A January invitation to random groups by Ollivier Y.

By Ollivier Y.

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Counting one-relator groups On a very different topic, consideration of generic-case rather than worstcase behavior for algorithmic problems in group theory (most notably the isomorphism problem) led I. Kapovich, Myasnikov, Schupp and Shpilrain, in a series of closely related papers [KSS, KS, KS05, KMSS05, KMSS03], to the conclusion that generic elements are often nicely behaved. The frontier between properties of one-relator groups and properties of a typical word in the free group is faint; for this review we selected an application where the emphasis is really put on the group, namely, evaluation of the number of distinct one-relator groups up to isomorphism.

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.

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.

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