Cogrowth and spectral gap of generic groups

Details Cogrowth and spectral gap of generic groups Cogrowth and spectral gap of generic groups

2004-01-06

arxiv.org/abs/math/0401048v2

Mathematics

Group Theory

2nd version: full redaction, 24 pages

Scientific paper

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.

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