floer homology, dynamics and groups leonid

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, ω) of a smooth connectedsymplectic manifold (M2m, ω). 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, 417–438.

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 Fω = −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 byφF . Hamiltonian diffeomorphisms form a group which is denoted by Ham(M, ω)and which is the main object of our study.

We start with the following problem. Let (M, ω) 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, ω).

Polterovich (2002) develops an approach to this problem for some symplecticmanifolds with π2 = 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, ω) be a closed symplectic manifold with π2 = 0. Thenthe group Ham(M, ω) has no torsion.

The proof is given in Section 2.2 below. Note that the assumption on π2 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, ω) be a closed symplectic manifold with π2 = 0. LetΓ be any group generated by elements of finite order. Then every homomorphismΦ: Γ → Ham(M, ω) is trivial: Φ ≡ 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)k−1x1)

which is clearly of finite order, and the matrix A ⊕ 1k−2 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 1−1 −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 π2 = 0 is trivial.

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

Each such Γ 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 Γ be an infinite normalsubgroup of SL(k,Z) for k ≥ 3. Then Γ is of finite index in SL(k,Z). Moreover,every infinite normal subgroup of Γ is of finite index in Γ.

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

Let Γ ⊂ SL(k,Z), k ≥ 2, be a normal subgroup of finite index. In general itmay happen that all elements of Γ 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 Γl ⊂ SL(k,Z)as the kernel of the natural homomorphism SL(k,Z) → SL(k,Z/lZ). Clearly,Γl is a normal subgroup of finite index. It turns out that Γl 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, ω) for any symplecticmanifold (M, ω). 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, ω) with π2(M) = 0. Then the lift ω of ω to the universal cover M ofM is exact: ω = dλ. We say that (M, ω) is symplectically hyperbolic if π2(M) = 0and ω admits a primitive λ 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 Γ ⊂ SL(k,Z), k ≥ 3, be an infinite normal subgroup. Let(M, ω) be a closed symplectically hyperbolic manifold. Then every homomor-phism Φ: Γ → Ham(M, ω) is trivial: Φ ≡ 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;

− ‖g−1‖ = ‖g‖;− ‖gh‖ ≤ ‖g‖ + ‖h‖

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

limn→+∞

‖gn‖n

= 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 Γ be a finitely generated group. Let S be a symmetric finitegenerating set of Γ. This means that

s ∈ S ⇐⇒ s−1 ∈ S

and every element g ∈ Γ 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


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 −1g−1, [ 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 ı:H → SL(3,Z) with

ı( f ) =

1 0 00 1 00 1 1

, ı(g) =

1 0 01 1 00 0 1

, ı(h) =

1 0 00 1 01 0 1

is a monomorphism. It follows that ı(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 ⊕ 1k−3. Hence the same conclusion holds true forSL(k,Z) with k ≥ 3.

We will need the following proposition which is a particular case of a muchstronger theorem by Lubotzky et al. (2000).

PROPOSITION 1.7. Let Γ be an infinite normal subgroup of SL(k,Z) with k ≥ 3.Then Γ has a distorted element.

Proof. Let ı:H → SL(k,Z) be the monomorphism described above. Put φ =

ı( f ), ψ = ı(g) and θ = ı(h). Since Γ is a normal subgroup of finite index, thereexists N ∈ N so that φN and ψN lie in Γ. It was already mentioned that [φnN , ψnN] =

θn2N2for all n ∈ N. This yields that the element θN2 ∈ Γ is distorted:

‖(θN2)n2‖ ≤ const · n.

1.4. THE NO-DISTORTION THEOREM

The next result provides a restriction on Hamiltonian actions of finitely generatedgroups on a closed symplectically hyperbolic manifold in terms of distortion.

THEOREM 1.8. Assume that (M, ω) is a closed symplectically hyperbolic man-ifold. Consider any finitely generated group Γ. Let g ∈ Γ be an element which isdistorted with respect to the word norm on Γ. Then Φ(g) = 1 for any homomor-phism Φ: Γ → Ham(M, ω). In particular, if Γ is a finitely generated subgroup ofHam(M, ω), every element of Γ \ 1 is undistorted with respect to the word normon Γ.

422 L. POLTEROVICH

The proof based on Floer theory will be given in Section 2.4 below.

