Nekrashevych V. Self-similar automata groups

The thesis studies properties of self-similar groups and its connection with automata theory, iterations of virtual endomorphisms, operator algebras and dynamical systems. A theory of contracting groups and their limit spaces is constructed. Algorithms for solution of the word problem in groups and semigroups of asynchronous automata are constructed. It is proved that the word problem for contracting groups is of polynomial complexity. An answer to a question of R.Grigorchuk on classification of the groups Gw up to isomorphism is given. Iterated monodromy groups are introduced and it is shown that the limit space of the iterated monodromy group of an expanding map coincides with their Julia set. A problem of R.Pink on computation of the Galois groups of iterated extensions of the function field is solved. It is proved that the Cuntz-Pimsner algebra, which is naturally associated with a self-similar group is simple.