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

TL;DR

This paper generalizes a discreteness criterion for purely loxodromic free Kleinian groups, establishing trace inequalities linked to hyperbolic displacements and employing optimization techniques from Karush-Kuhn-Tucker theory.

Contribution

It introduces a new trace inequality for free Kleinian groups and applies KKT optimization methods to hyperbolic geometry, expanding understanding of group discreteness criteria.

Findings

Established a trace inequality involving group elements and their conjugates.

Derived a polynomial with a real root determining the inequality constant.

Applied KKT theory to hyperbolic geometric problems.

Abstract

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 $\Gamma=\langle\xi_1,\xi_2,\dots,\xi_n\rangle$ for $n\geq 2$, the inequality $$\left|\text{trace}^2(\xi_i)-4\right|+\left|\text{trace}(\xi_i\xi_j\xi_i^{-1}\xi_j^{-1})-2\right|\geq 2\sinh^2\left(\frac{1}{4}\log\alpha_n\right)$$ 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 $(2n-1)^2$ 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.