K-theory of C*-algebras arising from commuting Hilbert bimodules and invariant ideals

We study the K-theory of the Cuntz-Nica-Pimsner C*-algebra of a rank-two product system that is an extension determined by an invariant ideal of the coefficient algebra. We use a construction of Deaconu and Fletcher that describes the Cuntz-Nica-Pimsner C*-algebra of the product system in terms of two iterations of Pimsner's original construction of a C*-algebra from a right-Hilbert bimodule. We apply our results to the product system built from two commuting surjective local homeomorphisms of a totally disconnected space, where the Cuntz-Nica-Pimsner C*-algebra is isomorphic to the C*-algebra of the associated rank-two Deaconu--Renault groupoid. We then apply a theorem of Spielberg about stable finiteness of an extension to obtain sufficient conditions for stable finiteness of the C*-algebra of the Deaconu-Renault groupoid.