2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/69785We 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.2nd version: full redaction, 24 pagesGroup Theory20P05; 20F69; 20F06text