By Miller G.A.
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A. Random quotients by words. Theorem 38 describes what happens when quotienting a hyperbolic group by random elements in it. Another possible generalization of Theorem 11 is to quotient by random words in the generators. t. one and the same generating set, as used notably in [Gro03]. Of course, the unavoidable consequence of the model being independent on the initial group is that the critical density will depend on this group. t. the generating set a1 , . . , am considered: basically, if this probability behaves like (2m)−αt for large times t, the critical density will be α.
Shukhov’s “not too many” relators condition is strikingly reminiscent of a density condition. f. Random walk in a typical group. A group together with a generating set defines a random walk, which consists in starting at e and, at each step, multiplying by one of the generators or its inverse, chosen at random (this is the simple random walk on the Cayley graph). A foundational paper of this theory is that of Kesten [Kes59], see also [Gri80, GH97, Woe00] for some reviews. One quantity containing a lot of information about the random walk is the spectral radius of the random walk operator (see [Kes59]).
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. It is not easy to follow the details of [Oll04, Oll05a, Ch93] closely enough to obtain an explicit necessary rate of growth for the i ’s, although i+1 Cst.