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Journal articles on the topic 'Discrete hyperbolic geometry'

Author: Grafiati

Published: 5 June 2025

Last updated: 15 July 2025

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1

Prishlyak, Oleksandr. "Regular Octagons in Hyperbolic Geometry." In the world of mathematics, no. 1 (2) (2024): 88–103. https://doi.org/10.17721/1029-4171.2024/2.10.

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When constructing hyperbolic structures on closed surfaces, one can use hyperbolic geometry (Lobachevsky geometry) on the plane. To do this, the surface must be represented as a 2n-gon on the hyperbolic plane, and a discrete group action, which is a subgroup of the movements of the hyperbolic plane, must be defined, for which the 2n-gon serves as a fundamental domain. If such a surface is a double torus (an oriented surface of genus 2), it can be obtained by gluing opposite sides of an octagon. In fact, the Lobachevsky plane is divided into octagons. The presence of symmetries simplifies calculations. Therefore, a natural problem arises regarding the partitioning into regular octagons. Additionally, it is important to provide examples of such octagons by specifying the coordinates of their vertices in one of the models of hyperbolic geometry. The models of the upper half-plane and the Poincaré model on the unit disk are used, for which the Riemannian metric is defined (the formula for finding the lengths of arcs of curves). We describe the main properties of hyperbolic lines and the group of movements (the group of isometric transformations) of hyperbolic geometry on the plane using fractional-linear transformations of the complex plane with real coefficients.

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2

Apanasov, Boris, and Xiangdong Xie. "Geometrically Finite Complex Hyperbolic Manifolds." International Journal of Mathematics 08, no. 06 (1997): 703–57. http://dx.doi.org/10.1142/s0129167x97000378.

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The aim of this paper is to study geometry and topology of geometrically finite complex hyperbolic manifolds, especially their ends, as well as geometry of their holonomy groups. This study is based on our structural theorem for discrete groups acting on Heisenberg groups, on the fiber bundle structure of Heisenberg manifolds, and on the existence of finite coverings of a geometrically finite manifold such that their parabolic ends have either Abelian or 2-step nilpotent holonomy. We also study an interplay between Kähler geometry of complex hyperbolic n-manifolds and Cauchy–Riemannian geometry of their boundary (2n-1)-manifolds at infinity, and this study is based on homotopy equivalence of manifolds and isomorphism of fundamental groups.

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3

Apanasov, Boris. "Quasiregular Mappings and Discrete Group Actions." Ukrainian Mathematical Bulletin 18, no. 4 (2021): 441–65. http://dx.doi.org/10.37069/1810-3200-2021-18-4-1.

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We develop a new tool based on quasiconformal dynamics and conformal dynamics of discrete group actions in 3-geometries to construct new types of quasiregular and quasisymmetric mappings in space. This tool has close relations to new effects in Teichmüller spaces of conformally flat structures on closed hyperbolic 3-manifolds/orbifolds and non-trivial hyperbolic 4-cobordisms, to the hyperbolic and conformal interbreedings as well as to non-faithful discrete representations of uniform hyperbolic 3-lattices. We demonstrate several applications of this tool and new types of quasiregular mappings in space. Leaving such applications to geometry and topology of manifolds to another our papers [10, 11], here we continue a series of applications of our constructions to long standing problems for quasiregular mappings in space, including M.A. Lavrentiev surjectivity problem, Pierre Fatou problem on radial limits and Matti Vuorinen injectivity and asymptotics problems for bounded quasiregular mappings in the unit 3-ball (cf. [4, 7-9]).

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4

Altmann, Kristina, and Ralf Gramlich. "On the hyperbolic unitary geometry." Journal of Algebraic Combinatorics 31, no. 4 (2009): 547–83. http://dx.doi.org/10.1007/s10801-009-0200-5.

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5

BURGER, MARC, and ALESSANDRA IOZZI. "Bounded cohomology and totally real subspaces in complex hyperbolic geometry." Ergodic Theory and Dynamical Systems 32, no. 2 (2011): 467–78. http://dx.doi.org/10.1017/s0143385711000393.

