On generalized word problem in groups of automorphisms of trees

Generalized word problem (or the subgroup membership problem) asks to decide algorithmically whether a word in the generators of a finitely generated group belongs to a given subgroup of the group. Beyond the question of decidability, it is interesting to determine, to which class of formal languages the language of words that defines a certain subgroup belongs, for various classes of groups. If a group is given by its action on a topological space, stabilizers of points, and the corresponding membership problem, are of special interest. In the talk, we will discuss different aspects of the generalized word problem on the example of groups of automorphisms of infinite trees, such as self-similar or automata groups.