On the genericity of Whitehead minimality
Résumé
We show that a finitely generated subgroups of a free group, chosen uniformly at random, is strictly Whitehead minimal. Whitehead minimality is one of the key elements of the solution of the orbit problem in free groups. The proofs strongly rely on automata theory and on the combinatorial tools used in the analysis of algorithms. The result we prove actually depends implicitly on the choice of a distribution on finitely generated subgroups, and we establish it for the two distributions which appear in the literature on random subgroups.
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