Algebraic Invariants for Group Actions on the Cantor Set

The algebraic invariants associated to the group actions on the Cantor set provide an interesting connection between the fields of dynamical systems and group theory. For instance, Giordano, Putnam and Skau have shown in [29] that the dimension group (see [24] for an introduction about dimension groups) of a minimal Z-action on the Cantor set completely determines its strong orbit equivalence class. Furthermore, the topological full group of such a system, which is known from Juschenko and Monod [38] to be amenable, determines its flip-conjugacy class (see [6] and [30] for more details). On the other hand, the amenability of the topological full groups of minimal Z-actions together with their properties shown in [41] by Matui make them the first known examples of infinite groups which are at the same time amenable, simple and finitely generated. Recently, another algebraic invariant, the group of automorphisms of actions on the Cantor set, has caught the eye of several researchers working in the field [13, 15, 16, 17, 14, 19, 20]. In [5], Boyle, Lind and Rudolph focused their attention on the group of automorphisms of subshifts of finite type, showing that these groups are always countable and residually finite. At the same time, they gave an example of a minimal Z-action on the Cantor set whose group of automorphisms contains Q, which implies that the automorphism group of a minimal action may be a non-residually finite group (recall that the Z-subshifts of finite type are not minimal). This leads to the natural question about the relation between the algebraic properties of the group of automorphisms and the dynamics of the system. Indeed, the residually finite property of the group of automorphisms of the subshifts of finite type is a consequence of the existence of periodic points.