Amenable minimal Cantor systems of free groups arising from diagonal actions

Abstract

We study amenable minimal Cantor systems of free groups arising from the diagonal actions of the boundary actions and certain Cantor systems. It is shown that every virtually free group admits continuously many amenable minimal Cantor systems whose crossed products are mutually non-isomorphic Kirchberg algebras in the UCT class (with explicitly determined K-theory). The technique developed in our study also enables us to compute the K-theory of certain amenable minimal Cantor systems. We apply it to the diagonal actions of the boundary actions and the products of the odometer transformations, and determine their K-theory. Then we classify them in terms of the topological full groups, continuous orbit equivalence, strong orbit equivalence, and the crossed products.

31 pages; Some proofs are simplified and shortened. Remark added (Remark 4.9). Section 6 is deleted. To appear in J. Reine Angew. Math

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• A class of $C^{\ast }$-algebras generalizing both graph algebras and homeomorphism $C^{\ast }$-algebras IV, pure infiniteness
• Realization of hyperbolic group C*-algebras as decreasing intersection of Cuntz algebras O_2

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• Realization of hyperbolic group C*-algebras as decreasing intersection of Cuntz algebras O_2
• On diagonal actions of free group on the Cantor set