Commuting free actions of groups G on the left and H on the right of a directed graph E give rise to an action of G on the quotient graph E/H, and hence to an action \alpha of G on the graph C*-algebra C*(E/H). Symmetrically, we also get an action \beta of H on C*(G\E). The crossed products are related by the Symmetric Imprimitivity Theorem of Pask and Raeburn, which provides a C*(E/H)\times_\alpha G - C*(G\E) \times_\beta H imprimitivity bimodule X(E); this generalizes the earlier asymmetric theorem of Kumjian and Pask in which only one group acts. The Rieffel-Morita equivalence implemented by X(E) can also be realized by tensoring together two bimodules which arise from the Kumjian-Pask theorem, and it is natural to ask whether this tensor-product bimodule is isomorphic to X(E).

In this talk I'll introduce a graph construction we are calling the hash product, which is a graph-level analogue of the bimodule tensor product, and show how it can be used to answer the question posed above.

This is a preliminary report on joint work with Astrid an Huef.