  Dynamical conditions that guarantee stability for discrete transformation
group $C^*$-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 $C^*$-algebra associated to the horocycle flow on the
corresponding compact homogeneous space of $SL(2,\mathbb{R})$.
