Asymptotic aspects of Schreier graphs and Hanoi Towers groups

We present relations between growth, growth of diameters and the rate of vanishing of the spectral gap in Schreier graphs of automaton groups. In particular, we introduce a series of examples, called Hanoi Towers groups since they model the well known Hanoi Towers Problem, that illustrate some of the possible types of behavior.Comment: 5 page

On the Word and Period Growth of some Groups of Tree Automorphisms

We generalize a class of groups defined by Rostislav Grigorchuk to a much larger class of groups, and provide upper and lower bounds for their word growth (they are all of intermediate growth) and period growth (under a small additional condition, they are periodic).Comment: to appear in Comm. Algebr

Self-describing sequences and the Catalan family tree

We introduce a transformation of finite integer sequences, show that every sequence eventually stabilizes under this transformation and that the number of fixed points is counted by the Catalan numbers. The sequences that are fixed are precisely those that describe themselves --- every term t is equal to the number of previous terms that are smaller than t. In addition, we provide an easy way to enumerate all these self-describing sequences by organizing them in a Catalan tree with a specific labelling system

Branch Groups

This is a long introduction to the theory of "branch groups": groups acting on rooted trees which exhibit some self-similarity features in their lattice of subgroups