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Measuring Pants Doan, Nhat Minh ; Parlier, Hugo ; E-print/Working paper (2020) We investigate the terms arising in an identity for hyperbolic surfaces proved by Luo and Tan, namely showing that they vary monotonically in terms of lengths and that they verify certain convexity ... [more ▼] We investigate the terms arising in an identity for hyperbolic surfaces proved by Luo and Tan, namely showing that they vary monotonically in terms of lengths and that they verify certain convexity properties. Using these properties, we deduce two results. As a first application, we show how to deduce a theorem of Thurston which states, in particular for closed hyperbolic surfaces, that if a simple length spectrum "dominates" another, then in fact the two surfaces are isometric. As a second application, we show how to find upper bounds on the number of pairs of pants of bounded length that only depend on the boundary length and the topology of the surface. [less ▲] Detailed reference viewed: 159 (68 UL)Geometric simplicial embeddings of arc-type graphs Parlier, Hugo ; in J. Korean Math. Soc. (2020), 57(5), 1103--1118 Detailed reference viewed: 74 (0 UL)Counting curves, and the stable length of currents ; Parlier, Hugo ; in J. Eur. Math. Soc. (JEMS) (2020), 22(6), 1675--1702 Detailed reference viewed: 80 (0 UL)Short closed geodesics with self-intersections ; Parlier, Hugo in Math. Proc. Cambridge Philos. Soc. (2020), 169(3), 623--638 Detailed reference viewed: 81 (0 UL)The maximum number of systoles for genus two Riemann surfaces with abelian differentials ; Parlier, Hugo in COMMENTARII MATHEMATICI HELVETICI (2019), 94(2), 399-437 In this article, we provide bounds on systoles associated to a holomorphic 1-form omega on a Riemann surface X. In particular, we show that if X has genus two, then, up to homotopy, there are at most 10 ... [more ▼] In this article, we provide bounds on systoles associated to a holomorphic 1-form omega on a Riemann surface X. In particular, we show that if X has genus two, then, up to homotopy, there are at most 10 systolic loops on (X, omega) and, moreover, that this bound is realized by a unique translation surface up to homothety. For general genus g and a holomorphic 1-form omega with one zero, we provide the optimal upper bound, 6g - 3, on the number of homotopy classes of systoles. If, in addition, X is hyperelliptic, then we prove that the optimal upper bound is 6g - 5. [less ▲] Detailed reference viewed: 32 (0 UL)The geometry of flip graphs and mapping class groups ; Parlier, Hugo in Trans. Amer. Math. Soc. (2019), 372(6), 3809--3844 Detailed reference viewed: 79 (0 UL)Delaunay Triangulations of Points on Circles ; ; Parlier, Hugo et al E-print/Working paper (2018) Delaunay triangulations of a point set in the Euclidean plane are ubiquitous in a number of computational sciences, including computational geometry. Delaunay triangulations are not well defined as soon ... [more ▼] Delaunay triangulations of a point set in the Euclidean plane are ubiquitous in a number of computational sciences, including computational geometry. Delaunay triangulations are not well defined as soon as 4 or more points are concyclic but since it is not a generic situation, this difficulty is usually handled by using a (symbolic or explicit) perturbation. As an alternative, we propose to define a canonical triangulation for a set of concyclic points by using a max-min angle characterization of Delaunay triangulations. This point of view leads to a well defined and unique triangulation as long as there are no symmetric quadruples of points. This unique triangulation can be computed in quasi-linear time by a very simple algorithm. [less ▲] Detailed reference viewed: 60 (10 UL)Interrogating surface length spectra and quantifying isospectrality Parlier, Hugo in MATHEMATISCHE ANNALEN (2018), 370(3-4), 1759-1787 This article is about inverse spectral problems for hyperbolic surfaces and in particular how length spectra relate to the geometry of the underlying surface. A quantitative answer is given to the ... [more ▼] This article is about inverse spectral problems for hyperbolic surfaces and in particular how length spectra relate to the geometry of the underlying surface. A quantitative answer is given to the following: how many questions do you need to ask a length spectrum to determine it completely? In answering this, a quantitative upper bound is given on the number of isospectral but non-isometric surfaces of a given genus. [less ▲] Detailed reference viewed: 34 (0 UL)Once Punctured Disks, Non-Convex Polygons, and Pointihedra Parlier, Hugo ; in ANNALS OF COMBINATORICS (2018), 22(3), 619-640 We explore several families of flip-graphs, all related to polygons or punctured polygons. In particular, we consider the topological flip-graphs of once punctured polygons which, in turn, contain all ... [more ▼] We explore several families of flip-graphs, all related to polygons or punctured polygons. In particular, we consider the topological flip-graphs of once punctured polygons which, in turn, contain all possible geometric flip-graphs of polygons with a marked point as embedded sub-graphs. Our main focus is on the geometric properties of these graphs and how they relate to one another. In particular, we show that the embeddings between them are strongly convex (or, said otherwise, totally geodesic). We find bounds on the diameters of these graphs, sometimes using the strongly convex embeddings and show that the topological flip-graph is Hamiltonian. These graphs relate to different polytopes, namely to type D associahedra and a family of secondary polytopes which we call pointihedra. [less ▲] Detailed reference viewed: 14 (0 UL)Modular flip-graphs of one-holed surfaces Parlier, Hugo ; in EUROPEAN JOURNAL OF COMBINATORICS (2018), 67 We study flip-graphs of triangulations on topological surfaces where distance is measured by counting the number of necessary flip operations between two triangulations. We focus on surfaces of positive ... [more ▼] We study flip-graphs of triangulations on topological surfaces where distance is measured by counting the number of necessary flip operations between two triangulations. We focus on surfaces of positive genus g with a single boundary curve and n marked points on this curve and consider triangulations up to homeomorphism with the marked points as their vertices. Our results are bounds on the maximal distance between two triangulations. Our lower bounds assert that these distances grow at least like 5n/2 for all g >= 1. Our upper bounds grow at most like [4 - 1/(4g)]n for g >= 2, and at most like 23n/8 for the bordered torus. (C) 2017 Elsevier Ltd. All rights reserved. [less ▲] Detailed reference viewed: 19 (0 UL)The Genus of Curve, Pants and Flip Graphs Parlier, Hugo ; in Discrete Comput. Geom. (2018), 59(1), 1--30 Detailed reference viewed: 3 (0 UL)Simultaneous Flips on Triangulated Surfaces ; Parlier, Hugo in MICHIGAN MATHEMATICAL JOURNAL (2018), 67(3), 451-464 We investigate a type of distance between triangulations on finite-type surfaces where one moves between triangulations by performing simultaneous flips. We consider triangulations up to homeomorphism ... [more ▼] We investigate a type of distance between triangulations on finite-type surfaces where one moves between triangulations by performing simultaneous flips. We consider triangulations up to homeomorphism, and our main results are upper bounds on the distance between triangulations that only depend on the topology of the surface. [less ▲] Detailed reference viewed: 14 (0 UL)The maximum number of systoles for genus two Riemann surfaces with abelian differentials ; Parlier, Hugo E-print/Working paper (2017) This article explores the length and number of systoles associated to holomorphic $1$-forms on surfaces. In particular, we show that up to homotopy, there are at most $10$ systolic loops on such a genus ... [more ▼] This article explores the length and number of systoles associated to holomorphic $1$-forms on surfaces. In particular, we show that up to homotopy, there are at most $10$ systolic loops on such a genus two surface and that the bound is realized by a unique translation surface up to homothety. We also provide sharp upper bounds on the the number of homotopy classes of systoles for a holomorphic $1$-form with a single zero in terms of the genus. [less ▲] Detailed reference viewed: 105 (7 UL)Chromatic numbers for the hyperbolic plane and discrete analogs Parlier, Hugo ; E-print/Working paper (2017) We study colorings of the hyperbolic plane, analogously to the Hadwiger-Nelson problem for the Euclidean plane. The idea is to color points using the minimum number of colors such that no two points at ... [more ▼] We study colorings of the hyperbolic plane, analogously to the Hadwiger-Nelson problem for the Euclidean plane. The idea is to color points using the minimum number of colors such that no two points at distance exactly $d$ are of the same color. The problem depends on $d$ and, following a strategy of Kloeckner, we show linear upper bounds on the necessary number of colors. In parallel, we study the same problem on $q$-regular trees and show analogous results. For both settings, we also consider a variant which consists in replacing $d$ with an interval of distances. [less ▲] Detailed reference viewed: 39 (5 UL)Geometric filling curves on surfaces ; Parlier, Hugo ; in Bull. Lond. Math. Soc. (2017), 49(4), 660--669 Detailed reference viewed: 4 (0 UL)Flip-graph moduli spaces of filling surfaces Parlier, Hugo ; in J. Eur. Math. Soc. (JEMS) (2017), 19(9), 2697--2737 Detailed reference viewed: 13 (0 UL)Distances in domino flip graphs Parlier, Hugo ; in Amer. Math. Monthly (2017), 124(8), 710--722 Detailed reference viewed: 13 (0 UL)On the homology length spectrum of surfaces ; Parlier, Hugo in Int. Math. Res. Not. IMRN (2017), (8), 2367--2401 Detailed reference viewed: 15 (0 UL)Arc and curve graphs for infinite-type surfaces ; ; Parlier, Hugo in Proc. Amer. Math. Soc. (2017), 145(11), 4995--5006 Detailed reference viewed: 13 (0 UL)Counting curves, and the stable length of currents ; Parlier, Hugo ; E-print/Working paper (2016) Let $\gamma_0$ be a curve on a surface $\Sigma$ of genus $g$ and with $r$ boundary components and let $\pi_1(\Sigma)\curvearrowright X$ be a discrete and cocompact action on some metric space. We study ... [more ▼] Let $\gamma_0$ be a curve on a surface $\Sigma$ of genus $g$ and with $r$ boundary components and let $\pi_1(\Sigma)\curvearrowright X$ be a discrete and cocompact action on some metric space. We study the asymptotic behavior of the number of curves $\gamma$ of type $\gamma_0$ with translation length at most $L$ on $X$. For example, as an application, we derive that for any finite generating set $S$ of $\pi_1(\Sigma)$ the limit $$\lim_{L\to\infty}\frac 1{L^{6g-6+2r}}\{\gamma\text{ of type }\gamma_0\text{ with }S\text{-translation length}\le L\}$$ exists and is positive. The main new technical tool is that the function which associates to each curve its stable length with respect to the action on $X$ extends to a (unique) continuous and homogenous function on the space of currents. We prove that this is indeed the case for any action of a torsion free hyperbolic group. [less ▲] Detailed reference viewed: 41 (2 UL) |
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