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AbstractWe characterize representations of finitely generated discrete groups into (the connected component of) the isometry group of a complex hyperbolic space via the pullback of the bounded Kähler class.

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6

Brown, Matthew R., Gary L. Ebert, and Deirdre Luyckx. "On the geometry of regular hyperbolic fibrations." European Journal of Combinatorics 28, no. 6 (2007): 1626–36. http://dx.doi.org/10.1016/j.ejc.2006.07.006.

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7

Chu, Kenneth C. K. "On the Geometry of the Moduli Space of Real Binary Octics." Canadian Journal of Mathematics 63, no. 4 (2011): 755–97. http://dx.doi.org/10.4153/cjm-2011-026-1.

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Abstract The moduli space of smooth real binary octics has five connected components. They para- metrize the real binary octics whose defining equations have 0, … , 4 complex-conjugate pairs of roots respectively. We show that each of these five components has a real hyperbolic structure in the sense that each is isomorphic as a real-analytic manifold to the quotient of an open dense subset of 5- dimensional real hyperbolic space by the action of an arithmetic subgroup of Isom. These subgroups are commensurable to discrete hyperbolic reflection groups, and the Vinberg diagrams of the latter are computed.

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8

Gramlich, Ralf. "On the hyperbolic symplectic geometry." Journal of Combinatorial Theory, Series A 105, no. 1 (2004): 97–110. http://dx.doi.org/10.1016/j.jcta.2003.10.005.

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9

LANG, URS. "INJECTIVE HULLS OF CERTAIN DISCRETE METRIC SPACES AND GROUPS." Journal of Topology and Analysis 05, no. 03 (2013): 297–331. http://dx.doi.org/10.1142/s1793525313500118.

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Injective metric spaces, or absolute 1-Lipschitz retracts, share a number of properties with CAT(0) spaces. In the '60s Isbell showed that every metric space X has an injective hull E (X). Here it is proved that if X is the vertex set of a connected locally finite graph with a uniform stability property of intervals, then E (X) is a locally finite polyhedral complex with finitely many isometry types of n-cells, isometric to polytopes in [Formula: see text], for each n. This applies to a class of finitely generated groups Γ, including all word hyperbolic groups and abelian groups, among others. Then Γ acts properly on E(Γ) by cellular isometries, and the first barycentric subdivision of E(Γ) is a model for the classifying space [Formula: see text] for proper actions. If Γ is hyperbolic, E(Γ) is finite dimensional and the action is cocompact. In particular, every hyperbolic group acts properly and cocompactly on a space of non-positive curvature in a weak (but non-coarse) sense.

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10

Höfer, R. "On the geometry of hyperbolic cycles." aequationes mathematicae 56, no. 1-2 (1998): 169–80. http://dx.doi.org/10.1007/s000100050053.

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11

Fiorini, Rodolfo A. "Towards Advanced Quantum Cognitive Computation." International Journal of Software Science and Computational Intelligence 9, no. 1 (2017): 1–19. http://dx.doi.org/10.4018/ijssci.2017010101.

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Computational information conservation theory (CICT) can help us to develop competitive applications and even advanced quantum cognitive computational application and systems towards deep computational cognitive intelligence. CICT new awareness of a discrete HG (hyperbolic geometry) subspace (reciprocal space, RS) of coded heterogeneous hyperbolic structures, underlying the familiar Q Euclidean (direct space, DS) system surface representation can open the way to holographic information geometry (HIG) to recover lost coherence information in system description and to develop advanced quantum cognitive systems. This paper is a relevant contribution towards an effective and convenient “Science 2.0” universal computational framework to achieve deeper cognitive intelligence at your fingertips and beyond.

