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

Dynamical conditions that guarantee stability for discrete transformation group $C^{\ast }$-algebras are determined. The results are applied to the case of some discrete subgroups of $SL(2,\mathbb{R})$ acting on the plane with the origin removed by means of matrix multiplication of vectors. In the case of cocompact subgroups, further properties of such crossed products are deduced from properties of the $C^{\ast }$-algebra associated to the horocycle flow on the corresponding compact homogeneous space of $SL(2,\mathbb{R})$.