For any countable group G acting by homeomorphisms on the Cantor set X, we can associate a larger group [[G]] consisting of homeomorphisms that locally coincide with elements of G. The group [[G]] is dubbed "the full group" of (X,G). The full group as an abstract group is a complete invariant for continuous orbit equivalence of (X,G). Thus, [[G]] possesses a great deal of information about the underlying dynamical system. These groups were recently used to construct the first examples of infinite simple finitely generated amenable groups [Juschenko-Monod].

In the talk we will give a survey of known algebraic properties of full groups, present a complete description of generators and defining relations for full groups associated with minimal Z-subshifts, and give an elementary proof that these groups cannot be finitely presented. The talk is based on joint work with Grigorchuk.