Hyperbolic spaces from self-similar group actions

Self-similar group actions (or self-similar groups) have proved to be interesting mathematical objects from the point of view of group theory and from the point of view of many other fields of mathematics (operator algebras, holomorphic dynamics, automata theory, etc). See the works [BGN02, GNS00, Gri00, Sid98, BG00, Nek02a], where different aspects of self-similar groups are studied. An important class of self-similar group actions are contracting actions. Contracting groups have many nice properties. For example, the word problem is solvable in a contracting group in a polynomial time [Nek]. The author has shown (see [Nek02b]) that a naturally defined topological space, called the limit space, is associated with every contracting self-similar action. This topological space is metrizable and finite-dimensional. It was discovered later (see [Nek02a]) that with many topological dynamical systems (like iterations of a rational function) a self-similar group is associated. This self-similar group (called the iterated monodromy group) is often contracting and in fact contains all the essential dynamics of the original dynamical system. In particular, if a map is expanding, then its iterated monodromy group is contracting and the limit space of the iterated monodromy group is homeomorphic to the Julia set of the map.