Random tree-automorphisms

We study random automorphisms of binary trees of depth n. The order of such an automorphism is 2^K for a random integer K=K_n. We show that K_n/n converges to a constant which we specify explicitly. The proof uses a simple combinatorial random-tree description. This description is developed further to study random words and subgroups generated by random elements.

The methods generalize to many groups acting on trees. In the case of symmetric p-groups (Sylow-p subgroups of the symmetric group) we get an answer to an old question of Turan. Open questions will be presented. No background in probability or groups is assumed. This is joint work with M. Abert.