Lengthening deformations of singular hyperbolic tori

Abstract

Abstract. Let S be a torus with a hyperbolic metric admitting one puncture or cone singularity. We describe which infinitesimal deformations of S lengthen (or shrink) all closed geodesics. We also study how the answer degenerates when S becomes Euclidean, i.e. very small.Résumé. Soit S un tore muni d'une métrique hyperbolique admettant un trou ou une singularité conique. Nous décrivons quelles déformations infinitésimales de S allongent (ou raccourcissent) toutes les géodésiques fermées. Nousétudions aussi comment la réponseà cette question dégénère lorsque S devient euclidienne, c'est-à-dire très petite.

Coarse and Fine Geometry of the Thurston Metric

“…The idea of the proof is to recognize lengths and intersection numbers of curves on X from features of the unit sphere in T X T(S). Analogous estimates for the shape of the cone of lengthening deformations of a hyperbolic one-holed torus were established in [Gué15]. In fact, Theorem 1.4 was known to Guéritaud and can be derived from those estimates [Gué16].…”

Strip maps of small surfaces are convex

“…• In the case of the punctured torus, we can extend Theorem 1.3 to singular hyperbolic metrics (Theorem 4.1), replacing the boundary component with a cone point of angle θ ∈ (0, 2π). Proposition 1.1 was already extended to that singular context in [2]. Theorem 4.2 also treats the intermediate case of a cusped metric (θ = 0).…”

“…The increment at the edge γ, or DA ′ , is ψ(pDA ′ ) − ψ(pDA ′ ) = (D − A ′ )(sinh a + sinh d), an infinitesimal loxodromy with axis perpendicular to 2 Moreover, all four infinitesimal translation axes run through p, because all four vectors have vanishing third coordinate; but we will not use this fact.…”

Margulis spacetimes via the arc complex

“…The group GL 2 (Z) acts on X , transitively on the vertices, via the mapping class group of the once-holed torus. We refer to [23] or [25] for more details about adm(ρ) and its closure in this case.…”