Pascal Weil

ATELIER A SPECTS G ÉOM ÉTRIQUES , ASYMPTOTIQUES ET COMBINATOIRES DE LA TH ÉORIE DES
GROUPES ET LEURS APPLICATIONS (GAGTA) 15-19 AO ÛT 2010
W ORKSHOP ON “G EOMETRIC , A SYMPTOTIC , C OMBINATORIAL G ROUP T HEORY WITH
A PPLICATIONS (GAGTA)”
AUGUST 15-19, 2010
On statistical properties of subgroups of free groups
Pascal Weil
Labortoire bordelais de recherche en informatique (LaBRI)
Unversité de Bordeaux I
351, Cours de la Libération
33405 Talence Cedex
FRANCE
[email protected]
Statistical properties of subgroups of free groups are often studied with the assumption that a
subgroup is given by a randomly chosen k-tuple (k fixed) of generators of length at most n, where
n is allowed to tend to infinity ; see the work of Arzhantseva and Ol’shanskii, and more recently
of Jitsukawa, Kapovich, Miasnikov, Schupp, Shpilrain and others. In previous work, Bassino,
Martino, Nicaud, Ventura and Weil (or a subset thereof) have studied the impact of replacing this
distribution by another, and in particular by generating uniformly at random Stallings graphs of
a given size. I will present an extension of this study, discussing other notions of generation of
a k-tuple of generators which escape the top-heavy bias of the generation of words of length
at most n (exponentially generically, these words have a length close to n). I will examine the
genericity of the property of Whitehead minimality (whether the image of H under every nonlength-preserving automorphism has a larger Stallings graph) under these various distributions.