Short geodesics and end invariants

2000/06/01 by Yair N. Minsky, Minsky, Yair N.
Mathematics · #30F40 (Primary) 57M50 (Secondary) #Advanced Algebra and Geometry #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.GT #msc:30F40 #msc:57M50

paper · pdf · doi:10.48550/arxiv.math/0006002

19 pages, 2 figures. To appear in Proceedings of RIMS Comprehensive Research on Complex Dynamical Systems and Related Fields

arxiv created 2000/06/01 · openalex publication_date 2000/06/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This expository article discusses some connections between the geometry of a hyperbolic 3-manifold homotopy-equivalent to a surface, and the combinatorial properties of its end invariants. In particular a necessary and sufficient condition is stated for the manifold to have arbitrarily short geodesics, in terms of a sequence of coefficients called subsurface projection distances, which are analogous in some ways to continued-fraction coefficients. (The proof of sufficiency appeared in math.GT/9907070)

Citations

• Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle
• Markoff triples and quasifuchsian groups

Related

• Effective length-projection bounds for hyperbolic 3-manifolds diffeomorphic to S×ℝ
• Combinatorial and Geometrical Aspects of Hyperbolic 3-Manifolds
• The length spectra of arithmetic hyperbolic 3-manifolds and their totally geodesic surfaces
• Shortening all the simple closed geodesics on surfaces with boundary
• A geometric property of closed geodesics on hyperbolic surfaces