Cogrowth and spectral gap of generic groups

Yann Ollivier (1)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

We prove that that for all $\eps$, having cogrowth exponent at most $1/2+\eps$ (in base $2m-1$ with $m$ the number of generators) is a generic property of groups in the density model of random groups. This generalizes a theorem of Grigorchuk and Champetier. More generally we show that the cogrowth of a random quotient of a torsion-free hyperbolic group stays close to that of this group. This proves in particular that the spectral gap of a generic group is as large as it can be.

Domaines: Théorie des groupes [math.GR]

Yann Ollivier. Cogrowth and spectral gap of generic groups. Annales de l'Institut Fourier, 2005, 55, pp.289-317. ⟨hal-00138230⟩