Algebraic & Geometric Topology

On growth rate and contact homology

Anne Vaugon

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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.

Article information

Algebr. Geom. Topol., Volume 15, Number 2 (2015), 623-666.

Received: 25 March 2012
Revised: 21 September 2014
Accepted: 28 September 2014
First available in Project Euclid: 28 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R17: Symplectic and contact topology
Secondary: 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.) 57M50: Geometric structures on low-dimensional manifolds

contact geometry Reeb vector field contact homology growth rate


Vaugon, Anne. On growth rate and contact homology. Algebr. Geom. Topol. 15 (2015), no. 2, 623--666. doi:10.2140/agt.2015.15.623.

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