Erratum to: A Variational Approach to Givental’s Nonlinear Maslov Index

We give a uniform lower bound for the polynomial complexity of all Reeb flows on the spherization (S * M, ξ) over a closed manifold. Our measure for the dynamical complexity of Reeb flows is slow volume growth, a polynomial version of topological entropy, and our uniform bound is in terms of the polynomial growth of the homology of the based loops space of M . As an application, we extend the Bott-Samelson theorem from geodesic flows to Reeb flows: If (S * M, ξ) admits a periodic Reeb flow, or, more generally, if there exists a positive Legendrian loop of a fibre S * q M , then M is a circle or the fundamental group of M is finite and the integral cohomology ring of the universal cover of M is the one of a compact rank one symmetric space.Proof of (i). We will see in the proof of (ii) that if π 1 (M) has subexponential growth, then M belongs to the list in (ii), and π 1 (M) has polynomial growth of order 0, 1, 3, or 4.Proof of (ii). A main ingredient of the proof is the following Lemma 3.3. Consider a closed orientable 3-manifold M. If π 1 (M) has subexponential growth, then M admits a geometric structure modeled on one of the four geometries Ë 3 , Ë 2 × Ê, 3 , Nil .Proof. The proof can be extracted from [5], and is repeated here for the readers convenience. We distinguish several cases. Case 1: M is not prime. This means that M can be written as a connected sum M = M 1 #M 2 with both π 1 (M 1 ) and π 1 (M 2 ) non-trivial. By the Seifert-Van Kampen Theorem, π 1 (M) is the free product π 1 (M 1 ) * π 1 (M 2 ). It follows from the existence of normal forms for free products that π 1 (M 1 ) * π 1 (M 2 ) contains a free subgroup of rank 2 unless π 1 (M 1 ) = π 1 (M 2 ) = 2 , see Exercise-with-hints 19 in Sec. 4.1 of [56]. Our hypothesis on π 1 (M) thus implies π 1 (M 1 ) = π 1 (M 2 ) = 2 , and so M = ÊP 3 # ÊP 3 . This manifold has a geometric structure modeled on the geometry Ë 2 × Ê, see [79, p. 457].Case 2: M is prime, but not irreducible. Then M = S 2 × S 1 , see [42, Proposition 1.4] or [45, Lemma 3.13]. In particular, M has a geometric structure modeled on Ë 2 × Ê.Case 3: M is irreducible. We distinguish two subcases:

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Slow volume growth for Reeb flows on spherizations and contact Bott–Samelson theorems

Frauenfelder

Labrousse

Schlenk

2015

J. Topol. Anal.

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Quasimorphisms on contactomorphism groups and contact rigidity

Borman

Zapolsky

2015

Geom. Topol.

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We build homogeneous quasi-morphisms on the universal cover of the contactomorphism group for certain prequantizations of monotone symplectic toric manifolds. This is done using Givental's nonlinear Maslov index and a contact reduction technique for quasimorphisms. We show how these quasi-morphisms lead to a hierarchy of rigid subsets of contact manifolds. We also show that the nonlinear Maslov index has a vanishing property, which plays a key role in our proofs. Finally we present applications to orderability of contact manifolds and Sandon-type metrics on contactomorphism groups.

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Erratum to: A Variational Approach to Givental’s Nonlinear Maslov Index

Cited by 2 publications

References 15 publications

Slow volume growth for Reeb flows on spherizations and contact Bott–Samelson theorems

Slow volume growth for Reeb flows on spherizations and contact Bott–Samelson theorems

Quasimorphisms on contactomorphism groups and contact rigidity

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