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Abstract

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.