Jörgensen’s inequality and purely loxodromic two-generator free Kleinian groups

Abstract

Let ξ and η be two noncommuting isometries of the hyperbolic 3 -space H3 so that Γ = ⟨ξ, η⟩ is a purely loxodromic free Kleinian group. For γ ∈ Γ and z ∈ H3 , let dγ z denote the hyperbolic distance between z and γ(z) . Let z1 and z2 be the midpoints of the shortest geodesic segments connecting the axis of ξ to the axes of ηξη−1 and η−1ξη , respectively. In this manuscript, it is proved that if dγ z2 < 1.6068... for every γ ∈ {η, ξ−1ηξ, ξηξ−1} and dηξη−1 z2 ≤ dηξη−1 z1 , then |trace2(ξ) − 4| + |trace(ξηξ−1η−1) − 2| ≥ 2 sinh2 1 log α = 1.5937.... Above α = 24.8692...is the unique real root of the polynomial 21x4 − 496x3 − 654x2 + 24x + 81 that is greater than 9 . Generalizations of this inequality for finitely generated purely loxodromic free Kleinian groups are also proposed.