A January invitation to random groups by Ollivier Y.

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By Ollivier Y.

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This calls for a few comments: Of course Z/2Z occurs for even . a. for an elaboration on this). 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).

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. i is likely to work. 2. or [Gro01a, Oll-c]); in particular, therein property (T ) is used to gain uniform control on the critical densities of all successive quotients.

Up to now, the main application of random groups is the construction by Gromov [Gro03] of a finitely presentable group whose Cayley graph (quasi-)contains an infinite family of expanding graphs and which contradicts the Baum-Connes conjecture with coefficients [HLS02]. 1. 2. ), using the same tools together with a construction of Rips [Rip82], allowed in [OW-a] to construct Kazhdan groups whose outer automorphism group contains an arbitrary countable group, answering a question of Paulin (in the list of open problems in [HV89]).

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