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FLOER HOMOLOGY, DYNAMICS AND GROUPS

LEONID POLTEROVICHTel Aviv University

Abstract. We discuss some recent results on algebraic properties of the group of Hamil-tonian diffeomorphisms of a symplectic manifold. We focus on two topics which lie at theinterface between Floer theory and dynamics:

1. Restrictions on Hamiltonian actions of finitely generated groups, including a Hamil-tonian version of the Zimmer program dealing with actions of lattices;

2. Quasi-morphisms on the group of Hamiltonian diffeomorphisms.

The unifying theme is the study of distortion of cyclic and one-parameter subgroups withrespect to various metrics on the group of Hamiltonian diffeomorphisms.

In the present lectures we discuss some recent results on algebraic properties ofthe group of Hamiltonian diffeomorphisms Ham(M, ) of a smooth connectedsymplectic manifold (M2m, ). We focus on two topics which lie at the interfacebetween Floer theory and dynamics, where by dynamics we mean the study ofasymptotic behavior of Hamiltonian diffeomorphisms under iterations:

Restrictions on Hamiltonian actions of finitely generated groups, and in par-ticular a Hamiltonian version of the Zimmer program dealing with actions oflattices (Polterovich, 2002);

Quasi-morphisms on Ham, including the Calabi quasi-morphism introducedin a joint work with Michael Entov (Entov and Polterovich, 2003).

The unifying theme is the study of distortion of cyclic and one-parametersubgroups with respect to various metrics on the group of Hamiltonian dif-feomorphisms. We refer to Hofer and Zehnder (1994), McDuff and Salamon(1995; 2004), and Polterovich (2001) for symplectic preliminaries.

1. Hamiltonian actions of finitely generated groups

1.1. THE GROUP OF HAMILTONIAN DIFFEOMORPHISMS

Recall that symplectic manifolds appear as phase spaces of classical mechanics.An important principle of classical mechanics is that the energy of a system de-

417

2006 Springer. Printed in the Netherlands.

P. Biran et al. (eds.), Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, 417438.

418 L. POLTEROVICH

termines its evolution. The energy (or Hamiltonian function) Ft(x) := F(x, t) isa smooth function on M R. Here t is the time coordinate. Define the time-dependent Hamiltonian vector field sgrad Ft by the point-wise linear equationisgrad F = dF. The evolution of the system is described by the flow ft on Mgenerated by the Hamiltonian vector field sgrad Ft. We always assume that theunion of the supports of Ft, t R, is contained in a compact subset of M. Thisguarantees that the evolution is well defined. We will refer to the time-one-map f1of this flow as to the Hamiltonian diffeomorphism generated by F and denote it byF . Hamiltonian diffeomorphisms form a group which is denoted by Ham(M, )and which is the main object of our study.

We start with the following problem. Let (M, ) be a closed symplecticmanifold.

PROBLEM 1.1. Find restrictions on Hamiltonian actions of finitely generatedgroups on M, and, in particular, on finitely generated subgroups of Ham(M, ).

Polterovich (2002) develops an approach to this problem for some symplecticmanifolds with 2 = 0 which is based on Floer theory. Below we discuss (withan outline of proofs) some sample results in this direction. The selection wasmade with the idea to avoid as much as possible the use of sophisticated algebraicmachinery (the only exception is the Margulis finiteness theorem).

1.2. THE NO-TORSION THEOREM

THEOREM 1.2. Let (M, ) be a closed symplectic manifold with 2 = 0. Thenthe group Ham(M, ) has no torsion.

The proof is given in Section 2.2 below. Note that the assumption on 2 is es-sential: the 2-sphere admits isometries (rotations) of finite order. As an immediateconsequence of the theorem, we get the following result.

COROLLARY 1.3. Let (M, ) be a closed symplectic manifold with 2 = 0. Let be any group generated by elements of finite order. Then every homomorphism: Ham(M, ) is trivial: 1.

A classical example of a group generated by elements of finite order is thegroup SL(k,Z) with k 2. Indeed, it is shown in Newman, (1972, Theorem VII.3)that SL(k,Z) is generated by the matrix of the transformation

(x1, . . . , xk) (x2, . . . , xk, (1)k1x1)

which is clearly of finite order, and the matrix A 1k2 with

A =

(1 10 1

)

.

FLOER HOMOLOGY, DYNAMICS AND GROUPS 419

The matrix A, in turn, can be represented as the product of finite order matrices:

A =

(0 11 0

)

(

0 11 1

)

.

Hence SL(k,Z) with k 2 is generated by elements of finite order. Corollary 1.3yields the following result.

COROLLARY 1.4. Every Hamiltonian action of SL(k,Z) on a closed symplecticmanifold with 2 = 0 is trivial.

