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

Stallings Graphs, Algebraic Extensions and Primitive Elements in F2