A conjecture on the lengths of filling pairs
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A conjecture on the lengths of filling pairs Bidyut Sanki1
· Arya Vadnere2
Received: 27 September 2019 / Accepted: 11 November 2020 © Springer Nature B.V. 2020
Abstract A pair (α, β) of simple closed geodesics on a closed and oriented hyperbolic surface Mg of genus g is called a filling pair if the complementary components of α ∪ β on Mg are simply connected. The length of a filling pair is defined to be the sum of their individual lengths. In Aougab and Huang (Algebr Geom Topol 15:903–932, 2015), Aougab–Huang conjectured m that the length of any filling pair on Mg is at least 2g , where m g is the perimeter of the regular right-angled hyperbolic (8g − 4)-gon. In this paper, we prove a generalized isoperimetric inequality for disconnected regions and we prove the Aougab–Huang conjecture as a corollary. Keywords Hyperbolic surfaces · Filling pairs · Generalized systole · Isoperimetric inequality · Gauss-Bonnet theorem Mathematics Subject Classification 57K20 · 51M16 · 51M25
1 Introduction Let Mg be a closed and oriented hyperbolic surface of genus g. A pair (α, β), of simple closed curves on Mg is called a filling pair if the complement of their union α ∪ β in Mg is a disjoint union of topological discs. It is assumed that the curves α and β are in minimal position, i.e., the geometric intersection number i(α, β) is equal to |α ∩ β| (see Section 1.2.3 in [5]). To a filling pair one can associate a natural number k, the number of topological discs in the complement Mg \(α ∪ β). A filling pair (α, β) is minimal when k = 1. For a minimal filling pair (α, β) of Mg , the geometric intersection number is given by i(α, β) = 2g − 1 (see Lemma 2.1 in [2]).
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Arya Vadnere [email protected] Bidyut Sanki [email protected]
1
Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, Uttar Pradesh 208016, India
2
Chennai Mathematical Institute, Siruseri, Tamil Nadu 603103, India
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Geometriae Dedicata
The set of all closed and oriented hyperbolic surfaces of genus g ≥ 2, up to isometry, is called the moduli space of genus g and is denoted by Mg . The length of a filling pair (α, β) on a hyperbolic surface Mg ∈ Mg is defined by the sum of their individual lengths: L Mg (α, β) = l Mg (α) + l Mg (β), where l Mg (α) denotes the length of the geodesic representative in the free homotopy class [α] of α on Mg . If (α, β) is a filling pair of a hyperbolic surface Mg ∈ Mg , then we assume that α and β are simple closed geodesics. When we cut Mg open along a minimal filling pair, we obtain a hyperbolic (8g − 4)-gon with area 4π(g − 1) which is equal to the area of the surface Mg . The length of the filling pair is equal to half of the perimeter of this (8g − 4)-gon. It is known that among all hyperbolic n-gons with a fixed area, the regular n-gon has the least perimeter (see Bezdek [3]). In particular, we see that a regular right-angled (8g −4)-gon, denoted by Pg , has the least perimeter among all (8g − 4)-gons with fixed area 4π(g − 1). Thus, if m g is the perimeter of a hyperbolic regular right-angled
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