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

TL;DR

This paper investigates the stability and properties of $C^*$-algebras arising from actions of discrete subgroups of $SL(2, eal)$ on the punctured plane, linking dynamical conditions to algebraic features.

Contribution

It establishes dynamical criteria for stability of these $C^*$-algebras and explores their structure for cocompact subgroups via horocycle flow properties.

Findings

Dynamical conditions for stability of group $C^*$-algebras.

Application to subgroups of $SL(2, eal)$ acting on the plane.

Properties of crossed products related to horocycle flow.

Abstract

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 $SL(2,\mathbb{R})$.