# A January invitation to random groups - download pdf or read online

By Ollivier Y.

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So in particular, taking a free group for G0 and iterating Theorem 40 we get: Proposition 43 – Let Fm be the free group on m generators a1 , . . , am . Let ( i )i∈N be a sequence of integers. Let d < − log2m ρ(Fm ) and, for each i, let Ri be a set of random words of length i at density d as in Theorem 40. Let R = Ri and let G = Fm / R be the (infinitely presented) random group so obtained. Then, if the i ’s grow fast enough, with probability arbitrarily close to 1 the group G is a direct limit of non-elementary hyperbolic groups, and in particular it is infinite.

Next come some small cancellation conditions. By the way, actually as d approaches 1/2, we have arbitrarily large cancellation (which refutes the expression “small cancellation on average” sometimes applied to this theory—we indeed measure cancellation on average, but it is not small), as results from the next proposition. Recall that, given a group presentation, a piece is a word which appears as a subword of two different relators in the presentation, or as a subword at two different positions in the same relator (relators are considered as cyclic words and up to inversion).

B. Dimension of the group. A consequence of the isoperimetric inequality holding for any reduced van Kampen diagram is that the Cayley 2-complex associated with the presentation is aspherical [Gro93, Oll04], so that the group has geometric (hence cohomological) dimension 2 as stated in Theorem 11. The Euler characteristic of the group is thus simply 1 − m + (2m − 1)d . In particular, since this Euler characteristic is positive for large , we get the following quite expected property (at least for d > 0, but this also holds at density 0 thanks to Theorem 18): Proposition 16 – With overwhelming probability, a random group in the density model is not free.