Isometries of length $1$ in purely loxodromic free Kleinian groups and trace inequalities

In this paper, we prove a generalization of a discreteness criteria for a large class of subgroups of PSL${}_2(\mathbb{C})$. In particular, we show that for a given finitely generated, purely loxodromic, free Kleinian group $\mathrm{\Gamma }=⟨{\xi }_1,{\xi }_2,\dots ,{\xi }_n⟩$ for $n\ge 2$, the inequality $$\mid {\text{trace}}^2({\xi }_i)-4\mid +\mid \text{trace}({\xi }_i{\xi }_j{\xi }_i^{-1}{\xi }_j^{-1})-2\mid \ge 2{\sinh }^2(\frac{1}{4}\log {\alpha }_n)$$ holds for some ${\xi }_i$ and ${\xi }_j$ for $i\mathrm{\ne }j$ in $\mathrm{\Gamma }$ 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 $(2n-1{)}^2$ of a fourth degree, integer coefficient polynomial obtained by solving a family of optimization problems via Karush-Kuhn-Tucker theory. The use of this theory in the context of hyperbolic geometry is another novelty of this work.