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<p>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 “Teichmüller deformation.”</p>

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Reduction theory for Fuchsian groups with cusps

  • 期刊论文

  • 发表日期：

    2026年7月11日

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  • Adam Abrams,
  • Svetlana Katok,
  • Ilie Ugarcovici

摘要

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 “Teichmüller deformation.”