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 PSL2(C). In particular, given a finitely generated purely loxodromic free Kleinian group Γ = ⟨ξ1, ξ2,. . ., ξn⟩ for n ≥ 2, we show that |trace2(ξi) - 4| + |trace(ξiξjξ-1i ξ-1j) - 2| ≥ 2 sinh2 (14 log αn) for some ξi and ξj for i ̸= j in Γ provided that certain conditions on the hyperbolic displacements given by ξi, ξj and their length 3 conjugates formed by the generators are satisfied. Above, the constant αn 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 the Karush-Kuhn-Tucker theory. The use of this theory in the context of hyperbolic geometry is another novelty of this work. [ABSTRACT FROM AUTHOR]