arXiv subjects
Publications and source records attributed to Hugo Parlier.
Let $X$ be a closed hyperbolic surface. We prove that the number of topological types of primitive closed geodesics of length at most $L$ is asymptotic to \[ \frac{1}{|\Isom(X)|}\frac{e^L}{2L}. \] as $L$ grows. Thus Huber's asymptotic remains unchanged after quotienting by topological type, up to the finite symmetry factor coming from the isometry group of $X$.
Hyperbolic surfaces are a fundamental object in mathematics and play an increasingly important role in computational geometry and topology. A key ingredient in the design of efficient algorithms on such surfaces is the availability of a geometric discretization of controlled complexity. In this paper, we present the first algorithm for constructing e-nets on hyperbolic surfaces starting from a fundamental polygon representation. Our approach is based on Delaunay refinement and relies on maintaining Delaunay triangulations through edge flips. The size of an e-net cannot be bounded solely as a function of the genus because of the presence of arbitrarily long collars around short geodesics. To overcome this difficulty, we introduce the notion of a pseudo e-net, which decomposes the surface into e-thin cylinders together with a Delaunay triangulation over an e-net of the remaining thick part. As applications, we obtain algorithms for computing the length spectrum of an e-thick hyperbolic surface and for computing the systole from a pseudo log(sqrt(2))-net. These results demonstrate that Delaunay-based discretizations provide a practical and versatile framework for algorithmic computations on hyperbolic surfaces.
This article introduces and studies the entropy spectrum of a hyperbolic surface, that is the set of entropies of its subsurfaces. The main results are that the entropy spectrum is a reverse well-ordered multiset, with finite multiplicities, and that there is a quantifiable gap around the value $1$. This gap comes from a counting result on the number of non-filling geodesics which in turn comes from explicit estimates on the number of curves on surfaces with boundary in terms of geometric data. The geometric data includes lengths of boundary geodesics and so-called boundary width, which measures maximal distance to the boundary but can also be interpreted in terms of the topology of the surface and the systole.
This paper is about closed hyperbolic surface amalgams with a focus on the growth of the number of closed geodesics. As in the case of surfaces, we show that topological and volume entropies coincide, but we show stark differences in how they behave according to geometric data with upper and lower bounds on the number of closed geodesics which depend on the length of the systole and the length of the pasting curves. In particular, we show that the entropy can increase exponentially in terms of the pasting length in the absence of a lower bound on the systole.
We determine the crossing number of polynomial size curve systems on standard surfaces, in terms of the genus, up to high precision.
We establish graded versions of Bridgeman's dilogarithm identity for hyperbolic cone surfaces, including surfaces with only cusps and cone points, and provide applications to the study of orthogeodesics.
Mathematics is often perceived as difficult or inaccessible, yet meaningful engagement can arise in unexpected places. In this article we describe a multi-year exploration of mathematical outreach through games, puzzles, exhibitions, and artistic activities. Starting from a small science festival exhibit, our work developed into a broad collection of experiences showcased at game festivals, museums, and the World Expos in Dubai (2021/2022) and Osaka (2025). We discuss the principles that shaped these activities -- simplicity, atmosphere, mediation, and progressive depth -- and how mathematical ideas from topology, geometry, and logic can be incorporated into playful and creative formats. The story illustrates how low-threshold engagement and carefully designed experiences can open doors to research-level mathematics for audiences of all ages and backgrounds.
This article gives a short proof that all ideal polygons admit a short orthogeodesic decomposition. Specifically, all $n$-gons admit an orthogeodesic decomposition with orthogeodesics all of length at most $\sim 2 \log(n)$, and this is roughly optimal.
This article describes a project called ReShape in which we created and designed a crowdsourced art initiative, inspired and powered by mathematics.
We study crossing numbers for systoles of congruence surfaces. Taken as a family of curves on a family of surfaces, we show that the growth rate of their intersection is optimally small among all sets of curves of the same cardinality lying on the same topological surface.
