Recent zbMATH articles in MSC 57Shttps://zbmath.org/atom/cc/57S2024-11-01T15:51:55.949586ZWerkzeugCoxeter groups and the Davis complexhttps://zbmath.org/1544.200622024-11-01T15:51:55.949586Z"Schroeder, Timothy A."https://zbmath.org/authors/?q=ai:schroeder.timothy-aSummary: We take a constructive and active look at group theory, by focusing on the action of finitely presented groups on CW-complexes. In particular, we focus on the action of Coxeter groups on the so-called Davis complex. Students are invited to participate in several constructions and investigate the group theoretic and geometric properties of the Davis complex. Students are encouraged to check the references on the included concepts and definitions, especially the italicized words.
For the entire collection see [Zbl 1388.20001].On the diffeomorphism groups of foliated manifoldshttps://zbmath.org/1544.530222024-11-01T15:51:55.949586Z"Narmanov, A. Ya."https://zbmath.org/authors/?q=ai:narmanov.abdigappar-yakubovich"Sharipov, A. S."https://zbmath.org/authors/?q=ai:sharipov.a-sAssume that \(M\) is a manifold and \(\mathrm{Diff}(M)\) is the set of all diffeomorphisms of \(M\) onto itself. We know that \(\mathrm{Diff}(M)\) is a group with respect to the composition and inverting mappings.
If \(M\) is a finite-dimensional manifold, then the isometry group \(I(M)\) of the Riemannian manifold \(M\) is a Lie group. Let \(M\) be a smooth connected Riemannian manifold of dimension \(n\), \(0< k <n\). A foliation \(F\) of dimension \(k\) (or a foliation of codimension \(n-k\)) is a partition of \(M\) into linearly connected subsets \(L_\alpha\subset M\) satisfying the following properties:
\begin{itemize}
\item[(\(F_1\))] \(\bigcup_{\alpha\in B}L_\alpha=M\);
\item[(\(F_2\))] if \(\alpha\neq \beta\), then \(L_\alpha\cap L_\beta=\emptyset\) for all \(\alpha, \beta\in B\);
\item[(\(F_3\))] for any point \(p\in M\), there exist a neighborhood \(U\) and a chart \((x^1,\dots,x^k,y^1,\dots,y^{n-k})\) such that for each leaf \(L_\alpha\), the linear connected components of the set \(U_p\cap L_\alpha\) are determined by the equations \(y^1=\mathrm{const},\cdots,y^{n-k}=\mathrm{const}\).
\end{itemize}
Now, by \((M, F)\) denote a smooth manifold \(M\) of dimension \(n\), \(0<k<n\), on which a smooth \(k\)-dimensional foliation \(F\) is defined. We denote by \(\mathrm{Diff}_F(M)\) the set of all \(C^r\)-diffeomorphisms of the foliated manifold \((M, F)\) for some fixed \(r\ge 0\). The set \(\mathrm{Diff}_F(M)\) is a group with respect to the operations of composition and inverting mappings. Note that \(\mathrm{Diff}_F(M) \le \mathrm{Diff}(M)\). The authors prove that \(\mathrm{Diff}_F(M)\) is a topological group with the \(F\)-compact-open topology.
The authors consider some examples of foliated manifolds and examine some subgroups of the group of diffeomorphisms of the foliated manifold.
Reviewer: Ali Morassaei (Zanjan)On a conjecture of Stolz in the toric casehttps://zbmath.org/1544.530432024-11-01T15:51:55.949586Z"Wiemeler, Michael"https://zbmath.org/authors/?q=ai:wiemeler.michaelSummary: In 1996 \textit{S. Stolz} [Math. Ann. 304, No. 4, 785--800 (1996; Zbl 0856.53033)] conjectured that a string manifold with positive Ricci curvature has vanishing Witten genus. Here we prove this conjecture for toric string Fano manifolds and for string torus manifolds admitting invariant metrics of non-negative sectional curvature.A tale of two symmetries: embeddable and non-embeddable group actions on surfaceshttps://zbmath.org/1544.570192024-11-01T15:51:55.949586Z"Peterson, Valerie"https://zbmath.org/authors/?q=ai:peterson.valerie-j"Wootton, Aaron"https://zbmath.org/authors/?q=ai:wootton.aaronSummary: Plainly speaking, a compact Riemann surface, \(S\), can be thought of as the layer of glaze on a \(g\)-holed doughnut. A group of symmetries of \(S\) is a group that acts on \(S\) while preserving some of its underlying structure. We provide an easily understood exposition of the modern techniques used to determine which groups can act as symmetry groups on a compact Riemann surface \(S\) of genus \(g\ge 2\). We then illustrate these techniques by providing necessary and sufficient conditions for the existence of an \(A_4\)-action on such a surface (in terms of a signature), and through explicit geometric construction show which of these actions are embeddable.
For the entire collection see [Zbl 1388.20001].A diameter gap for quotients of the unit spherehttps://zbmath.org/1544.570272024-11-01T15:51:55.949586Z"Gorodski, Claudio"https://zbmath.org/authors/?q=ai:gorodski.claudio"Lange, Christian"https://zbmath.org/authors/?q=ai:lange.christian"Lytchak, Alexander"https://zbmath.org/authors/?q=ai:lytchak.alexander"Mendes, Ricardo A. E."https://zbmath.org/authors/?q=ai:mendes.ricardo-a-eThis paper establishes the following ```gap theorem'' on the diameter of a quotient space of the unit \(n\)-sphere \(S^n\) by isometric group actions.
\textbf{Theorem:} There exists an \(\epsilon>0\) so that for any dimension \(n \geq 2\), and any group \(G\) of isometries of \(S^{n}\), either
\[
\mathrm{diam}(S^n/ G) = 0 \mbox{ or } \mathrm{diam}(S^n / G) \geq \epsilon.
\]
The restriction on dimension \(n \geq 2\) cannot be removed, for we can consider cyclic actions on the circle \(S^1\) by rotations. The quotient \(S^1 / G\) has arbitrarily small diameter as rotation order increases.
\textit{S. J. Greenwald} established a lower bound \(\epsilon(n)\) (depending on dimension) for \(\mathrm{diam}(S^n / G)\) in [Indiana Univ. Math. J. 49, No. 4, 1449--1479 (2000; Zbl 0984.57019)]. More specific lower bounds can be found when the action of \(G\) on \(S^n\) is restricted to specific classes such as free actions or the action of finite Coxeter groups. Recent work by \textit{B. Green} [Duke Math. J. 169, No. 3, 551--578 (2020; Zbl 1481.20047)] shows that \(\epsilon(n)\) converges to \(\frac{\pi}{2}\) as \(n \rightarrow \infty\), when \(G\) is finite.
The significance of the authors' theorem is that their fixed lower bound \(\epsilon\) works for a unit sphere of \textit{any} dimension \(n\), and \textit{any} subgroup of isometries \(G \subseteq \mathrm{SO}(n+1)\). While the value of \(\epsilon\) has not been explicitly determined, the proof of the main theorem does give some insight on a potential range of values for \(\epsilon\).
There is no loss of generality in assuming that \(G\) is compact, as \(G\) may be replaced with its closure \(\overline{G}\) in \(\mathrm{SO}(n+1)\). The proof heavily utilizes representation theoretic results on the induced representation of \(G\) on \(\mathbb{R}^{n+1}\), and the authors reduce the problem to a series of cases, and carefully treat each case.
In summary, this paper is well written, contains significant results, and suggests topics for further exploration.
Reviewer: Scott Thuong (Pittsburg)