JournalscmhVol. 81 , No. 3DOI 10.4171/cmh/64

Growth exponent of generic groups

  • Yann Ollivier

    École Normale Supérieure de Lyon, France
Growth exponent of generic groups cover

Abstract

In [GrH97], Grigorchuk and de la Harpe ask for conditions under which some group presentations have growth rate close to that of the free group with the same number of generators. We prove that this property holds for a generic group (in the density model of random groups). Namely, for every positive \eps\eps, the property of having growth exponent at least 1ϵ1-\epsilon (in base 2m12m-1 where mm is the number of generators) is generic in this model. In particular this extends a theorem of Shukhov [Shu99]. More generally, we prove that the growth exponent does not change much through a random quotient of a torsion-free hyperbolic group.