TY - JOUR
T1 - Growth of maps, distortion in groups and symplectic geometry
AU - Polterovich, Leonid
PY - 2002
Y1 - 2002
N2 - In the present paper we study two sequences of real numbers associated to a symplectic diffeomorphism: • The uniform norm of the differential of its n-th iteration; • The word length of its n-th iteration, where we assume that our diffeomorphism lies in a finitely generated group of symplectic diffeomorphisms. We find lower bounds for the growth rates of these sequences in a number of situations. These bounds depend on the symplectic geometry of the manifold rather than on the specific choice of a diffeomorphism. They are obtained by using recent results of Schwarz on Floer homology. As an application, we prove non-existence of certain non-linear symplectic representations for finitely generated groups.
AB - In the present paper we study two sequences of real numbers associated to a symplectic diffeomorphism: • The uniform norm of the differential of its n-th iteration; • The word length of its n-th iteration, where we assume that our diffeomorphism lies in a finitely generated group of symplectic diffeomorphisms. We find lower bounds for the growth rates of these sequences in a number of situations. These bounds depend on the symplectic geometry of the manifold rather than on the specific choice of a diffeomorphism. They are obtained by using recent results of Schwarz on Floer homology. As an application, we prove non-existence of certain non-linear symplectic representations for finitely generated groups.
UR - http://www.scopus.com/inward/record.url?scp=0036027234&partnerID=8YFLogxK
U2 - 10.1007/s00222-002-0251-x
DO - 10.1007/s00222-002-0251-x
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AN - SCOPUS:0036027234
VL - 150
SP - 655
EP - 686
JO - Inventiones Mathematicae
JF - Inventiones Mathematicae
SN - 0020-9910
IS - 3
ER -