Hochschild and cyclic homology of the crossed product of algebraic irrational rotational algebra by finite subgraoups of $SL(2,\mathbb Z)$

Abstract: Let $\Gamma \subset SL(2, \mathbb Z)$ be a finite subgroup acting on the irrational rotational algebra $\mathcal A_\theta$ via the restriction of the canonical action of $SL(2,\mathbb Z)$. Consider the crossed product algebra $\mathcal A_\theta^{alg} \rtimes \Gamma$ obtained by the restriction of the $\Gamma$ action on the algberaic irrational rotational algebra. In this paper we prove many results on the homology group of the crossed product algebra $\mathcal A_\theta^{alg} \rtimes \Gamma$. We also analyse the case of the smooth crossed product algebra, $\mathcal A_\theta \rtimes \Gamma$ and calculate some of its homology groups.