We introduce a spectrum of monotone coarse invariants for metric measure spaces called Poincaré profiles. The two extremes of this spectrum determine the growth of the space, and the separation profile as defined by Benjamini--Schramm--Timár. In this paper we focus on properties of the Poincaré profiles of groups with polynomial growth, and of hyperbolic spaces, where we deduce a connection between these profiles and conformal dimension. As applications, we use these invariants to show the non-existence of coarse embeddings in a variety of examples.
We define Poincaré profiles of Dirichlet type for graphs of bounded degree, in analogy with the Poincaré profiles (of Neumann type) defined in [HMT19]. The obvious first definition yields nothing of interest, but an alternative definition yields a spectrum of profiles which are quasi-isometry invariants and monotone with respect to subgroup inclusion. Moreover, in the extremal cases $p=1$ and $p=\infty$, they detect the Følner function and the growth function respectively.
We exhibit finitely generated groups with prescribed Poincaré profiles. It can be prescribed for functions between $n/\log n$ and linear, and is sharp for functions at least $n/(\log\log n)$. Those profiles were introduced by Hume, Mackay and Tessera in 2019 as a generalization of the separation profile, defined by Benjamini, Schramm and Timár in 2012. The family of groups used is based on a construction of Brieussel and Zheng. As applications, we show that there exists bounded degrees graphs of asymptotic dimension one that do not coarsely embed in any finite product of bounded degrees trees,…
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.
Poincare series of Klein groups, Coxeter polynomials, the Burau representation and Milnor invariants
对于二元多面体群的Poincare级数和Coxeter多项式，得到了与Euclid算法和正交多项式理论相近的命题：广义Ebeling公式，分解成分歧的连分数，Christoffel-Darboux恒等式，组合公式.关于亚历山大-康威多项式的因式分解的已知结果允许将庞加莱级数和考克斯特多项式与弦链的Burau表示和Milnor不变量连接起来。一个使用A 'Campo链路和Coxeter链路的重构。利用S. M. Gusein-Zade，F. Delgado和A.坎皮罗
Let $(W,S)$ be a Coxeter system of finite rank and let $J,K\subset S$. We study the rationality of the Poincaré series of the set of representatives of minimal length of $(W_J,W_K)$-double cosets of $W$: we conclude that it depends mostly on the rationality of the Poincaré series of the normalizers of finite parabolic subgroups of $W$. For affine Weyl groups, we prove that all these series are rational and we give some explicit examples.
We give a new self-contained proof of Poincaré's Polyhedron Theorem on presentations of discontinuous groups of isometries of a Riemann manifold of constant curvature. The proof is not based on the theory of covering spaces, but only makes use of basic geometric concepts. In a sense one hence obtains a proof that is of a more constructive nature than most known proofs.
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.
Let $W^c(A_n)$ be the set of fully commutative elements of the Coxeter group $W(A_n)$. Let $$ a_n(q)= \sum_{w \in W^c(A_n)} q^{l(w)} . $$ We compute $a_n(q)$.
We propose and prove an identity relating the Poincaré polynomials of stabilizer subgroups of the affine Weyl group and of the corresponding stabilizer subgroups of the Weyl group.
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.
