Mostow rigidity for Fuchsian buildings

2004/07/23 by Xiangdong Xie, Xie, Xiangdong
Engineering · Mathematics · #20F65 #51E24 #57M20 #Combinatorics #Cross-ratio #Crystallography #Engineering #FOS: Mathematics #Fuchsian group #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Homeomorphism (graph theory) #Homotopy and Cohomology in Algebraic Topology #Ideal (ethics) #Isomorphism (crystallography) #Mathematical Dynamics and Fractals #Mathematics #Philosophy #Pure mathematics #Rigidity (electromagnetism) #Structural engineering #math.GR #math.GT #msc:20F65 #msc:51E24 #msc:57M20

paper · pdf · doi:10.48550/arxiv.math/0407394

58 pages

arxiv created 2004/07/23 · openalex publication_date 2004/07/23 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We show that if a homeomorphism between the ideal boundaries of two Fuchsian buildings preserves the combinatorial cross ratio almost everywhere, then it extends to an isomorphism between the Fuchsian buildings. It follows that Mostow rigidity holds for Fuchsian buildings: if a group acts properly and cocompactly on two Fuchsian buildings X and Y, then X and Y are equivariantly isomorphic.

Citations

• Buildings of Spherical Type and Finite BN-Pairs
• Lectures on Analysis on Metric Spaces
• The density at infinity of a discrete group of hyperbolic motions
• Quasiconformal maps in metric spaces with controlled geometry

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