Isometries of length 1 in free Kleinian groups and trace inequalities

In this thesis, a generalization of a discreteness criteria for a large class of subgroups of PSL2(C) is proven. In particular, given a finitely generated, purely loxodromic, free Kleinian group Γ = 〈ξ1, ξ2, . . . , ξn〉 for n ≥ 2, the inequality |trace^2(ξi) − 4| + |trace(ξiξjξi^-1ξj^-1) − 2| ≥ 2 sinh^2(1/4 log(αn)) holds for some ξi and ξj for i not equal 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 Karush-Kuhn-Tucker theory. The use of this theory in the context of hyperbolic geometry is another novelty of this work.