On distributions of subgroups of free groups

Frédérique Bassino (1) , Armando Martino (2) , Cyril Nicaud (3) , Enric Ventura (4) , Pascal Weil (5)

1 LIPN - Laboratoire d'Informatique de Paris-Nord
2 School of Mathematics
3 LIGM - Laboratoire d'Informatique Gaspard-Monge
4 Department of Applied Mathematics III
5 LaBRI - Laboratoire Bordelais de Recherche en Informatique

We study and compare two natural distributions of finitely generated sub- groups of free groups. One is based on the random generation of tuples of reduced words; that is the one classically used by group theorists. The other relies on Stallings' graphical representation of subgroups and in spite of its naturality, it was only recently consid- ered. The combinatorial structures underlying both distributions are highly amenable to the methods of analytic combinatorics. We use these methods to point at the differences between these distributions. It is particularly interesting that certain important properties of subgroups that are generic in one distribution, turn out to be negligible in the other.

Domaines: Algorithme et structure de données [cs.DS]; Théorie des groupes [math.GR]

Frédérique Bassino, Armando Martino, Cyril Nicaud, Enric Ventura, Pascal Weil. On distributions of subgroups of free groups. Workshop on Analytic Algorithmics and Combinatorics (ANALCO) 2010, Jan 2010, United States. à paraître. ⟨hal-00433210v1⟩