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12

Garoufalidis, Stavros, and Alan W. Reid. "Constructing 1-cusped isospectral non-isometric hyperbolic 3-manifolds." Journal of Topology and Analysis 10, no. 01 (2017): 1–25. http://dx.doi.org/10.1142/s1793525318500024.

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We construct infinitely many examples of pairs of isospectral but non-isometric [Formula: see text]-cusped hyperbolic [Formula: see text]-manifolds. These examples have infinite discrete spectrum and the same Eisenstein series. Our constructions are based on an application of Sunada’s method in the cusped setting, and so in addition our pairs are finite covers of the same degree of a 1-cusped hyperbolic 3-orbifold (indeed manifold) and also have the same complex length spectra. Finally we prove that any finite volume hyperbolic 3-manifold isospectral to the figure-eight knot complement is homeomorphic to the figure-eight knot complement.

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13

Hamenstädt, Ursula. "Hyperbolic relatively hyperbolic graphs and disk graphs." Groups, Geometry, and Dynamics 10, no. 1 (2016): 365–405. http://dx.doi.org/10.4171/ggd/352.

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14

Springborn, Boris. "Ideal Hyperbolic Polyhedra and Discrete Uniformization." Discrete & Computational Geometry 64, no. 1 (2019): 63–108. http://dx.doi.org/10.1007/s00454-019-00132-8.

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15

Blok, Rieuwert J. "The generating rank of the symplectic grassmannians: Hyperbolic and isotropic geometry." European Journal of Combinatorics 28, no. 5 (2007): 1368–94. http://dx.doi.org/10.1016/j.ejc.2006.05.013.

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16

Kallosh, Renata, та Andrei Linde. "SL(2,ℤ) cosmological attractors". Journal of Cosmology and Astroparticle Physics 2025, № 04 (2025): 045. https://doi.org/10.1088/1475-7516/2025/04/045.

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Abstract We study cosmological theory where the kinetic term and potential have SL(2,ℤ) symmetry. Potentials have a plateau at large values of the inflaton field, where the axion forms a flat direction. Due to the underlying hyperbolic geometry and special features of SL(2,ℤ) potentials, the theory exhibits an α-attractor behavior: its cosmological predictions are stable with respect to significant modifications of the SL(2,ℤ) invariant potentials. We present a supersymmetric version of this theory in the framework of D3 induced geometric inflation. The choice of α is determined by underlying string compactification. For example, in a CY compactification with T 2, one has 3α = 1, the lowest discrete Poincaré disk target for LiteBIRD.

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17

Le Boudec, Adrien. "Locally compact lacunary hyperbolic groups." Groups, Geometry, and Dynamics 11, no. 2 (2017): 415–54. http://dx.doi.org/10.4171/ggd/402.

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18

Mutanguha, Jean Pierre. "Hyperbolic immersions of free groups." Groups, Geometry, and Dynamics 14, no. 4 (2020): 1253–75. http://dx.doi.org/10.4171/ggd/580.

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19

Almeida, Igor, and Nikolay Gusevskii. "Moduli of triples of points in quaternionic hyperbolic geometry." Linear Algebra and its Applications 686 (April 2024): 33–63. http://dx.doi.org/10.1016/j.laa.2024.01.006.

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20

NATANZON, SERGEY, and ANNA PRATOUSSEVITCH. "HYPERBOLIC GROUPS AND NON-COMPACT REAL ALGEBRAIC CURVES." Transformation Groups 26, no. 2 (2021): 631–40. http://dx.doi.org/10.1007/s00031-021-09644-1.

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AbstractIn this paper we study the spaces of non-compact real algebraic curves, i.e. pairs (P, τ), where P is a compact Riemann surface with a finite number of holes and punctures and τ: P → P is an anti-holomorphic involution. We describe the uniformisation of non-compact real algebraic curves by Fuchsian groups. We construct the spaces of non-compact real algebraic curves and describe their connected components. We prove that any connected component is homeomorphic to a quotient of a finite-dimensional real vector space by a discrete group and determine the dimensions of these vector spaces.

