Automorphisms of one-rooted trees: Growth, circuit structure, and acyclicity

On amenability of automata groups

“…In this Section we recall some standard facts about automata, see [GNS00] and [Sid00] for further details.…”

“…The class of groups generated by bounded automata was defined by Sidki in [Sid00] (see [BN03] for an interpretation of these groups in terms of fractal geometry). Most of the well-studied examples of groups of finite automata belong to this class.…”

Free subgroups in groups acting on rooted trees

“…For the notion of bounded automorphisms see Definition 4.3 of our paper and the articles [Sid00], [BN03], [BKN10]. It is known that groups generated by bounded automorphisms defined by finite automata have no free subgroups [Sid04] and that they are even amenable [BKN10].…”

“…A version of the growth function  g .n/ (given in Definition 2.2) was defined and studied by S. Sidki in [Sid00]. It follows from the results of [Sid00] that if g is finite state then the function  g .n/ has either exponential or polynomial growth. The set P d .X/ of all finite state automorphisms of X for which  g .n/ is bounded by a polynomial of degree d is a group.…”

“…The following inductive description of elements of P d .X/ is given in [Sid00]. We denote by P 1 .X/ the group of finitary automorphisms of X , i.e., the automorphisms g for which the sequence  g .n/ is eventually zero.…”