DGM}(H_{G}))$ (resp. $(\_)_{*}:=Ω_{G,\rm c}(\_)$) is then defined in the category of oriented $G$-manifolds an…
In this paper we define and develop the theory of the cohomology of a profinite group relative to a collection of closed subgroups. Having made the relevant definitions we establish a robust theory of cup products and use this theory to define profinite Poincaré duality pairs. We use the theory of groups acting on profinite trees to give Mayer-Vietoris sequences, and apply this to give results concerning decompositions of 3-manifold groups. Finally we discuss the relationship between discrete duality pairs and profinite duality pairs, culminating in the result that profinite completion of the …
We study the finite generation of homotopy groups of closed manifolds and finite CW-complexes by relating it to the cohomology of their fundamental groups. Our main theorems are as follows: when $X$ is a finite CW-complex of dimension $n$ and $π_1(X)$ is virtually a Poincaré duality group of dimension $\geq n-1$, then $π_i(X)$ is not finitely generated for some $i$ unless $X$ is homotopy equivalent to the Eilenberg--MacLane space $K(π_1(X),1)$; when $M$ is an $n$-dimensional closed manifold and $π_1(M)$ is virtually a Poincaré duality group of dimension $\ge n-1$, then for some $i\leq [n/2]$, …
Canonical Kummerian Kernels in 1-Smooth Pro-$p$ Poincaré Duality Groups of Dimension Three
本文研究三维1-光滑Pro-$p$庞加莱对偶群的典范库默尔核，证明其有限生成时必为平凡群，结合相关定理与分类得到上同调性质及维数限制，明确非$\theta$-阿贝尔群的相关结论。
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