Volume 15, issue 2 (2015)

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On growth rate and contact homology

Anne Vaugon

Algebraic & Geometric Topology 15 (2015) 623–666

A conjecture of Colin and Honda states that the number of periodic Reeb orbits of universally tight contact structures on hyperbolic manifolds grows exponentially with the period, and they speculate further that the growth rate of contact homology is polynomial on nonhyperbolic geometries. Along the line of the conjecture, for manifolds with a hyperbolic component that fibers on the circle, we prove that there are infinitely many nonisomorphic contact structures for which the number of periodic Reeb orbits of any nondegenerate Reeb vector field grows exponentially. Our result hinges on the exponential growth of contact homology, which we derive as well. We also compute contact homology in some nonhyperbolic cases that exhibit polynomial growth, namely those of universally tight contact structures on a circle bundle nontransverse to the fibers.

contact geometry, Reeb vector field, contact homology, growth rate
Mathematical Subject Classification 2010
Primary: 57R17
Secondary: 53C15, 57M50
Received: 25 March 2012
Revised: 21 September 2014
Accepted: 28 September 2014
Published: 22 April 2015
Anne Vaugon
Laboratoire de Mathématiques Jean Leray
2, rue de la Houssinière
BP 92208
44322 Nantes Cedex 3
Département de Mathématiques d’Orsay
Université Paris-Sud
Bâtiment 425
Faulté des Sciences d’Orsay
91405 Orsay Cedex