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Universal length bounds for non‐simple closed geodesics on hyperbolic surfaces
Author(s) -
Basmajian Ara
Publication year - 2013
Publication title -
journal of topology
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.447
H-Index - 31
eISSN - 1753-8424
pISSN - 1753-8416
DOI - 10.1112/jtopol/jtt005
Subject(s) - geodesic , mathematics , surface (topology) , cusp (singularity) , bounded function , constant (computer programming) , generalization , simple (philosophy) , intersection (aeronautics) , combinatorics , uniform boundedness , mathematical analysis , geometry , philosophy , epistemology , computer science , engineering , programming language , aerospace engineering
We investigate the relationship, in various contexts, between a closed geodesic with self‐intersection number k (for brevity, called a k ‐geodesic) and its length. We show that for a fixed compact hyperbolic surface, the short k ‐geodesics have length comparable with the square root of k . On the other hand, if the fixed hyperbolic surface has a cusp and is not the punctured disc, then the short k ‐geodesics have length comparable with log k . The length of a k ‐geodesic on any hyperbolic surface is known to be bounded from below by a constant that goes to infinity with k . In this paper, we show that the optimal constants { M k } are comparable with log k leading to a generalization of the well‐known fact that length less than 4 log ( 1 + 2 ) implies simple. Finally, we show that for each natural number k , there exists a hyperbolic surface where the constant M k is realized as the length of a k ‐geodesic. This was previously known for k = 1, where M 1 is the length of the figure eight on the thrice punctured sphere.