Parabolic isometries of visible CAT(0) spaces and metrics on moduli space

2013/02/06 by Yunhui Wu, Wu, Yunhui
Mathematics · #Combinatorics #Computer science #Dynamical Systems (math.DS) #FOS: Mathematics #Genus #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Geometry #Geometry and complex manifolds #Isometry (Riemannian geometry) #Isometry group #Mathematical analysis #Mathematics #Metric (unit) #Metric space #Moduli space #Pure mathematics #Space (punctuation) #Surface (topology) #Translation (biology) #math.DS #math.GT

paper · pdf · doi:10.48550/arxiv.1302.1618

16 pages, comments are welcome

arxiv created 2013/02/06 · openalex publication_date 2013/02/06 · arxiv updated 2013/02/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that the translation length of any parabolic isometry on a complete semi-uniformly visible CAT(0) space is always zero. As a consequence, we will classify the isometries on visible CAT(0) spaces in terms of translation lengths. We will also show that the moduli space \mathbbM(S_g,n) of surface S_g,n of g genus with n punctures admits no complete visible CAT(0) Riemannian metric if 3g+n≥ 5, which answers the Brock-Farb-McMullen question in the visible case.

Citations

• Manifolds of negative curvature
• The Moduli Space of Riemann Surfaces is Kahler Hyperbolic
• On non-expansive mappings of Banach spaces
• Metric Spaces of Non-Positive Curvature
• Van Kampen’s embedding obstruction for discrete groups

Related

• Simple length spectra as moduli for hyperbolic surfaces and rigidity of length identities
• The space of metric structures on hyperbolic groups
• Almost-isometry between Teichmüller metric and length-spectra metric on moduli space
• Quasi-isometric embeddings for shrinking maps from surfaces into the moduli space
• CAT(0) metrics on contractible manifolds