Random generation of finitely generated subgroups of a free group

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

1 LIGM - Laboratoire d'Informatique Gaspard-Monge
2 LaBRI - Laboratoire Bordelais de Recherche en Informatique

We give an efficient algorithm to randomly generate finitely generated subgroups of a given size, in a finite rank free group. Here, the size of a subgroup is the number of vertices of its representation by a reduced graph such as can be obtained by the method of Stallings foldings. Our algorithm randomly generates a subgroup of a given size $n$, according to the uniform distribution over size $n$ subgroups. In the process, we give estimates of the number of size $n$ subgroups, of the average rank of size $n$ subgroups, and of the proportion of such subgroups that have finite index. Our algorithm has average case complexity $\O(n)$ in the RAM model and $\O(n^2\log^2n)$ in the bitcost model.

Mots clés: subgroups of free groups; random generation

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

Frédérique Bassino, Cyril Nicaud, Pascal Weil. Random generation of finitely generated subgroups of a free group. International Journal of Algebra and Computation, 2008, 18 (1), pp.375-405. ⟨10.1142/S0218196708004482⟩. ⟨hal-00164584⟩