Université de Fribourg

Systoles and kissing numbers of finite area hyperbolic surfaces

Fanoni, Federica ; Parlier, Hugo

In: Algebraic & Geometric Topology, 2016, vol. 15, no. 6, p. 3409–3433

We study the number and the length of systoles on complete finite area orientable hyperbolic surfaces. In particular, we prove upper bounds on the number of systoles that a surface can have (the so-called kissing number for hyperbolic surfaces). Our main result is a bound which only depends on the topology of the surface and which grows subquadratically in the genus.

Université de Fribourg

Curve graphs on surfaces of infinite type

Fossas, Ariadna ; Parlier, Hugo

In: Annales Academiae Scientiarum Fennicae Mathematica, 2015, vol. 40, p. 793–801

We define and study analogs of curve graphs for infinite type surfaces. Our definitions use the geometry of a fixed surface and vertices of our graphs are infinite multicurves which are bounded in both a geometric and a topological sense. We show that the graphs we construct are generally connected, infinite diameter and infinite rank.

Université de Fribourg

A short note on short pants

Parlier, Hugo

In: Canadian Mathematical Bulletin, 2015, vol. 57, no. 4, p. 870–876

It is a theorem of Bers that any closed hyperbolic surface admits a pants decomposition consisting of curves of bounded length where the bound only depends on the topology of the surface. The question of the quantification of the optimal constants has been well studied and the best upper bounds to date are linear in genus, a theorem of Buser and Seppälä. The goal of this note is to give a short...

Université de Fribourg

Convexity of strata in diagonal pants graphs of surfaces

Aramayona, Javier ; Lecuire, Cyril ; Parlier, Hugo ; Shackleton, Kenneth J.

In: Publicacions Matemàtiques, 2012, vol. 57, no. 1, p. 219-237

We prove a number of convexity results for strata of the diagonal pants graph of a surface, in analogy with the extrinsic geometric properties of strata in the Weil-Petersson completion. As a consequence, we exhibit convex flat subgraphs of every possible rank inside the diagonal pants graph.

Université de Fribourg

Constructing metrics on a 2-torus with a partially prescribed stable norm

Makover, Eran ; Parlier, Hugo ; Sutton, Craig J.

In: Manuscripta Mathematica, 2012, vol. 139, no. 3-4, p. 515-534

A result of Bangert states that the stable norm associated to any Riemannian metric on the 2-torus T ² is strictly convex. We demonstrate that the space of stable norms associated to metrics on T ² forms a proper dense subset of the space of strictly convex norms on R2{/span> . In particular, given a strictly convex norm || · ||∞ on R2{/span> we construct a...

Université de Fribourg

Bers' constants for punctured spheres And hyperelliptic surfaces

Balacheff, Florent ; Parlier, Hugo

In: Journal of Topology and Analysis, 2012, vol. 4, no. 3, p. 271

Université de Fribourg

Short loop decompositions of surfaces and the geometry of jacobians

Balacheff, Florent ; Parlier, Hugo ; Sabourau, Stéphane

In: Geometric and functional analysis, 2012, vol. 22, no. 1, p. 37-73

Given a Riemannian surface, we consider a naturally embedded graph which captures part of the topology and geometry of the surface. By studying this graph, we obtain results in three different directions.First, we find bounds on the lengths of homologically independent curves on closed Riemannian surfaces. As a consequence, we show that for any l Î (0, 1) there exists a constant C λ such that...

Université de Fribourg

Small filling sets of curves on a surface

Anderson, James W. ; Parlier, Hugo ; Pettet, Alexandra

In: Topology and its Applications, 2011, vol. 158, no. 1, p. 84-92

We show that the asymptotic growth rate for the minimal cardinality of a set of simple closed curves on a closed surface of genus g which fill and pairwise intersect at most Kgreater-or-equal, slanted1 times is View the MathML source as g→∞. We then bound from below the cardinality of a filling set of systoles by g/log(g). This illustrates that the topological condition that a set of curves...