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

Abstract

In this paper, we prove a generalization of a discreteness criteria for a large class of subgroups of PSL 2(C). In particular, given a finitely generated purely loxodromic free Kleinian group Gamma = for n >= 2, we show that |trace(2) (xi(i)) - 4| + |trace(xi(i)xi(j)xi(-1)(i) xi(-1)(j)) - 2| >= 2 sinh(2) ( 1/4 log alpha(n)) for some xi(i) and xi(j) for i not equal 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 the Karush-Kuhn-Tucker theory. The use of this theory in the context of hyperbolic geometry is another novelty of this work.