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.

Tree languages and branched groups

“…Even more strikingly, the derived subgroup of the "twisted twin" group G is not closed in the tree topology, equivalently does not contain G ∩ ker π m for any m. Thus its set of portraits is regular, but the group is not finitely constrained. This answers an open question in [20].…”

“…The relations between a self-similar group G and the collection of portraits of its elements have already been considered in the literature, in particular by Siegenthaler in [22] and his 2008 doctoral thesis; see also [18,19,24,26]. In [20], Penland and Šunić prove that the closures of certain self-similar groups of rooted trees that satisfy an algebraic law do not have a regular language of portraits.…”

“…The article [20] by Penland and Šunić considers closely related questions; in particular, they point to self-similar branched groups whose language of portraits enjoy various properties. The definition of regular tree language that we gave is the most general; another, more restrictive accepting condition is known as the Büchi condition, and an example of a group whose language is regular but not Büchi is {g ∈ Isom T : g(v) and v differ in finitely many positions for all v ∈ X * }.…”