Invariant random subgroups of groups acting on rooted trees

Free subgroups in groups acting on rooted trees

We show that if a group G acting faithfully on a rooted tree T has a free subgroup, then either there exists a point w of the boundary @T and a free subgroup of G with trivial stabilizer ofw, or there existsw 2 @T and a free subgroup ofG fixingw and acting faithfully on arbitrarily small neighborhoods of w. This can be used to prove the absence of free subgroups for different known classes of g...

The Automorphism Tower of Groups Acting on Rooted Trees

The group of isometries Aut(Tn) of a rooted n-ary tree, and many of its subgroups with branching structure, have groups of automorphisms induced by conjugation in Aut(Tn). This fact has stimulated the computation of the group of automorphisms of such well-known examples as the group G studied by R. Grigorchuk, and the group Γ̈ studied by N. Gupta and the second author. In this paper, we pursue t...

Dimension and randomness in groups acting on rooted trees

We explore the structure of the p-adic automorphism group Γ of the infinite rooted regular tree. We determine the asymptotic order of a typical element, answering an old question of Turán. 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...

Dimension and Randomness in Groups Acting on Rooted Trees

Dimension and Randomness in Groups Acting on Rooted Trees