By W. Waliszewski

Warsaw 1965 Rozprawy Matematyczne XLV. Sm.4to., 40pp., unique published wraps. Uncut. VG, mild soiling.

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Critical densities for various properties A bunch of properties are now known to hold for random groups. This ranges from group combinatorics (small cancellation properties) to algebra (freeness of subgroups) to geometry (boundary at infinity, growth exponent, CAT(0)-ness) to probability (random walk in the group) to representation theory on the Hilbert space (property (T ), Haagerup property). Some of the properties studied here are intrinsic to the group, others depend on a marked set of generators or on the standard presentation through which the random group was obtained.

But they probably extend to small positive values of d, the determination of which is an interesting problem. The first such theorem [AO96] is the following: Theorem 17 – With overwhelming probability in the few-relator model of random groups with m generators and n relators, any subgroup generated by m − 1 elements is free. ). When the m − 1 generators of the subgroup are chosen among the m standard generators of the group, this is a particular case of Theorem 6. The group itself is not free and more precisely [Ar97]: Theorem 18 – In the few-relator model of random groups with n relators, no finite-index subgroup of the group is free.

Let 0 d 1 and let W be the set of all (2m) words of length in ’s. Let R be the random set obtained by picking (2m)d times at the a±1 i random a word in W . Typical elements in a group 51 Let G = G0 / R be the random quotient so obtained. Then with overwhelming probability: • If d < dcrit , then G is (non-elementary) hyperbolic. • If d > dcrit , then G = {e}. Once again the spirit of the density model is to kill a number of words equal to some power d of the total number of words (2m) . Note that dcrit < 1/2, even when G0 is the free group (the difference with Theorem 11 being that we use plain words instead of reduced ones).