A January invitation to random groups by Ollivier Y.

By Ollivier Y.

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G. the case of a constant number of relators (the fewrelator model of Def. 1—but not the one of Def. 4). 3. 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).

As briefly mentioned above, the torsionfreeness assumption can be relaxed to a “harmless torsion” one demanding that the centralizers of torsion elements are either finite, or virtually Z, or the whole group [Oll04]. But in [Oll05b] we give an example of a hyperbolic group with “harmful” torsion, for which Theorem 40 does not hold; moreover its random quotients actually exhibit three genuinely different phases instead of the usual two. Theorem 41 – Let G0 = (F4 × Z/2Z) F4 equipped with its natural generating set, where denotes a free product.

The Euler characteristic of the group is thus simply 1 − m + (2m − 1)d . In particular, since this Euler characteristic is positive for large , we get the following quite expected property (at least for d > 0, but this also holds at density 0 thanks to Theorem 18): Proposition 16 – With overwhelming probability, a random group in the density model is not free. Consideration of the Euler characteristic also implies that, for fixed m, the “dimension” d of the set of relations of the group is well-defined by its algebraic structure.

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