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arXiv · 2507.16958

Reduction theory for Fuchsian groups with cusps

Adam Abrams·Svetlana Katok·Ilie Ugarcovici
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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."

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Adam Abrams, Svetlana Katok, Ilie Ugarcovici. 2025-07-22. Reduction theory for Fuchsian groups with cusps. https://doi.org/10.1007/s10711-026-01102-0

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