A lower bound for the volumes of complements of periodic geodesics

Abstract

Every closed geodesic γ on a surface has a canonically associated knotγ in the projective unit tangent bundle. We study, for γ filling, the volume of the associated knot complement with respect to its unique complete hyperbolic metric.

Hyperbolicity of link complements in Seifert-fibered spaces

“…In section 3 we prove Theorem 1.3 and Theorem 1.2. In section 4 by using results in [HP18] and [RM17] we prove some volume bounds.…”

“…Given α 1 and α 2 be a filling closed geodesics on Σ g,1 and Σ 1,2 respectively. Let α 1 s α 2 be the closed geodesic homotopic to a closed curve obtained by surgering α 1 and α 2 along a simple arc meeting transversely one boundary component in each surface, see [RM17,Subsec. 4.2].…”

“…Similarly to [RM17], given any geodesic multi-curve γ and any continuous lift s γ, one has a combinatorial lower bound for the volume of M where v 3 the volume of the regular ideal tetrahedron.…”

“…The volume invariant has been studied in the particular case of canonical lifts of geodesics in the projective unit tangent bundle P T 1 (Σ) of a hyperbolic surface Σ or the modular orbifold. Upper bounds have been found in terms of the geodesic length in [BPS17] and a combinatorial lower bound by the second author in [RM17].…”

Hyperbolicity of links complements in Seifert fibered spaces

“…Similarly to [RM17], given any geodesic multi-curve γ and any continuous lift s γ, one has a combinatorial lower bound for the volume of M s γ . Recall that a pants decomposition on an orbifold O, is a maximal family of disjoint simple closed geodesics on the underlying topological surface Σ O which do not intersect the singular points of O.…”