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.
This video is part of the New York Group Theory Cooperative‘s group theory seminar series.
