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Fundamental group and contractible closed geodesics
Author(s) -
Nabutovsky Alexander
Publication year - 1996
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/(sici)1097-0312(199612)49:12<1257::aid-cpa1>3.0.co;2-9
Subject(s) - mathematics , contractible space , geodesic , riemannian manifold , class (philosophy) , bounded function , group (periodic table) , fundamental group , constant (computer programming) , pure mathematics , combinatorics , manifold (fluid mechanics) , order (exchange) , uniform boundedness , mathematical analysis , physics , computer science , mechanical engineering , finance , quantum mechanics , artificial intelligence , engineering , economics , programming language
We prove the existence of a nonempty class of finitely presented groups with the following property: If the fundamental group of a compact Riemannian manifold M belongs to this class, then there exists a constant c ( M ) > 1 such that for any sufficiently large x the number of contractible closed geodesics on M of length not exceeding x is greater than c(M) x . In order to prove this result, we give a lower bound for the number of contractible closed geodesics of length ≤ x on a compact Riemannian manifold M in terms of the resource‐bounded Kolmogorov complexity of the word problem for π 1 ( M ), thus answering a question posed by Gromov. © 1996 John Wiley & Sons, Inc.