Freiheitssatz and phase transition for the density model of random groups

Tsung-Hsuan Tsai (1)

1 IRMA - Institut de Recherche Mathématique Avancée

Magnus' Freiheitssatz states that if a group is defined by a presentation with $m$ generators and a single relator containing the last generating letter, then the first $m−1$ letters freely generate a free subgroup. We study an analogue of this theorem in the Gromov density model of random groups, showing a phase transition phenomenon at density $d_r = \min{1/2, 1 − log_{2m−1} (2r − 1)}$ with $1 ≤ r ≤ m−1$: we prove that for a random group with $m$ generators at density $d$, if $d < d_r$ then the first $r$ letters freely generate a free subgroup; whereas if $d > d_r$ then the first $r$ letters generate the whole group.

Domaines: Théorie des groupes [math.GR]

Tsung-Hsuan Tsai. Freiheitssatz and phase transition for the density model of random groups. 2021. ⟨hal-03455318⟩