Reduction theory for Fuchsian groups with cusps

TL;DR

This paper develops a reduction theory for Fuchsian groups with cusps, constructing a natural extension of boundary maps with a finite attractor, confirming a conjecture by Don Zagier.

Contribution

It introduces a new reduction map for Fuchsian groups with cusps, demonstrating its domain of bijectivity and finite attractor, based on a novel fundamental polygon construction.

Findings

The reduction map has a domain of bijectivity.

The natural extension has a finite rectangular attractor.

The results confirm Zagier's conjecture.

Abstract

We study a family of Bowen-Series-like maps associated to any finitely generated Fuchsian group of the first kind with at least one cusp. These maps act on the boundary of the hyperbolic plane in a piecewise manner by generators of the group. We show that the two-dimensional natural extension (reduction map) of the boundary map has a domain of bijectivity and global attractor with a finite rectangular structure, confirming a conjecture of Don Zagier. Our work is based on the construction of a special fundamental polygon, related to the free product structure of the group, whose marking is preserved by "Teichm\"uller deformation."