Abstract :
[en] 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.
Funding text :
Received by the editors February 26, 2020, and, in revised form, September 15, 2020, and December 9, 2020. 2020 Mathematics Subject Classification. Primary 32G15, 57K20, 37D20; Secondary 30F10, 30F60, 53C23, 57M50. Key words and phrases. Geometric identities, hyperbolic surfaces, pairs of pants. Research of the first author was supported by FNR PRIDE15/10949314/GSM. Research of the second author was partially supported by ANR/FNR project SoS, INTER/ANR/16/11554412/SoS, ANR-17-CE40-0033. Research of the third author was partially supported by R146-000-289-114.
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