Dependence and algebraicity over subgroups of free groups

Abstract

An element g of a free group F depends on a subgroup H < F , or is algebraic over H, if it satisfies a univariate equation over H. Equivalently, when H is finitely-generated, the rank of H, g is at most the rank of H. We study the elements that depend on subgroups of F , the subgroups they generate and the equations that they satisfy.