Fedorova M. Automaton actions of free product of groups

A series of new faithful actions of a free product of a finite number of finite groups by finite automaton permutations is constructed. Several properties of the Schreier graphs on the levels and the orbital Schreier graphs corresponding to the constructed faithful actions of free products of finite groups are established. The question of the planarity of such graphs is studied. With the help of the introduced notions of Schreier embedding and equivalence, we present a natural sufficient condition for the faithfulness of a group action on a set in terms of Schreier graphs. In the dissertation, we constructively prove that the class of groups that act faithfully on regular rooted trees by finite state automorphisms (in the other words the class of groups generated by finite initial automata over finite alphabet) is closed under free products. A new method of constructing faithful images of free groups by finite-automaton permutations is proposed. It is shown how to transform an automaton over a binary alphabet, known as an adding machine, into new finite initial automata that define a free group of rank 2.