Stallings Graphs, Algebraic Extensions and Primitive Elements in F2

TL;DR

This paper explores primitive elements in the free group of rank two using Stallings graphs, provides new proofs for known results, and constructs a counterexample to a conjecture on algebraic extensions.

Contribution

It offers new proofs for primitive element classifications and presents a counterexample to a conjecture relating Stallings graphs to algebraic extensions.

Findings

New proofs for primitive element classifications

Counterexample to the conjecture on algebraic extensions

Insights into Stallings graph representations

Abstract

This paper studies the free group of rank two from the point of view of Stallings core graphs. The first half of the paper examines primitive elements in this group, giving new and self-contained proofs for various known results about them. In particular, this includes the classification of bases of this group. The second half of the paper is devoted to constructing a counterexample to a conjecture by Miasnikov, Ventura and Weil, which seeks to characterize algebraic extensions in free groups in terms of Stallings graphs.