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21

Benz, Walter. "Cayley–Klein’s model of dimension-free hyperbolic geometry via projective mappings." Aequationes mathematicae 75, no. 3 (2008): 226–38. http://dx.doi.org/10.1007/s00010-007-2899-1.

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22

Froese, Richard, David Hasler, and Wolfgang Spitzer. "Transfer matrices, hyperbolic geometry and absolutely continuous spectrum for some discrete Schrödinger operators on graphs." Journal of Functional Analysis 230, no. 1 (2006): 184–221. http://dx.doi.org/10.1016/j.jfa.2005.04.004.

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23

Hull, Michael. "Small cancellation in acylindrically hyperbolic groups." Groups, Geometry, and Dynamics 10, no. 4 (2016): 1077–119. http://dx.doi.org/10.4171/ggd/377.

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24

Ballas, Samuel, and Ludovic Marquis. "Properly convex bending of hyperbolic manifolds." Groups, Geometry, and Dynamics 14, no. 2 (2020): 653–88. http://dx.doi.org/10.4171/ggd/558.

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25

Bowers, Philip L., and Kenneth Stephenson. "The set of circle packing points in the Teichmüller space of a surface of finite conformal type is dense." Mathematical Proceedings of the Cambridge Philosophical Society 111, no. 3 (1992): 487–513. http://dx.doi.org/10.1017/s0305004100075575.

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W. Thurston initiated interest in circle packings with his provocative suggestion at the International Symposium in Celebration of the Proof of the Bieberbach Conjecture (Purdue University, 1985) that a result of Andreev[2] had an interpretation in terms of circle packings that could be applied systematically to construct geometric approximations of classical conformal maps. Rodin and Sullivan [11] verified Thurston's conjecture in the setting of hexagonal packings, and more recently Stephenson [12] has announced a proof for more general combinatorics. Inspired by Thurston's work and motivated by the desire to discover and exploit discrete versions of classical results in complex variable theory, Beardon and Stephenson [4, 5] initiated a study of the geometry of circle packings, particularly in the hyperbolic setting. This topic is a recent example among many of the beautiful and sometimes unexpected interplay between Geometry, Topology, and Cornbinatorics that is evident in much of the topological research of the past decade, and that has its roots in the seminal work of the great geometrically-minded mathematicians – Riemann, Klein, Poincaré – of the last century. A somewhat surprising example of this interplay concerns us here; namely, the fact that the combinatorial information encoded in a simplicial triangulation of a topological surface can determine its geometry.

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26

Kombrink, Sabrina. "Renewal theorems for processes with dependent interarrival times." Advances in Applied Probability 50, no. 4 (2018): 1193–216. http://dx.doi.org/10.1017/apr.2018.56.

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Abstract In this paper we develop renewal theorems for point processes with interarrival times ξ(Xn+1Xn…), where (Xn)n∈ℤ is a stochastic process with finite state space Σ and ξ:ΣA→ℝ is a Hölder continuous function on a subset ΣA⊂Σℕ. The theorems developed here unify and generalise the key renewal theorem for discrete measures and Lalley's renewal theorem for counting measures in symbolic dynamics. Moreover, they capture aspects of Markov renewal theory. The new renewal theorems allow for direct applications to problems in fractal and hyperbolic geometry, for instance to the problem of Minkowski measurability of self-conformal sets.

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27

Bader, Uri, Alex Furman, and Roman Sauer. "Efficient subdivision in hyperbolic groups and applications." Groups, Geometry, and Dynamics 7, no. 2 (2013): 263–92. http://dx.doi.org/10.4171/ggd/182.

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28

Coulon, Rémi, and Vincent Guirardel. "Automorphisms and endomorphisms of lacunary hyperbolic groups." Groups, Geometry, and Dynamics 13, no. 1 (2018): 131–48. http://dx.doi.org/10.4171/ggd/488.

