On algebraic properties of topological full groups

We discuss the algebraic structure of the topological full group [[T]] of a Cantor minimal system (X,T). We show that [[T]] has a structure similar to a union of permutational wreath products of the group Z. This allows us to prove that the topological full groups are locally embeddable into finite groups, give an elementary proof of the fact that the group [[T]]^′ is infinitely presented, and provide explicit examples of maximal locally finite subgroups of [[T]]. We also show that the commutator subgroup [[T]]^′, which is simple and finitely-generated for minimal subshifts, is decomposable into a product of two locally finite groups, and that [[T]] and [[T]]^′ possess continuous ergodic invariant random subgroups. Bibliography: 36 titles. (paper)