arXiv subjects

Hugo Parlier

Publications and source records attributed to Hugo Parlier.

At least 37 records · Page 2Linked to original sources

arXiv2023

Counting geodesics between surface triangulations

Given a surface $Σ$ equipped with a set $P$ of marked points, we consider the triangulations of $Σ$ with vertex set $P$. The flip-graph of $Σ$ whose vertices are these triangulations, and whose edges correspond to flipping arcs appears in the study of moduli spaces and mapping class groups. We consider the number of geodesics in the flip-graph of $Σ$ between two triangulations as a function of their distance. We show that this number grows exponentially provided the surface has enough topology, and that in the remaining cases the growth is polynomial.

math.GT↗

arXiv2023

A shorter note on shorter pants

This note is about variations on a theorem of Bers about short pants decompositions of surfaces. It contains a version for surfaces with boundary but also a slight improvement on the best known bound for closed surfaces.

math.GT↗

arXiv2023

Universal families of arcs and curves on surfaces

The main goal of this paper is to investigate the minimal size of families of curves on surfaces with the following property: a family of simple closed curves $Γ$ on a surface realizes all types of pants decompositions if for any pants decomposition of the surface, there exists a homeomorphism sending it to a subset of the curves in $Γ$. The study of such universal families of curves is motivated by questions on graph embeddings, joint crossing numbers and finding an elusive center of moduli space. In the case of surfaces without punctures, we provide an exponential upper bound and a superlinear lower bound on the minimal size of a family of curves that realizes all types of pants decompositions. We also provide upper and lower bounds in the case of surfaces with punctures which we can consider labelled or unlabelled, and investigate a similar concept of universality for triangulations of polygons, where we provide bounds which are tight up to logarithmic factors.

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arXiv2022

Computing a Dirichlet domain for a hyperbolic surface

The goal of this paper is to exhibit and analyze an algorithm that takes a given closed orientable hyperbolic surface and outputs an explicit Dirichlet domain. The input is a fundamental polygon with side pairings. While grounded in topological considerations, the algorithm makes key use of the geometry of the surface. We introduce data structures that reflect this interplay between geometry and topology and show that the algorithm finishes in polynomial time, in terms of the initial perimeter and the genus of the surface.

cs.CG↗

arXiv2022

The shortest non-simple closed geodesics on hyperbolic surfaces

This article explores closed geodesics on hyperbolic surfaces. We show that, for sufficiently large $k$, the shortest closed geodesics with at least $k$ self-intersections, taken among all hyperbolic surfaces, all lie on an ideal pair of pants and have length $2\arccosh(2k+1)$.

math.GT↗

arXiv2022

Quadratis Puzzles

We introduce a family of reconfiguration puzzles arising from ideas in geometry and topology. We present their construction from square-tiled shapes, discuss some of the underlying mathematics and describe how they are naturally associated to puzzle spaces which can be explored through visualization.

math.GT↗

arXiv2022

Quasisymmetric maps, shears, lambda lengths and flips

We study quasisymmetric maps, which act on the boundary of the hyperbolic plane, by looking at their action on the Farey triangulation. Our main results identify exactly which quasisymmetric maps correspond to pinched lambda lengths in terms of shearing coordinates and, separately, in terms of (simultaneous) flip distance of the Farey triangulation and its image by the map. These extend and clarify previous results of Penner and Sullivan.

math.GT↗

arXiv2022

The coarse geometry of hexagon decomposition graphs

We define and study graphs associated to hexagon decompositions of surfaces by curves and arcs. One of the variants is shown to be quasi-isometric to the pants graph, whereas the other variant is quasi-isometric to (a Cayley graph of) the mapping class group.

math.GT↗

arXiv2021

The topological types of length bounded multicurves

This article discusses inequalities on lengths of curves on hyperbolic surfaces. In particular, a characterization is given of which topological types of curves and multicurves always have a representative that satisfies a length inequality that holds over all of moduli space.

math.GT↗

arXiv2021

Bounded geometry with no bounded pants decomposition

We construct a quasiconformally homogeneous hyperbolic Riemann surface-other than the hyperbolic plane-that does not admit a bounded pants decomposition. Also, given a connected orientable topological surface of infinite type with compact boundary components, we construct a complete hyperbolic metric on the surface that has bounded geometry but does not admit a bounded pants decomposition.

math.GT↗

arXiv2021

The simple separating systole for hyperbolic surfaces of large genus

In this note we show that the expected value of the separating systole of a random surface of genus $g$ with respect to Weil-Petersson volume behaves like $2\log g $ as the genus goes to infinity. This is in strong contrast to the behavior of the expected value of the systole which, by results of Mirzakhani and Petri, is independent of genus.

math.GT↗

arXiv2021

Systoles and diameters of hyperbolic surfaces

In this article we explore the relationship between the systole and the diameter of closed hyperbolic orientable surfaces. We show that they satisfy a certain inequality, which can be used to deduce that their ratio has a (genus dependent) upper bound.

math.GT↗

arXiv2021

Ideally, all infinite type surfaces can be triangulated

We show that any surface of infinite type admits an ideal triangulation. Furthermore, we show that a set of disjoint arcs can be completed into a triangulation if and only if, as a set, they intersect every simple closed curve a finite number of times.

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arXiv2020

Minimal Delaunay triangulations of hyperbolic surfaces

Motivated by recent work on Delaunay triangulations of hyperbolic surfaces, we consider the minimal number of vertices of such triangulations. First, we will show that every hyperbolic surface of genus $g$ has a simplicial Delaunay triangulation with $O(g)$ vertices, where edges are given by distance paths. Then, we will construct a class of hyperbolic surfaces for which the order of this bound is optimal. Finally, to give a general lower bound, we will show that the $Ω(\sqrt{g})$ lower bound for the number of vertices of a simplicial triangulation of a topological surface of genus $g$ is tight for hyperbolic surfaces as well.

cs.CG↗

arXiv2020

Flip graphs for infinite type surfaces

We associate to triangulations of infinite type surface a type of flip graph where simultaneous flips are allowed. Our main focus is on understanding exactly when two triangulations can be related by a sequence of flips. A consequence of our results is that flip graphs for infinite type surfaces have uncountably many connected components.

math.GT↗

arXiv2020

Prime orthogeodesics, concave cores and families of identities on hyperbolic surfaces

We prove and explore a family of identities relating lengths of curves and orthogeodesics of hyperbolic surfaces. These identities hold over a large space of metrics including ones with hyperbolic cone points, and in particular, show how to extend a result of the first author to surfaces with cusps. One of the main ingredients in the approach is a partition of the set of orthogeodesics into sets depending on their dynamical behavior, which can be understood geometrically by relating them to geodesics on orbifold surfaces. These orbifold surfaces turn out to be exactly on the boundary of the space in which the underlying identity holds.

math.GT↗

arXiv2020

Geodesic and orthogeodesic identities on hyperbolic surfaces

The lengths of geodesics on hyperbolic surfaces satisfy intriguing equations, known as identities, relating these lengths to geometric quantities of the surface. This paper is about a large family of identities that relate lengths of closed geodesics and orthogeodesics to boundary lengths or number of cusps. These include, as particular cases, identities due to Basmajian, to McShane and to Mirzakhani and Tan-Wong-Zhang. In stark contrast to previous identities, the identities presented here include the lengths taken among all closed geodesics.

math.GT↗

arXiv2020

Measuring pants

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.

math.GT↗