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Some small cancellation properties of random groups

Yann Ollivier
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Abstract

We work in the density model of random groups. We prove that they satisfy an isoperimetric inequality with sharp constant $1-2d$ depending upon the density parameter $d$. This implies in particular a property generalizing the ordinary $C'$ small cancellation condition, which could be termed ``macroscopic small cancellation''. This also sharpens the evaluation of the hyperbolicity constant $δ$. As a consequence we get that the standard presentation of a random group at density $d<1/5$ satisfies the Dehn algorithm and Greendlinger's Lemma, and that it does not for $d>1/5$.

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Yann Ollivier. 2004-09-14. Some small cancellation properties of random groups. https://arxiv.org/abs/math/0409226

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