By Ollivier Y.
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This quantity offers an account of either the standard and modular illustration idea of the symmetric teams. the diversity of purposes of this concept is immense, various from theoretical physics, via combinatorics, to the learn of the polynomial id algebras; and new makes use of are nonetheless being stumbled on.
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Additional info for A January invitation to random groups
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.