Groupoids, equivalence bibundles and bimodules for noncommutative solenoids

Abstract

Let $p$ be a prime number and $\mathcal{S}_p$ the $p$-solenoid. For $\alpha\in \mathbb{R}\times \mathbb{Q}_p$ we consider in this paper a naturally associated action groupoid $S_\alpha:=\mathbb{Z} [1/p]\ltimes_\alpha \mathcal{S}_p \rightrightarrows \mathcal{S}_p$ whose $C^*-$algebra is a model for the noncommutative solenoid $\mathcal{A}_\alpha^\mathscr{S}$ studied by Latremoli\`ere and Packer. Following the geometric ideas of Connes and Rieffel to describe the Morita equivalences of noncommutative torus using the Kronecker foliation on the torus, we give an explicit description of the geometric/topologic equivalence bibundle for groupoids $S_\alpha$ and $S_\beta$ whenever $\alpha,\beta\in \mathbb{R}\times \mathbb{Q}_p$ are in the same orbit of the $GL_2(\mathbb{Z}[1/p])$ action by linear fractional transformations. As a corollary, for $\alpha,\beta\in \mathbb{R}\times \mathbb{Q}_p$ as above we get an explicit description of the imprimitivity bimodules for the associated noncommutative solenoids.