Jorgensen's Inequality and Purely Loxodromic 2-Generator Free Kleinian Groups

Abstract

Let $ξ$ and $η$ be two non--commuting isometries of the hyperbolic $3$--space $\mathbb{H}^3$ so that $Γ=\langleξ,η\rangle$ is a purely loxodromic free Kleinian group. For $γ\inΓ$ and $z\in\mathbb{H}^3$, let $d_γz$ denote the distance between $z$ and $γ\cdot z$. Let $z_1$ and $z_2$ be the mid-points of the shortest geodesic segments connecting the axes of $ξ$, $ηξη^{-1}$ and $η^{-1}ξη$, respectively. In this manuscript it is proved that if $d_γz_2<1.6068...$ for every $γ\in\{η, ξ^{-1}ηξ, ξηξ^{-1}\}$ and $d_{ηξη^{-1}}z_2\leq d_{ηξη^{-1}}z_1$, then \[ |\text{trace}^2(ξ)-4|+|\text{trace}(ξηξ^{-1}η^{-1})-2|\geq 2\sinh^2\left(\tfrac{1}{4}\logα\right) = 1.5937.... \] Above $α=24.8692...$ is the unique real root of the polynomial $21 x^4 - 496 x^3 - 654 x^2 + 24 x + 81$ that is greater than $9$. Also generalisations of this inequality for finitely generated purely loxodromic free Kleinian groups are conjectured.