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Dynamical conditions that guarantee stability for discrete transformationgroup $C^*$-algebras are determined. The results are applied to the case ofsome discrete subgroups of $SL(2,\mathbb{R})$ acting on the plane with theorigin removed by means of matrix multiplication of vectors. In the case ofcocompact subgroups, further properties of such crossed products are deducedfrom properties of the $C^*$-algebra associated to the horocycle flow on thecorresponding compact homogeneous space of $SL(2,\mathbb{R})$.

On $C^*$-algebras associated to actions of discrete subgroups of $SL(2,\mathbb{R})$ on the punctured plane