We show that all amenable, minimal actions of a large class of nonamenable countable groups on compact metric spaces have dynamical comparison. This class includes all nonamenable hyperbolic groups, many HNNextensions, nonamenable Baumslag-Solitar groups, a large class of amalgamated free groups, lattices in many Lie groups, A 2 -groups, as well as direct products of the above with arbitrary countable groups. As a consequence, crossed products by amenable, minimal and topologically free actions of such groups on compact metric spaces are Kirchberg algebras in the UCT class, and are therefore classified by K-theory.
For a given discrete group G, we apply results of Kirchberg on exact and injective tensor products of C * -algebras to give an explicit description of the minimal exact correspondence crossed-product functor and the maximal injective crossed-product functor for G in the sense of Buss, Echterhoff and Willett. In particular, we show that the former functor dominates the latter.
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