On the genericity of Whitehead minimality

Frédérique Bassino (1) , Cyril Nicaud (2) , Pascal Weil (3)

1 LIPN - Laboratoire d'Informatique de Paris-Nord
2 LIGM - Laboratoire d'Informatique Gaspard-Monge
3 LaBRI - Laboratoire Bordelais de Recherche en Informatique

We show that a finitely generated subgroup of a free group, chosen uniformly at random, is strictly Whitehead minimal with overwhelming probability. Whitehead minimality is one of the key elements of the solution of the orbit problem in free groups. The proofs strongly rely on combinatorial tools, notably those of analytic combinatorics. 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.

Mots clés: free groups; subgroups; Whitehead minimality

Domaines: Complexité [cs.CC]; Théorie des groupes [math.GR]; Combinatoire [math.CO]

Frédérique Bassino, Cyril Nicaud, Pascal Weil. On the genericity of Whitehead minimality. Journal of Group Theory, 2016, 19 (1), pp.137-159. ⟨hal-00919489v2⟩