| Summary: | <p>In the present work we investigate various aspects of free groups and their automorphisms, utilizing Stallings' folding techniques introduced in 1983. These techniques have provided new insights into several classical results and have been widely applied over the years to study various properties of free groups and to generalize results to wider families of groups.</p>
<p>We apply folding techniques to uncover a fine property of the classical Whitehead's algorithm for recognizing primitive elements and free factors in a free group. This property is then used to the study of the complex of free factors, a crucial object on which Out(F_n) acts. We also make use of these techniques to investigate echelon subgroups of F_n, a particular type of subgroup which is relevant to the study of fixed-point subgroups of automorphisms of F_n.</p>
<p>Next we turn our attention to the problem of studying equations in a free group. Given two free groups H≤F and an element g ∈ F, we study the ideal of the equations w(x) in H* ⟨x⟩ with g as a solution. In particular, we focus on the relationship between equations and their degree, providing an algorithm that determines the minimum integer d that appears as the degree of an equation in a given ideal. More generally, we study the subset of the equation in a given ideal of a fixed degree d. This is done using two different techniques, one based on Stallings' folding operations and the other using context-free languages.</p>
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