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Measuring Pants Doan, Nhat Minh ; Parlier, Hugo ; E-print/Working paper (2020) We investigate the terms arising in an identity for hyperbolic surfaces proved by Luo and Tan, namely showing that they vary monotonically in terms of lengths and that they verify certain convexity ... [more ▼] We investigate the terms arising in an identity for hyperbolic surfaces proved by Luo and Tan, namely showing that they vary monotonically in terms of lengths and that they verify certain convexity properties. Using these properties, we deduce two results. As a first application, we show how to deduce a theorem of Thurston which states, in particular for closed hyperbolic surfaces, that if a simple length spectrum "dominates" another, then in fact the two surfaces are isometric. As a second application, we show how to find upper bounds on the number of pairs of pants of bounded length that only depend on the boundary length and the topology of the surface. [less ▲] Detailed reference viewed: 100 (55 UL)CARRIER GRAPHS FOR REPRESENTATIONS OF THE RANK TWO FREE GROUP INTO ISOMETRIES OF HYPERBOLIC THREE SPACE ; Xu, Binbin E-print/Working paper (2018) Carrier graphs were first introduced for closed hyperbolic 3-manifolds by White. In this paper, we first generalize this definition to carrier graphs for representations of a rank two free group into the ... [more ▼] Carrier graphs were first introduced for closed hyperbolic 3-manifolds by White. In this paper, we first generalize this definition to carrier graphs for representations of a rank two free group into the isometry group of hyperbolic three space. Then we prove the existence and the finiteness of minimal carrier graphs for those representations which are discrete, faithful and geometrically finite, and more generally, those that satisfy certain finiteness conditions first introduced by Bowditch. [less ▲] Detailed reference viewed: 53 (1 UL)A new identity for SL(2,C)-characters of the once punctured torus group Hu, Hengnan ; ; in Mathematical Research Letters (2015), 22(2), 485499 We obtain new variations of the original McShane identity for those SL(2,C)–representations of the once punctured torus group which satisfy the Bowditch conditions, and also for those fixed up to ... [more ▼] We obtain new variations of the original McShane identity for those SL(2,C)–representations of the once punctured torus group which satisfy the Bowditch conditions, and also for those fixed up to conjugacy by an Anosov mapping class of the torus and satisfying the relative Bowditch conditions. [less ▲] Detailed reference viewed: 111 (5 UL)New identities for small hyperbolic surfaces Hu, Hengnan ; in Bulletin of the London Mathematical Society (2014), 46(5), 10211031 Luo and Tan gave a new identity for hyperbolic surfaces with/without geodesic boundary in terms of dilogarithms of the lengths of simple closed geodesics on embedded three-holed spheres or one-holed tori ... [more ▼] Luo and Tan gave a new identity for hyperbolic surfaces with/without geodesic boundary in terms of dilogarithms of the lengths of simple closed geodesics on embedded three-holed spheres or one-holed tori in Luo and Tan [‘A dilogarithm identity on moduli spaces of curves’, J. Differential Geom., Preprint, 2011, arXiv:1102.2133[math.GT]]. However, the identity was trivial for a hyperbolic one-holed torus with geodesic boundary. In this paper, we adapt the argument from Luo and Tan to give an identity for hyperbolic tori with one geodesic boundary or cusp in terms of dilogarithm functions on the set of lengths of simple closed geodesics on the torus. As a corollary, we are also able to express the Luo–Tan identity as a sum over all immersed three-holed spheres P which are embeddings when restricted to the interior of P [less ▲] Detailed reference viewed: 94 (5 UL) |
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