Recent zbMATH articles in MSC 30F60https://www.zbmath.org/atom/cc/30F602022-05-16T20:40:13.078697ZWerkzeugFrom hierarchical to relative hyperbolicityhttps://www.zbmath.org/1483.300852022-05-16T20:40:13.078697Z"Russell, Jacob"https://www.zbmath.org/authors/?q=ai:russell.jacobSummary: We provide a simple, combinatorial criteria for a hierarchically hyperbolic space to be relatively hyperbolic by proving a new formulation of relative hyperbolicity in terms of hierarchy structures. In the case of clean hierarchically hyperbolic groups, this criteria characterizes relative hyperbolicity. We apply our criteria to graphs associated to surfaces and prove that the separating curve graph of a surface is relatively hyperbolic when the surface has zero or two punctures. We also recover a celebrated theorem of Brock and Masur on the relative hyperbolicity of the Weil-Petersson metric on Teichmüller space for surfaces with complexity three.Intersection pairings for higher laminationshttps://www.zbmath.org/1483.300862022-05-16T20:40:13.078697Z"Le, Ian"https://www.zbmath.org/authors/?q=ai:le.ianSummary: One can realize higher laminations as positive configurations of points in the affine building [the author, Geom. Topol. 20, No. 3, 1673--1735 (2016; Zbl 1348.30023)]. The duality pairings of \textit{V. Fock} and \textit{A. Goncharov} [Publ. Math., Inst. Hautes Étud. Sci. 103, 1--211 (2006; Zbl 1099.14025)] give pairings between higher laminations for two Langlands dual groups \(G\) and \(G^{\vee}\). These pairings are a generalization of the intersection pairing between measured laminations on a topological surface.
We give a geometric interpretation of these intersection pairings in a wide variety of cases. In particular, we show that they can be computed as the minimal weighted length of a network in the building. Thus we relate the intersection pairings to the metric structure of the affine building. This proves several of the conjectures from [the author and \textit{E. O'Dorney}, Doc. Math. 22, 1519--1538 (2017; Zbl 1383.51009)]. We also suggest the next steps toward giving geometric interpretations of intersection pairings in general.
The key tools are linearized versions of well-known classical results from combinatorics, like Hall's marriage lemma, König's theorem, and the Kuhn-Munkres algorithm, which are interesting in themselves.On an Enneper-Weierstrass-type representation of constant Gaussian curvature surfaces in 3-dimensional hyperbolic spacehttps://www.zbmath.org/1483.300872022-05-16T20:40:13.078697Z"Smith, Graham"https://www.zbmath.org/authors/?q=ai:smith.graham-a|smith.graham-mSummary: For all \(k\in ]0,1[\), we construct a canonical bijection between the space of ramified coverings of the sphere of hyperbolic type and the space of complete immersed surfaces in 3-dimensional hyperbolic space of finite area and of constant extrinsic curvature equal to \(k\). We show, furthermore, that this bijection restricts to a homeomorphism over each stratum of the space of ramified coverings of the sphere.
For the entire collection see [Zbl 1473.53006].Anti-de Sitter geometry and Teichmüller theoryhttps://www.zbmath.org/1483.530022022-05-16T20:40:13.078697Z"Bonsante, Francesco"https://www.zbmath.org/authors/?q=ai:bonsante.francesco"Seppi, Andrea"https://www.zbmath.org/authors/?q=ai:seppi.andreaSummary: The aim of this chapter is to provide an introduction to Anti-de Sitter geometry, with special emphasis on dimension three and on the relations with Teichmüller theory, whose study has been initiated by the seminal paper of Geoffrey Mess in 1990. In the first part we give a broad introduction to Anti-de Sitter geometry in any dimension. The main results of Mess, including the classification of maximal globally hyperbolic Cauchy compact manifolds and the construction of the Gauss map, are treated in the second part. Finally, the third part contains related results which have been developed after the work of Mess, with the aim of giving an overview on the state-of-the-art.
For the entire collection see [Zbl 1470.57002].