Geodesics in hyperbolic 3-folds (Q1371327)

A Kleinian group \(G\) is a discrete, nonelementary subgroup of \(I\text{ som}^+(\mathbb{H}^3)\), where \(I\text{ som}^+(\mathbb{H}^3)\) is the group of orientation preserving isometries of the 3-hyperbolic space \(\mathbb{H}^3\). In the discussed paper, nonelementary means that the group \(G\) is not virtually Abelian. A hyperbolic 3-orbifold (manifold) is the orbit space \(\mathbb{H}^3/G\) of a (torsion-free) Kleinian group \(G\). The paper is concerned with the length \(l()\) of intersecting and nonsimple closed geodesics in hyperbolic 3-manifolds and 3-orbifolds. By a geodesic \(\alpha\) the authors mean a complete geodesic in the induced metric of constant curvature \(-1\). A closed geodesic is simple if it is embedded or a power of a closed embedded geodesics; otherwise is nonsimple. There are proved: Theorem 1. If \(\alpha_1\) and \(\alpha_2\) are closed geodesics in a hyperbolic 3-fold that intersect at an angle \(\phi\) where \(0<\phi<\pi\), then \[ \sinh(l(\alpha_1))\sinh(l(\alpha_2)) \sin^{3/4}(\phi)\geq l^2_1, \] where \(0.121\leq l_1\). The exponent of \(\sin(\phi)\) cannot be replaced by any constant greater than \(4/3\). Theorem 2. If \(\alpha\) is a closed geodesics in a hyperbolic 3-fold with a selfintersection of angle \(\phi\) where \(0<\phi<\pi\), then \[ \sinh(l(\alpha)) \sin(\phi)\geq l_2, \] where \(0.122\leq l_2\). The exponent of \(\sin(\phi)\) cannot be replaced by any constant greater than 1. The authors mentioned that the above theorems are motivated by a result due to \textit{A. Beardon} [Theorem 11.6.8 in The geometry of discrete groups (1983; Zbl 0528.30001)] which gives sharp estimates for the case of closed geodesics on Riemann surfaces. The metohds of proofs used the theory of Kleinian groups.