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

Authors: İLKER SAVAŞ YÜCE

Abstract: Let $\xi$ and $\eta$ be two noncommuting 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 $z\in\hyp$, let $d_{\gamma}z$ denote the hyperbolic distance between $z$ and $\gamma(z)$. Let $z_1$ and $z_2$ be the midpoints of the shortest geodesic segments connecting the axis of $\xi$ to the axes of $\eta\xi\eta^{-1}$ 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 %\begin{equation*} $|\text{trace}^2(\xi)-4|+|\text{trace}(\xi\eta\xi^{-1}\eta^{-1})-2|\geq 2\sinh^2\left(\tfrac{1}{4}\log\alpha\right) = 1.5937....$ %\end{equation*} 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 $9$. Generalizations of this inequality for finitely generated purely loxodromic free Kleinian groups are also proposed.

Keywords: Free Kleinian groups, Jörgensen's inequality, the $\log 3$ theorem, loxodromic isometries, hyperbolic displacements

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