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By Ollivier Y.

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As a corollary, we get that the critical density for the new group G is arbitrarily close to that for G0 . So we could take a new random quotient of G, at least if we knew that G is torsion-free. ), but the results of [Oll04] imply that if G0 is of geometric dimension 2 then so is G. 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.

B. Harmful torsion. As briefly mentioned above, the torsionfreeness assumption can be relaxed to a “harmless torsion” one demanding that the centralizers of torsion elements are either finite, or virtually Z, or the whole group [Oll04]. But in [Oll05b] we give an example of a hyperbolic group with “harmful” torsion, for which Theorem 40 does not hold; moreover its random quotients actually exhibit three genuinely different phases instead of the usual two. Theorem 41 – Let G0 = (F4 × Z/2Z) F4 equipped with its natural generating set, where denotes a free product.

Am lies in the interval (ρ(G0 ); ρ(G0 ) + ε). The same theorem holds for quotients by random reduced words, and, very likely [Oll-e], for quotients by random elements of the ball as in Theorem 38. As a corollary, we get that the critical density for the new group G is arbitrarily close to that for G0 . So we could take a new random quotient of G, at least if we knew that G is torsion-free. ), but the results of [Oll04] imply that if G0 is of geometric dimension 2 then so is G. 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 , .

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A January invitation to random groups by Ollivier Y.


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