In this talk, I will prove a generalization of a discreteness criteria for a large class of subgroups of $\operatorname{PSL}_{2}(\mathbb{C})$. In particular, I will show that for a given finitely generated, purely loxodromic, free Kleinian group $\Gamma=\left\langle\xi_{1}, \xi_{2}, \ldots, \xi_{n}\right\rangle$ for $n \geq 2$, the inequality
$$ \left|\operatorname{trace}^{2}\left(\xi_{i}\right)-4\right|+\left|\operatorname{trace}\left(\xi_{i} \xi_{j} \xi_{i}^{-1} \xi_{j}^{-1}\right)-2\right| \geq 2 \sinh ^{2}\left(\frac{1}{4} \log \alpha_{n}\right) $$
holds for some $\xi_{i}$ and $\xi_{j}$ for $i \neq j$ in $\Gamma$ provided that certain conditions on the hyperbolic displacements given by $\xi_{i}, \xi_{j}$ and their length 3 conjugates formed by the generators are satisfied. Above, the constant $\alpha_{n}$ turns out to be the real root strictly larger than $(2 n-1)^{2}$ of a fourth degree, integer coefficient polynomial obtained by solving a family of optimization problems via Karush-Kuhn-Tucker theory. This result is a joint work with A. Nedim Narman.

References

[1] Beardon AF. The Geometry of Discrete Groups. Berlin, Germany: Springer-Verlag, 1983.

[2] Borwein, JM, Lewis A. Convex Analysis and Nonlinear Optimization (Second Edition) CMSSpringer, ISBN-13 : 978-0387-29570-1

[3] Culler M, Shalen PB. Paradoxical decompositions, 2-generator Kleinian groups, and volumes of hyperbolic 3-manifolds. J Am Math Soc 1992; 5: 231-288.

[4] Jörgensen T. On discrete groups of Möbius transformations. Am J Math 1976; 98: 739-749.

[5] Yüce, İS. Two-generator free Kleinian groups and hyperbolic displacements, Algebraic and Geometric Topology 14:6 (2014), 3141-3184.

[6] Yüce, İS. Symmetric decompositions of free Kleinian groups and hyperbolic displacements, Communications in Analysis and Geometry, 26:6 (2018), 1375-1447.

[7] Yüce, İS. Jörgensen's Inequality and Purely Loxodromic 2-Generator Free Kleinian Groups, Turkish Journal of Mathematics, 43:2 (2019), 833-861.