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29

Mason, A. W., R. W. K. Odoni, and W. W. Stothers. "Almost all Bianchi groups have free, non-cyclic quotients." Mathematical Proceedings of the Cambridge Philosophical Society 111, no. 1 (1992): 1–6. http://dx.doi.org/10.1017/s0305004100075101.

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Let d be a square-free positive integer and let O (= Od) be the ring of integers of the imaginary quadratic number field ℚ(-d). The groups PSL2(O) are called the Bianchi groups after Luigi Bianchi who made the first important contribution 1 to their study in 1892. Since then they have attracted considerable attention particularly during the last thirty years. Their importance stems primarily from their action as discrete groups of isometries on hyperbolic 3-space, H3. As a consequence they play an important role in hyperbolic geometry, low-dimensional topology together with the theory of discontinuous groups and automorphic forms. In addition they are of particular significance in the class of linear groups over Dedekind rings of arithmetic type. Serre9 has proved that in this class the Bianchi groups (along with, for example, the modular group, PSL2(z), where z is the ring of rational integers) have an exceptionally complicated (non-congruence) subgroup structure.

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30

Kontorovich, Alex, and Christopher Lutsko. "Effective Counting in Sphere Packings." Journal of the Association for Mathematical Research 2, no. 1 (2024): 15–52. http://dx.doi.org/10.56994/jamr.002.001.002.

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Given a Zariski-dense, discrete group, Γ, of isometries acting on (n + 1)- dimensional hyperbolic space, we use spectral methods to obtain a sharp asymptotic formula for the growth rate of certain Γ-orbits. In particular, this allows us to obtain a best-known effective error rate for the Apollonian and (more generally) Kleinian sphere packing counting problems, that is, counting the number of spheres in such with radius bounded by a growing parameter. Our method extends the method of Kontorovich [Kon09], which was itself an extension of the orbit counting method of Lax-Phillips [LP82], in two ways. First, we remove a compactness condition on the discrete subgroups considered via a technical cut- off and smoothing operation. Second, we develop a coordinate system which naturally corresponds to the inversive geometry underlying the sphere counting problem, and give structure theorems on the arising Casimir operator and Haar measure in these coordinates.

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31

Sela, Z. "Structure and Rigidity in (Gromov) Hyperbolic Groups and Discrete Groups in Rank 1 Lie Groups II." Geometric And Functional Analysis 7, no. 3 (1997): 561–93. http://dx.doi.org/10.1007/s000390050019.

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32

Mateljevic, Miodrag, and Marek Svetlik. "Hyperbolic metric on the strip and the Schwarz lemma for HQR mappings." Applicable Analysis and Discrete Mathematics 14, no. 1 (2020): 150–68. http://dx.doi.org/10.2298/aadm200104001m.

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We give simple proofs of various versions of the Schwarz lemma for real valued harmonic functions and for holomorphic (more generally harmonic quasiregular, shortly HQR) mappings with the strip codomain. Along the way, we get a simple proof of a new version of the Schwarz lemma for real valued harmonic functions (without the assumption that 0 is mapped to 0 by the corresponding map). Using the Schwarz-Pick lemma related to distortion for harmonic functions and the elementary properties of the hyperbolic geometry of the strip we get optimal estimates for modulus of HQR mappings.

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33

Löh, Clara, and Cristinia Pagliantini. "Integral foliated simplicial volume of hyperbolic 3-manifolds." Groups, Geometry, and Dynamics 10, no. 3 (2016): 825–65. http://dx.doi.org/10.4171/ggd/368.

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34

Antolín, Yago, Ashot Minasyan, and Alessandro Sisto. "Commensurating endomorphisms of acylindrically hyperbolic groups and applications." Groups, Geometry, and Dynamics 10, no. 4 (2016): 1149–210. http://dx.doi.org/10.4171/ggd/379.

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35

Murillo, Plinio. "Systole of congruence coverings of arithmetic hyperbolic manifolds." Groups, Geometry, and Dynamics 13, no. 3 (2019): 1083–102. http://dx.doi.org/10.4171/ggd/515.

