Effective Equidistribution of Closed horocycles for Geometrically finite surfaces

TL;DR

This paper proves effective equidistribution of closed horocycles on certain hyperbolic surfaces with finitely generated fundamental groups, extending Sarnak's 1981 results and exploring applications in affine sieves.

Contribution

It extends Sarnak's classical equidistribution results to geometrically finite surfaces with critical exponent greater than 1/2, providing effective bounds.

Findings

Effective equidistribution established for hyperbolic surfaces with finitely generated fundamental groups.

Results applicable to affine sieve methods.

Generalization of classical results to a broader class of surfaces.

Abstract

For a complete hyperbolic surface whose fundamental group is finitely generated and has critical exponent bigger than 1/2, we obtain an effective equidistribution of closed horocycles in its unit tangent bundle. This extends a result of Sarnak in 1981 for surfaces of finite area. We also discuss its applications in Affine sieves.