Systole, entropie et espace des modules des surfaces hyperboliques de type fini

Keywords

Hyperbolic geometry, Computational geometry

Subject details

This topic focuses on the geometry of hyperbolic surfaces through a dual approach—mathematical (Riemannian geometry) and computational (algorithmic geometry). A hyperbolic surface Σ is a complete surface equipped with a Riemannian metric of curvature −1. It can be seen as the quotient of the hyperbolic plane ℍ² by a discrete group of isometries Γ, which in this thesis is assumed to be generated by a finite number (strictly greater than one) of generators. Therefore, the surface has a genus g ≥ 0 and possibly (if non-compact) a number n of cusps (ends of finite volume) and k of funnels (ends of infinite volume), with g + k + n ≥ 2. In this work, we assume n = 0, though interesting variations in the presence of cusps may also be studied. The systole sys(Σ) of Σ is the length of its shortest closed geodesic. Its entropy hₛ is the exponential growth rate of the number of closed geodesics. It also corresponds to the critical exponent of the Poincaré series above which the series converges. When Σ is compact and of genus g, the systole is bounded above by a universal constant times log g (see, for instance, [Par12] and the references therein), and the entropy satisfies hₛ = 1. If Σ has at least one funnel, then its systole can be arbitrarily large, and the entropy satisfies hₛ ∈ (0, 1), while potentially being arbitrarily close to 0 or 1 (see, for example, [McM]). The following two questions lie at the heart of this thesis topic. The first is well-known and notoriously difficult, but algorithmic tools could provide new approaches; the second is less well-known and offers more accessible avenues of investigation: 1 If Σ is compact of genus g, what is the maximum possible value of its systole? Is this maximum (or an approximation thereof) achieved by explicit metrics? 2 If Σ has genus g with k ≥ 1 funnels, can the entropy be bounded above by a function depending only on g, k, and sys(Σ), which tends to 0 as sys(Σ) grows? If so, can one obtain an optimal estimate for such a bound? Are there metrics that nearly achieve this inequality?