Tag - Random groups

We will use Gromov's density model of randomness. A random group at density d satisfies some property (of groups) P if the probability of occurrence of P tends to 1 as the length of relations goes to infinity. Julia Knight conjectured that the limit of the theories of random groups should converge to the theory of a free group. We will show that this is true for the universal theory of a random group at density d<1/16. Namely, every universal and every existential axiom of the free group is also true in a random group. Notice that a random group at density d<1/16 satisfies a small cancellation condition C'(1/8). We will also show that a random group at density d<1/2 is not a limit group (for a few relations model this was proved by Ho when the number of generators is less than the number of relations). These are joint results with R. Sklinos.