On growth rate and contact homology

Vaugon, Anne¹

¹Laboratoire de Mathématiques Jean Leray, 2, rue de la Houssinière, BP 92208, 44322 Nantes Cedex 3, France, Département de Mathématiques d’Orsay, Université Paris-Sud, Bâtiment 425, Faulté des Sciences d’Orsay, 91405 Orsay Cedex, France

Algebraic and Geometric Topology, Tome 15 (2015) no. 2, pp. 623-666

Résumé

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.

Détail

DOI : 10.2140/agt.2015.15.623

Classification : 57R17, 53C15, 57M50
Keywords: contact geometry, Reeb vector field, contact homology, growth rate

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