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36

Ivrissimtzis, Ioannis, Norbert Peyerimhoff, and Alina Vdovina. "Trivalent expanders, $(\Delta – Y)$-transformation, and hyperbolic surfaces." Groups, Geometry, and Dynamics 13, no. 3 (2019): 1103–31. http://dx.doi.org/10.4171/ggd/518.

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37

Baker, Owen, and Timothy Riley. "Cannon–Thurston maps, subgroup distortion, and hyperbolic hydra." Groups, Geometry, and Dynamics 14, no. 1 (2020): 255–82. http://dx.doi.org/10.4171/ggd/543.

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38

Berlai, Federico, and Bruno Robbio. "A refined combination theorem for hierarchically hyperbolic groups." Groups, Geometry, and Dynamics 14, no. 4 (2020): 1127–203. http://dx.doi.org/10.4171/ggd/576.

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39

Chemnitz, Robin, Maximilian Engel, and Péter Koltai. "Continuous-time extensions of discrete-time cocycles." Proceedings of the American Mathematical Society, Series B 11, no. 3 (2024): 23–35. http://dx.doi.org/10.1090/bproc/209.

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We consider linear cocycles taking values in S L d ( R ) \mathrm {SL}_d(\mathbb {R}) driven by homeomorphic transformations of a smooth manifold, in discrete and continuous time. We show that any discrete-time cocycle can be extended to a continuous-time cocycle, while preserving its characteristic properties. We provide a necessary and sufficient condition under which this extension is canonical in the sense that the base is extended to an associated suspension flow and that the discrete-time cocycle is recovered as the time-1 map of the continuous-time cocycle. Further, we refine our general result for the case of (quasi-)periodic driving. We use our findings to construct a non-uniformly hyperbolic continuous-time cocycle in S L 2 ( R ) \mathrm {SL}_{2}(\mathbb {R}) over a uniquely ergodic driving.

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40

Matsuzaki, Katsuhiko, Yasuhiro Yabuki, and Johannes Jaerisch. "Normalizer, divergence type, and Patterson measure for discrete groups of the Gromov hyperbolic space." Groups, Geometry, and Dynamics 14, no. 2 (2020): 369–411. http://dx.doi.org/10.4171/ggd/548.

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41

Piña, Eduardo, and Abimael Bengochea. "Hyperbolic Geometry for the Binary Collision Angles of the Three-Body Problem in the Plane." Qualitative Theory of Dynamical Systems 8, no. 2 (2009): 399–417. http://dx.doi.org/10.1007/s12346-010-0009-6.

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42

Cunha, Heleno, and Nikolay Gusevskii. "A note on trace fields of complex hyperbolic groups." Groups, Geometry, and Dynamics 8, no. 2 (2014): 355–74. http://dx.doi.org/10.4171/ggd/229.

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43

Kin, Eiko, and Dale Rolfsen. "Braids, orderings, and minimal volume cusped hyperbolic 3-manifolds." Groups, Geometry, and Dynamics 12, no. 3 (2018): 961–1004. http://dx.doi.org/10.4171/ggd/460.

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44

Franceschini, Federico. "A characterization of relatively hyperbolic groups via bounded cohomology." Groups, Geometry, and Dynamics 12, no. 3 (2018): 919–60. http://dx.doi.org/10.4171/ggd/463.

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45

Oregón-Reyes, Eduardo. "Properties of sets of isometries of Gromov hyperbolic spaces." Groups, Geometry, and Dynamics 12, no. 3 (2018): 889–910. http://dx.doi.org/10.4171/ggd/468.

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46

Sunderland, Matthew. "Linear progress with exponential decay in weakly hyperbolic groups." Groups, Geometry, and Dynamics 14, no. 2 (2020): 539–66. http://dx.doi.org/10.4171/ggd/554.

