The square model for random groups

Tomasz Odrzygóźdź¹

¹ Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-656 Warszawa, Poland

Colloquium Mathematicum, Tome 142 (2016) no. 2, pp. 227-254.

Résumé

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.

Détail

DOI : 10.4064/cm142-2-5

Keywords: introduce random group model called square model quotient group generators random set relations each which reduced word length nbsp prove just gromov model densities random group square model trivial overwhelming probability densities random group hyperbolic overwhelming probability moreover densities random group square model does have property inspired results triangular model prove densities square model random group overwhelming probability introduce abstract diagrams fixed edges prove generalization isoperimetric inequality