Symmetry And Group

A January invitation to random groups by Ollivier Y.

By Ollivier Y.

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Hk be elements of the free group Fm generating a subgroup H of infinite index. Then with overwhelming probability, the map from Fm to a random few-relator group is injective on H. Conversely, it is easily seen that subgroups of finite index do not embed. Of course this holds for elements h1 , . . , hk fixed in advance: it cannot be true that the quotient map is injective on all subgroups... d. Boundary and geometric properties of the Cayley graph. We refer to [GhH90, CDP90, BH99] for the notion of boundary of a hyperbolic space.

So, up to this latter technicality, there is a natural homomorphism ϕ : G3 → G from a random group in the triangular model to a random group in the density model, at the same density. This means that the triangular model is “less quotiented” than the density one. ˙ It is possible to prove [Zuk03] quite the same hyperbolicity theorem as for the density model: Theorem 29 – If d < 1/2, then with overwhelming probability a random group in the triangular model, at density d, is non-elementary hyperbolic.

Am . t. this generating set; let dcrit = − log2m ρ(G0 ) and let G be a quotient of G0 by random words at density d < dcrit as in Theorem 40. t. a1 , . . , 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.

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