## A January invitation to random groups by Ollivier Y.

By Ollivier Y.

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The contents of this publication were used in classes given via the writer. the 1st was once a one-semester direction for seniors on the collage of British Columbia; it used to be transparent that sturdy undergraduates have been completely in a position to dealing with hassle-free crew conception and its software to basic quantum chemical difficulties.

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Note that Proposition 10 already implies the conclusion for d < 1/12, for then the presentation satisfies the good old C (1/6) small cancellation condition. 1. ). a. the birthday paradox : in a class of more than 23 pupils there is a good chance that two of them share the same birthday. This is a simple combinatorial exercise. 32 A January 2005 invitation to random groups √ than N pigeons then we will put two pigeons in the same hole (very probably as N → ∞, provided that the assignment was made at random).

Theorem 27 – Let d > 1/3 and let G be a random group at density d and at lengths , + 1 and + 2. Then, with overwhelming probability, G has property (T ). ). This results from the necessity to have some relators of length a multiple of 3, as we explain now. This theorem is proven using an intermediate random group model better suited to apply the spectral criterion, the triangular model, which we now define. This model consists in taking relators of length only 3, but let˙ ˙ ting the number of distinct generators tend to infinity.

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). 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.