We identify a condition that prevents a hyperbolic space from being quasi-isometric to the curve complex of any non-sporadic surface. Our result applies to several hyperbolic complexes, including arc complexes, disk complexes, non-separating curve complexes, (hyperbolic) pants complexes, and to free splitting complexes of free groups.
In this paper, we provide upper and lower bounds on the crossing numbers of dense graphs on surfaces, which match up to constant factors. First, we prove that if $G$ is a dense enough graph with $m$ edges and $\Sigma$ is a surface of genus $g$, then any drawing of $G$ on $\Sigma$ incurs at least $\Omega \left(\frac{m^2}{g} \log ^2 g\right)$ crossings. The poly-logarithmic factor in this lower bound is new even in the case of complete graphs and disproves a conjecture of Shahrokhi, Sz\'ekely and Vrt'o from 1996. Then we prove a geometric converse to this lower bound: we provide an explicit family of hyperbolic surfaces such that for any graph $G$, sampling the vertices uniformly at random on this surface and connecting them with shortest paths yields $O\left(\frac{m^2}{g} \log ^2 g\right)$ crossings in expectation.
We study the order of lengths of closed geodesics on hyperbolic surfaces. Our first main result is that the order of lengths of curves determine a point in Teichm\"uller space. In an opposite direction, we identify classes of curves whose order never changes, independently of the choice of hyperbolic metric. We use this result to identify short curves with small intersections on pairs of pants.
Consider a surface $\Sigma$ with punctures that serve as marked points and at least one marked point on each boundary component. We build a filling surface $\Sigma_n$ by singling out one of the boundary components and denoting by $n$ the number of marked points it contains. We consider the triangulations of $\Sigma_n$ whose vertices are the marked points and the associated flip-graph $\mathcal{F}(\Sigma_n)$. Quotienting $\mathcal{F}(\Sigma_n)$ by the homeomorphisms of $\Sigma$ that fix the privileged boundary component results in a finite graph $\mathcal{MF}(\Sigma_n)$. Bounds on the diameter of $\mathcal{MF}(\Sigma_n)$ are available when $\Sigma$ is orientable and we provide corresponding bounds when $\Sigma$ is non-orientable. We show that the diameter of this graph grows at least like $5n/2$ and at most like $4n$ as $n$ goes to infinity. If $\Sigma$ is an unpunctured M\"obius strip, $\mathcal{MF}(\Sigma_n)$ coincides with $\mathcal{F}(\Sigma_n)$ and we prove that the diameter of this graph grows exactly like $5n/2$ as $n$ goes to infinity.
We prove that, as $m$ grows, any family of $m$ homotopically distinct closed curves on a surface induces a number of crossings that grows at least like $(m \log m)^2$. We use this to answer two questions of Pach, Tardos and Toth related to crossing numbers of drawings of multigraphs where edges are required to be non-homotopic. Furthermore, we generalize these results, obtaining effective bounds with optimal growth rates on every orientable surface.
We show a generalization of the crossing lemma for multi-graphs drawn on orientable surfaces in which pairs of edges are assumed to be drawn by non-homotopic simple arcs which pairwise cross at most $k$ times.
This paper explores the relationship between closed curves on surfaces and their intersections. Like Dehn-Thurston coordinates for simple curves, we explore how to determine closed curves using the number of times they intersect other curves. We construct and study $k$-equivalent curves: these are distinct curves that intersect all curves with $k$ self-intersection points the same number of times. We show that such curves must intersect all simple curves in the same way, but that all other possible implications fail. Our methods give a quantitative approach to a theorem of Otal which shows that curves are determined by their intersection with all other curves. In the opposite direction, we show that non-simple curves can only be distinguished by looking at their intersection with infinitely many curves.
This paper proves an elementary topological fact about closed curves on surfaces, namely that by carefully smoothing an intersection point, one can reduce self-intersection by exactly $1$. This immediately implies a positive answer to a problem first raised by Basmajian in the 1990s: among all closed geodesics of a hyperbolic surface that self-intersect at least $k$ times, does the shortest one self-intersect exactly $k$ times? The answer is also shown to be positive for arbitrary Riemannian metrics.