We introduce a new random group model called the square model: we quotient a free group on n generators by a random set of relations, each of which is a reduced word of length 4. We prove that, just as in the Gromov model, for densities > 1/2 a random group in the square model is trivial with overwhelming probability and for densities < 1/2 a random group is hyperbolic with overwhelming probability. Moreover, we show that for densities d < 1/3 a random group in the square model does not have Property (T). Inspired by the results for the triangular model, we prove that for densities < 1/4 in the square model, a random group is free with overwhelming probability. We also introduce abstract diagrams with fixed edges and prove a generalization of the isoperimetric inequality.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm142-2-5,
author = {Tomasz Odrzyg\'o\'zd\'z},
title = {The square model for random groups},
journal = {Colloquium Mathematicae},
volume = {144},
year = {2016},
pages = {227-254},
zbl = {06498816},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm142-2-5}
}
Tomasz Odrzygóźdź. The square model for random groups. Colloquium Mathematicae, Tome 144 (2016) pp. 227-254. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm142-2-5/