Consider now the case when SL(k,Z) is a normal subgroup of finite index.What can one say about Hamiltonian actions of ? In other words, we ask whetherthe phenomenon presented in Corollary 1.4 is robust from the viewpoint of grouptheory.

Each such is finitely generated (see de la Harpe, 2000). We will need alsothe following important result which is a particular case of the Margulis finitenesstheorem.

THEOREM 1.5 (Margulis, 1991; Zimmer, 1984). Let be an infinite normalsubgroup of SL(k,Z) for k 3. Then is of finite index in SL(k,Z). Moreover,every infinite normal subgroup of is of finite index in .

In contrast to this, the group SL(2,Z) has infinite normal subgroups of infiniteindex.

Let SL(k,Z), k 2, be a normal subgroup of finite index. In general itmay happen that all elements of have infinite order, and hence our argumentused in the proof of Corollary 1.4 does not work anymore. For instance, takeany integer l 2 and define the principal congruence subgroup l SL(k,Z)as the kernel of the natural homomorphism SL(k,Z) SL(k,Z/lZ). Clearly,l is a normal subgroup of finite index. It turns out that l has no torsion forl 3 (see Witte, 2001, Section 5.I). Another example is given by the commutatorsubgroup of SL(2,Z) which is isomorphic to the free group F2 with 2 generators(see Newman, 1972) and hence has no torsion.

Now we come to a well known and quite important point: the cases k = 2(the rank-one case) and k 3 (the higher-rank case) are dramatically different.The free group F2 admits a monomorphism to Ham(M, ) for any symplecticmanifold (M, ). Indeed, take two Hamiltonian diffeomorphisms f and g with acommon fixed point x M. It is easy to arrange that the differentials dx f and dxggenerate a free subgroup of linear transformations of TxM. Hence the subgroupgenerated by f and g is free.

In contrast to this, in the case k 3, there exist obstructions to Hamiltonianactions of infinite normal subgroups of SL(k,Z). We will focus for simplicity on

420 L. POLTEROVICH

the following special class of symplectic manifolds. Consider a closed symplecticmanifold (M, ) with 2(M) = 0. Then the lift of to the universal cover M ofM is exact: = d. We say that (M, ) is symplectically hyperbolic if 2(M) = 0and admits a primitive which is bounded with respect to a Riemannian metricon M coming from M. It is an easy exercise in hyperbolic geometry to show thatsurfaces of genus 2 endowed with the hyperbolic area form are symplecticallyhyperbolic. The same is true for their direct products. This class of symplecticmanifolds is a counterpart of Kahler hyperbolic manifolds considered in complexgeometry.

THEOREM 1.6. Let SL(k,Z), k 3, be an infinite normal subgroup. Let(M, ) be a closed symplectically hyperbolic manifold. Then every homomor-phism : Ham(M, ) is trivial: 1.

For the proof, we have to introduce the notion of distortion, which in a senseis a unifying theme for various topics discussed in these lectures.

1.3. DISTORTION IN NORMED GROUPS

Let G be a group endowed with a norm g, g G. The axioms of a norm are asfollows:

g > 0 if g 1, and 1 = 0; g1 = g; gh g + h

for all g, h G. We say that an element g G is distorted if

limn+

gnn

= 0.

Otherwise, g is called undistorted.Informally speaking, one can think of a cyclic subgroup generated by an

undistorted element as of a minimal geodesic in G. We will return to the notion ofdistortion many times throughout these lectures.

For instance, let be a finitely generated group. Let S be a symmetric finitegenerating set of . This means that

s S s1 S

and every element g can be written as

g = s1 sN , si S . (1)

Define the word norm g as the minimum of N over all decompositions (1). Notethat the word norms associated to different finite generating sets are mutually

FLOER HOMOLOGY, DYNAMICS AND GROUPS 421

equivalent, hence the property of g to be distorted/undistorted with respect to theword norm is well-defined.

Let us illustrate this notion in the following important example. The Heisen-berg groupH is the group with three generators f , g, h which satisfy the followingrelations: h = [ f , g] := f g f 1g1, [ f , h] = [g, h] = 1. It is not hard to check thathmn = [ f m, gn] for all m, n N. This yields hn2 const n for all n N. Inparticular, the element h is a distorted element of infinite order in H .

The Heisenberg group can be considered as a subgroup of SL(3,Z) (see, e.g.,de la Harpe, 2000, IV.A.8): the map :H SL(3,Z) with

( f ) =

1 0 00 1 00 1 1

, (g) =

1 0 01 1 00 0 1

, (h) =

1 0 00 1 01 0 1

is a monomorphism. It follows that (h) is a distorted element of infinite order inSL(3,Z). Note that SL(3,Z) naturally lies in SL(k,Z) for all k 3: we identifya matrix A SL(3,Z) with A 1k3. Hence the same conclusion holds true forSL(k,Z) with k 3.

We will need the foll