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

Abstract

Let $\xi$ and $\eta$ be two non--commuting isometries of the hyperbolic $3$--space $\mathbb{H}^3$ so that $\Gamma=\langle\xi,\eta\rangle$ is a purely loxodromic free Kleinian group. For $\gamma\in\Gamma$ and $z\in\mathbb{H}^3$, let $d_{\gamma}z$ denote the distance between $z$ and $\gamma\cdot z$. Let $z_1$ and $z_2$ be the mid-points of the shortest geodesic segments connecting the axes of $\xi$, $\eta\xi\eta^{-1}$ and $\eta^{-1}\xi\eta$, respectively. In this manuscript it is proved that if $d_{\gamma}z_2<1.6068...$ for every $\gamma\in\{\eta, \xi^{-1}\eta\xi, \xi\eta\xi^{-1}\}$ and $d_{\eta\xi\eta^{-1}}z_2\leq d_{\eta\xi\eta^{-1}}z_1$, then $$|\text{trace}^2(\xi)-4|+|\text{trace}(\xi\eta\xi^{-1}\eta^{-1})-2|\geq 2\sinh^2\left(\tfrac{1}{4}\log\alpha\right) = 1.5937....$$ Above $\alpha=24.8692...$ 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.Comment: A contradiction with Theorem 4.1 in v3, named as Theorem 4.2 in this version, arose while rephrasing of Theorem 4.2 in v3. This was fixed by restating Theorem 4.2, which was named as Theorem 4.3 in this version. Lemma 4.1 is due to the anonymous referee. Conjecture 4.2 was also restated accordingly. No changes occurred in the computations otherwise. 26 pages, 3 Figure