Related papers: Non-simple systoles on random hyperbolic surfaces …

We give upper bounds for $L^p$ norms of eigenfunctions of the Laplacian on compact hyperbolic surfaces in terms of a parameter depending on the growth rate of the number of short geodesic loops passing through a point. When the genus $g \to…

We exhibit closed hyperbolic manifolds with arbitrarily small systole in each dimension that are not quasi-arithmetic in the sense of Vinberg, and are thus not commensurable to those constructed by Agol, Belolipetsky--Thomson, and…

The systole of a closed Riemannian manifold is the minimal length of a non-contractible closed loop. We give a uniform lower bound for the systole for large classes of simple arithmetic locally symmetric orbifolds. We establish new bounds…

We present two constructions, both inspired by ideas from graph theory, of sequences random surfaces of growing area, whose systoles grow logarithmically as a function of their area. This also allows us to prove a new lower bound on the…

We present a new construction of embedded minimal surfaces in hyperbolic space with $3$ asymptotically totally geodesic ends and arbitrary finite genus.

We study uniformly random maps with a single face, genus $g$, and size $n$, as $n,g\rightarrow \infty$ with $g = o(n)$, in continuation of several previous works on the geometric properties of "high genus maps". We calculate the number of…

We find bounds for the length of the systole -- the shortest essential, non-peripheral closed curve -- for arithmetic punctured spheres with $n$ cusps, for $n=4$ through $n=12$, some of which were previously known due to Schmutz. This is…

The main goal of this article is to understand how the length spectrum of a random surface depends on its genus. Here a random surface means a surface obtained by randomly gluing together an even number of triangles carrying a fixed metric.…

We study the distribution of geometrically and topologically nearly geodesic random surfaces in a closed hyperbolic 3-manifold M. In particular, we describe PSL(2,R) invariant measures on the Grassmann bundle G(M) which arise as limits of…

The so-called {\it kissing number} for hyperbolic surfaces is the maximum number of homotopically distinct systoles a surface of given genus $g$ can have. These numbers, first studied (and named) by Schmutz Schaller by analogy with lattice…

The goal of the article is to provide different explicit quantifications of the non density of simple closed geodesics on hyperbolic surfaces. In particular, we show that within any embedded metric disk on a surface, lies a disk of radius…

Given a closed surface S of genus at least 2, we compare the symplectic structure of Taubes' moduli space of minimal hyperbolic germs with the Goldman symplectic structure on the character variety X(S, PSL(2,C)) and the affine cotangent…

We study large uniform random maps with one face whose genus grows linearly with the number of edges, which are a model of discrete hyperbolic geometry. In previous works, several hyperbolic geometric features have been investigated. In the…

Let $(M,g)$ be a closed, oriented, Riemannian manifold of dimension $m$. We call a systole a shortest non-contractible loop in $(M,g)$ and denote by $sys(M,g)$ its length. Let $SR(M,g)=\frac{{sys(M,g)}^m}{vol(M,g)}$ be the systolic ratio of…

The moduli spaces of hyperbolic surfaces of genus g with n geodesic boundary components are naturally symplectic manifolds. Mirzakhani proved that their volumes are polynomials in the lengths of the boundaries by computing the volumes…

The systole of a hyperbolic surface is bounded by a logarithmic function of its genus. This bound is sharp, in that there exist sequences of surfaces with genera tending to infinity that attain logarithmically large systoles. These are…

Periodic geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle $\mathrm{PSL}_2(\mathbb{Z})\backslash\mathrm{PSL}_2(\mathbb{R})$. The complement of any finite number of orbits is a…

In this paper we obtain a bound on the number of isometry classes of finite area hyperbolic surfaces which are length isospectral to a given surface depending only on the topological type of the surface and the length of the shortest closed…

We study the geometry of hyperbolic cone surfaces, possibly with cusps or geodesic boundaries. We prove that any hyperbolic cone structure on a surface of non-exceptional type is determined up to isotopy by the geodesic lengths of a finite…

Let $\Sigma$ be a closed hyperbolic surface. We study, for fixed $g$, the asymptotics of the number of those periodic geodesics in $\Sigma$ having at most length $L$ and which can be written as the product of $g$ commutators. The basic idea…