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47

Anan'in, Sasha, and Philipy Chiovetto. "A couple of real hyperbolic disc bundles over surfaces." Groups, Geometry, and Dynamics 14, no. 4 (2020): 1419–28. http://dx.doi.org/10.4171/ggd/585.

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48

Margenstern, Maurice. "About an Algorithmic Approach to Tilings {p,q} of the Hyperbolic Plane." JUCS - Journal of Universal Computer Science 12, no. (5) (2006): 512–50. https://doi.org/10.3217/jucs-012-05-0512.

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In this paper, we remind previous results about the tilings {p,q} of the hyperbolic plane. As proved in [Margenstern and Skordev 2003a], these tilings are combinatoric, a notion which we recall in the introduction. It turned out that in this case, most of these tilings also have the interesting property that the language of the splitting associated to the tiling is regular. In this paper, we investigate the consequence of the regularity of the language by providing algorithms to compute the path from a tile to the root of the spanning tree as well as to compute the coordinates of the neighbouring tiles. These algorithms are linear in the coordinate of the given node.

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49

Asselmeyer-Maluga, Torsten. "Braids, 3-Manifolds, Elementary Particles: Number Theory and Symmetry in Particle Physics." Symmetry 11, no. 10 (2019): 1298. http://dx.doi.org/10.3390/sym11101298.

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In this paper, we will describe a topological model for elementary particles based on 3-manifolds. Here, we will use Thurston’s geometrization theorem to get a simple picture: fermions as hyperbolic knot complements (a complement C ( K ) = S 3 \ ( K × D 2 ) of a knot K carrying a hyperbolic geometry) and bosons as torus bundles. In particular, hyperbolic 3-manifolds have a close connection to number theory (Bloch group, algebraic K-theory, quaternionic trace fields), which will be used in the description of fermions. Here, we choose the description of 3-manifolds by branched covers. Every 3-manifold can be described by a 3-fold branched cover of S 3 branched along a knot. In case of knot complements, one will obtain a 3-fold branched cover of the 3-disk D 3 branched along a 3-braid or 3-braids describing fermions. The whole approach will uncover new symmetries as induced by quantum and discrete groups. Using the Drinfeld–Turaev quantization, we will also construct a quantization so that quantum states correspond to knots. Particle properties like the electric charge must be expressed by topology, and we will obtain the right spectrum of possible values. Finally, we will get a connection to recent models of Furey, Stoica and Gresnigt using octonionic and quaternionic algebras with relations to 3-braids (Bilson–Thompson model).

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50

Arora, S., I. Castellano, G. Corob Cook, and E. Martínez-Pedroza. "Subgroups, hyperbolicity and cohomological dimension for totally disconnected locally compact groups." Journal of Topology and Analysis, March 13, 2021, 1–27. http://dx.doi.org/10.1142/s1793525321500254.

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This paper is part of the program of studying large-scale geometric properties of totally disconnected locally compact groups, TDLC-groups, by analogy with the theory for discrete groups. We provide a characterization of hyperbolic TDLC-groups, in terms of homological isoperimetric inequalities. This characterization is used to prove the main result of this paper: for hyperbolic TDLC-groups with rational discrete cohomological dimension [Formula: see text], hyperbolicity is inherited by compactly presented closed subgroups. As a consequence, every compactly presented closed subgroup of the automorphism group [Formula: see text] of a negatively curved locally finite [Formula: see text]-dimensional building [Formula: see text] is a hyperbolic TDLC-group, whenever [Formula: see text] acts with finitely many orbits on [Formula: see text]. Examples where this result applies include hyperbolic Bourdon’s buildings. We revisit the construction of small cancellation quotients of amalgamated free products, and verify that it provides examples of hyperbolic TDLC-groups of rational discrete cohomological dimension [Formula: see text] when applied to amalgamated products of profinite groups over open subgroups. We raise the question of whether our main result can be extended to locally compact hyperbolic groups if rational discrete cohomological dimension is replaced by asymptotic dimension. We prove that this is the case for discrete groups and sketch an argument for TDLC-groups.

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