Stably finite extensions of rank-two graph C*-algebras

We study stable finiteness of extensions of 2-graph C*-algebras determined by saturated hereditary sets of vertices. We use two iterations of the Pimsner-Voiculescu sequence to calculate the map in K-theory induced by the inclusion of a hereditary subgraph into the larger 2-graph it lives in. We then apply a theorem of Spielberg about stable finiteness of extensions to provide a sufficient condition for the C*-algebra of the larger 2-graph to be stably finite. We illustrate our results with examples.