Distribution of orbits in $\R^2$ of a finitely generated group of $\SL(2,\R)$

TL;DR

This paper investigates how finitely generated groups of linear transformations in 2 distribute their orbits over the plane, using new equidistribution results related to horocyclic flows on associated surfaces.

Contribution

It introduces novel equidistribution results for the horocyclic flow, advancing understanding of orbit distribution in linear group actions on 2.

Findings

Distribution of non-discrete orbits characterized

New equidistribution theorems established

Insights into linear group actions on 2 obtained

Abstract

In this work, we study the asymptotic distribution of the non discrete orbits of a finitely generated group acting linearly on $\R^2$. To do this, we establish new equidistribution results for the horocyclic flow on the unitary tangent bundle of the associated surface.