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Let $\Gamma \subset SL(2, \mathbb Z)$ be a finite subgroup acting on theirrational rotational algebra $\mathcal A_\theta$ via the restriction of thecanonical 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 paperwe prove many results on the homology group of the crossed product algebra$\mathcal A_\theta^{alg} \rtimes \Gamma$. We also analyse the case of thesmooth crossed product algebra, $\mathcal A_\theta \rtimes \Gamma$ andcalculate some of its homology groups.

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