A language hierarchy and Kitchens-type theorem for self-similar groups

Abstract

We generalize the notion of self-similar groups of infinite tree automorphisms to allow for groups which are defined on a tree but may not act faithfully on it. The elements of such a group correspond to labeled trees which may be recognized by a tree automaton (e.g. Rabin, Büchi, etc.), or considered as elements of a tree shift (e.g. of finite type, sofic) as in symbolic dynamics. We give examples to show how self-similar groups defined in this way can be separated into different tree language hierarchies. As the main result, extending the classical result of Kitchens on one-dimensional group shifts, we provide a sufficient condition for a self-similar group whose elements form a sofic tree shift to be a tree shift of finite type. As an application, we show that the closures of certain self-similar groups of rooted k-ary tree automorphisms that satisfy an algebraic law are not Rabin-recognizable, that is, they can not be described within the second order theory of k successors. In both the main result and in the application a crucial role is played by a distinguished branched subgroup structure of the groups under consideration.