Dimension and randomness in groups acting on rooted trees

We explore the structure of the p-adic automorphism group Gamma of the infinite rooted regular tree. We determine the asymptotic order of a typical element, answering an old question of Turan. We initiate the study of a general dimension theory of groups acting on rooted trees. We describe the relationship between dimension and other properties of groups such as solvability, existence of dense free subgroups and the normal subgroup structure. We show that subgroups of Gamma generated by three random elements are full-dimensional and that there exist finitely generated subgroups of arbitrary dimension. Specifically, our results solve an open problem of Shalev and answer a question of Sidki.