Recent zbMATH articles in MSC 57Shttps://zbmath.org/atom/cc/57S2023-11-13T18:48:18.785376ZWerkzeugDivision in group rings of surface groupshttps://zbmath.org/1521.200922023-11-13T18:48:18.785376Z"Avramidi, Grigori"https://zbmath.org/authors/?q=ai:avramidi.grigoriLet \(F_{n}\) be the free group on \(n\) generators. In the rational group ring \(\mathbb{Q}[F_{n}]\), there is a division algorithm analogous to polynomial long division that was discovered by \textit{P. M. Cohn} [Free ideal rings and localization in general rings. Cambridge: Cambridge University Press (2006; Zbl 1114.16001)]. A division algorithm is a process that lets one divide one element \(x\) by another non-zero element \(y\) with a remainder \(r\) whose ``size'' is smaller than that of \(y\). In \(\mathbb{Q}[F_{n}]\), the measure of ``size'' is the diameter of the support of the group ring element.
In this paper, the author shows that the same division algorithm is true when \(\Gamma\) is the fundamental group of a surface of sufficiently high genus. The main result is stated in Theorem 1: Let \(K\) be a field and let \(\Gamma\) be the fundamental group of a closed, orientable, surface of genus \(\geq e^{1000000}\). Suppose that \(x\) and \(y\) are elements in \(K[\Gamma]\) satisfying a non-trivial relation \(ax + by = 0\), and \(y \not =0\). Then there are \(q,r \in K[\Gamma]\) such that \(x= qy + r\) and \(|r| < |y|\) or \(r = 0\).
The division algorithm works somewhat more generally for groups acting on hyperbolic space \(\mathbb{H}^{n}\) with large infimum displacement. The author derives also an application of this fact to cohomological dimension of 2-relator groups acting on \(\mathbb{H}^{n}\) and to handle decompositions of hyperbolic \(n\)-manifolds.
Reviewer: Egle Bettio (Venezia)Entropy and affine actions for surface groupshttps://zbmath.org/1521.370282023-11-13T18:48:18.785376Z"Labourie, François"https://zbmath.org/authors/?q=ai:labourie.francoisSummary: We give a short and independent proof of a theorem of \textit{J. Danciger} and \textit{T. Zhang} [Geom. Funct. Anal. 29, No. 5, 1369--1439 (2019; Zbl 1427.57028)]: surface groups with Hitchin linear part cannot act properly on the affine space. The proof is fundamentally different and relies on ergodic methods.
{{\copyright} 2022 The Authors. \textit{Journal of Topology} is copyright {\copyright} London Mathematical Society.}Cohomogeneity one central Kähler metrics in dimension fourhttps://zbmath.org/1521.530522023-11-13T18:48:18.785376Z"Jeffres, Thalia"https://zbmath.org/authors/?q=ai:jeffres.thalia-d"Maschler, Gideon"https://zbmath.org/authors/?q=ai:maschler.gideon"Ream, Robert"https://zbmath.org/authors/?q=ai:ream.robertSummary: A Kähler metric is called central if the determinant of its Ricci endomorphism is constant; see [\textit{G. Maschler}, Trans. Am. Math. Soc. 355, No. 6, 2161--2182 (2003; Zbl 1043.53056)]. For the case in which this constant is zero, we study on 4-manifolds the existence of complete metrics of this type which have cohomogeneity one for three unimodular 3-dimensional Lie groups: SU(2), the group \(E(2)\) of Euclidean plane motions, and a quotient by a discrete subgroup of the Heisenberg group \(\mathrm{nil}_3\). We obtain a complete classification for SU(2), and some existence results for the other two groups, in terms of specific solutions of an associated ODE system.Multiplicativity and nonrealizable equivariant chain complexeshttps://zbmath.org/1521.550032023-11-13T18:48:18.785376Z"Rüping, Henrik"https://zbmath.org/authors/?q=ai:ruping.henrik"Stephan, Marc"https://zbmath.org/authors/?q=ai:stephan.marcAuthors' abstract: Let \(G\) be a finite \(p\)-group and \(\mathbb{F}\) a field of characteristic \(p\). We filter the cochain complex of a free \(G\)-space with coefficients in \(\mathbb{F}\) by powers of the augmentation ideal of \(\mathbb{F}G\). We show that the cup product induces a multiplicative structure on the arising spectral sequence and compute the \(E_1\)-page as a bigraded algebra. As an application, we prove that recent counterexamples of \textit{S. B. Iyengar} and \textit{M. E. Walker} [Acta Math. 221, No. 1, 143--158 (2018; Zbl 1403.13026)] to an algebraic version of Carlsson's conjecture [\textit{G. Carlsson}, Lect. Notes Math. 1217, 79--83 (1986; Zbl 0614.57023)] can not be realized topologically.
Reviewer: Hero Saremi (Sanandaj)Lower bound for Buchstaber invariants of real universal complexeshttps://zbmath.org/1521.570252023-11-13T18:48:18.785376Z"Shen, Qifan"https://zbmath.org/authors/?q=ai:shen.qifanSummary: In this article, we prove that Buchstaber invariant of 4-dimensional real universal complex is no less than 24 as a follow-up to the work of \textit{A. Ayzenberg} [Osaka J. Math. 53, No. 2, 377--395 (2016; Zbl 1339.05439) and ``The problem of Buchstaber number and its combinatorial aspects'', Preprint, \url{arXiv:1003.0637}] and \textit{Y. Sun} [Chin. Ann. Math., Ser. B 38, No. 6, 1335--1344 (2017; Zbl 1387.57052)]. Moreover, a lower bound for Buchstaber invariants of \(n\)-dimensional real universal complexes is given as an improvement of result of \textit{N. Yu. Erokhovets} [Proc. Steklov Inst. Math. 286, 128--187 (2014; Zbl 1317.52019); translation from Tr. Mat. Inst. Steklova 286, 144--206 (2014)].Discrete homogeneity and ends of manifoldshttps://zbmath.org/1521.570312023-11-13T18:48:18.785376Z"Chatyrko, Vitalij A."https://zbmath.org/authors/?q=ai:chatyrko.vitalij-a"Karassev, Alexandre"https://zbmath.org/authors/?q=ai:karassev.alexandreThe authors of the present paper define a topological space to be \textit {discrete homogeneous} if for any couple of discrete subsets of the same cardinality one can find a homeomorphism of the space sending the first subspace into the second one.
Furthermore, if any bijection of the two subspaces extends to a homeomorphism of the whole space, then the space is said to be \textit{strongly discrete homogeneous}.
Examples of strongly discrete homogeneous spaces are compact connected \(n\)-manifolds, for \(n\geq 2\), and the Euclidean spaces of dimension at least two.
The main problem addressed in the paper is to understand which metrizable manifolds have such properties, and the authors prove, with new techniques, the following nice old result, which relates the somehow local property of being homogeneous with the topological behaviour at infinity: ``For any \(n\geq 2\), a connected manifold \(M^n\) is strongly discrete homogeneous if and only if it is has at most one end''.
Reviewer: Daniele Ettore Otera (Vilnius)