In the introductory part of the talk we first go over the basic notions in one-sided symbolic dynamics. In particular, we define the shifts of finite type and sofic shifts, and discuss the representation of sofic shifts by finite, directed, labeled, graphs, that is, automata. We then recall the definition of group shifts, which are shifts that use finite groups as their alphabets and that are groups under the usual product defined by coordinates and we state the fundamental, if somewhat surprising, result of Kitchens claiming that all group shifts are shifts of finite type. Seemingly without much motivation, we then switch to a discussion of groups of automorphisms of regular, rooted trees. Time permitting, we swerve in yet another direction, and briefly discuss self-covers of the Riemann sphere and, in particular, those coming from post-critically finite polynomials and rational maps.

In the second part of the talk, we consider symbolic dynamics on regular, rooted trees. This is a, more or less, straightforward generalization of the previous situation, if we take the point of view that the one-sided shifts are defined over a ray, which is the Cayley graph of the free monoid of rank one, and the regular, rooted tree of degree \(d\) is the Cayley graph of the free monoid of rank \(d\). This enables us to define, by analogy, tree shifts of finite type and sofic tree shifts, and leads to representation of the latter by automata that can "read" labeled trees as inputs (Rabin automata). We then define group tree shifts and connect this notion to the usual notion of groups of tree automorphisms, self-similar groups, topological closures of groups of tree automorphisms, and the notion of a regular branch group and its relation to symbolic dynamics on trees. We state many results along the way. For instance:

 

1. A self-similar branch group is a group tree shift of finite type if and only if it is a closure of a group that branches over a level stabilizer.
2. Every sofic group tree-shift that is level transitive and self-replicating is, in fact, a tree shift of finite type.
3. The topological closure of a self-replicating regular branch group is a tree shift of finite type.

Concrete examples are provided and will usually be chosen from the words of complex dynamics -- iterated monodromy groups of post-critically finite polynomials and rational maps on the Riemann sphere.