Reduction theory for Fuchsian groups with cusps

2025/07/22 by Adam Abrams, Svetlana Katok, Abrams, Adam +3 · 2 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Boundary (topology) #Conjecture #Domain (mathematical analysis) #Extension (predicate logic) #Fuchsian group #Group (periodic table) #Hyperbolic geometry #Mathematical Dynamics and Fractals #Piecewise #Reduction (mathematics) #math.DS #msc:37D40 #msc:37E10

paper · pdf · doi:10.1007/s10711-026-01102-0

32 pages, 15 figures

openalex created_date 2025/10/07 · openalex publication_date 2026/07/11 · arxiv created 2026/07/29 · arxiv updated 2026/07/31 · openalex updated_date 2026/08/05

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 "Teichmüller deformation."

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