Proof of Theorem 1.6. Apply the theorem above to a distorted (with respectto the word norm) element g of infinite order lying in Γ. Such an element existsdue to Proposition 1.7. We see that g lies in the kernel of any homomorphismΦ: Γ → Ham(M, ω). Note that Ker(Φ) is a normal subgroup of Γ. It is infinitesince g has infinite order. The Margulis finiteness theorem (see Theorem 1.5above) guarantees that Ker(Φ) has finite index in Γ. Hence the quotient Γ/Ker(Φ)is finite, so Φ has finite image. Applying No-Torsion Theorem 1.2 we concludethat Φ ≡ 1.

1.5. THE ZIMMER PROGRAM

Infinite normal subgroups of SL(k,Z) are basic examples of much more generalfinitely generated groups called lattices (see Witte, 2001, for an introduction tolattices). The problems formulated below are already highly nontrivial for thefollowing class of lattices. Consider the semisimple Lie group

G = SL(k1,R) × · · · × SL(kd,R)

where ki ≥ 2 and d ∈ N. The number∑d

i=1(k1 − 1) is called the real rank ofG. A discrete subgroup Γ ⊂ G is a lattice if the Haar measure of G/Γ is finite.The proof of the fact that SL(k,Z) ⊂ SL(k,R) is a lattice is quite involved,see, e.g., Feres (1998, Appendix A.1) for a transparent exposition. A lattice Γ

is reducible if there exists a decomposition G = G1 × G2 and lattices Γi ⊂ Gi

such that the intersection Γ ∩ (Γ1 × Γ2) of the two subgroups has finite index inboth of them. Otherwise, Γ is irreducible. Every lattice in SL(k,R) is irreducible.A lattice is called uniform if G/Γ is compact, and nonuniform otherwise. It isnot hard to see that SL(k,Z) ⊂ SL(k,R) is nonuniform. The same holds for itsinfinite normal subgroups since they are of finite index by Margulis theorem. Animportant property of nonuniform lattices is the existence of “strongly” distortedelements of infinite order (see Section 2.5). This difficult theorem was establishedby Lubotzky, Mozes and Raghunathan in Lubotzky et al. (2000). Proposition 1.7above is an elementary manifestation of this fact.

Problem 1.1 goes back to Zimmer (1987) which contains a number of ex-citing conjectures about actions of lattices on manifolds. Roughly speaking, theZimmer program can be formulated as follows: Assume that the real rank of Gis at least two. Let Γ be an irreducible lattice in G. The Zimmer conjecture statesthat the image of every homomorphism of Γ to the group of (volume-preserving)diffeomorphisms of a closed connected manifold of sufficiently small dimension isfinite.

The Zimmer program gave rise to many remarkable developments, see, e.g.,Burger and Monod (1999; 2002), Farb and Shalen (1999), and Ghys (1999). Ghys


(1999) gave a highly nontrivial proof of the Zimmer conjecture in the case ofactions of lattices on the circle by (not necessarily volume-preserving) diffeo-morphisms. The two-dimensional case of the conjecture is an active field, see(Polterovich, 2002) and Franks and Handel (2003; 2004). We refer to Fisher(2003) for a discussion of recent progress in this direction. Below we addressa Hamiltonian version of the Zimmer program which deals with actions of latticesby Hamiltonian diffeomorphisms of closed symplectic manifolds.

PROBLEM 1.9. Find restrictions on Hamiltonian actions of lattices on closedsymplectic manifolds.

Theorem 1.6 should be considered as a step in this direction. We referthe reader to Polterovich (2002) and Franks and Handel (2003) for variousgeneralizations, and to Section 2.5 for a further discussion.

2. Floer theory in action

2.1. A BRIEF SKETCH OF FLOER THEORY

Here we present a very brief sketch of Floer theory (Floer, 1988, 1989a, 1989b)with an emphasize on the main tool we are going to use — spectral invariantsof Hamiltonian diffeomorphisms coming from filtered Floer homology. Theseinvariants were introduced and studied by Schwarz (2000) in the case whenthe cohomology class of the symplectic form vanishes on π2 of the symplecticmanifold, and by Oh (2002b) in the general case (see also Viterbo, 1992; Oh,1997, 1999, 2002a; Entov and Polterovich, 2003). A more detailed exposition ofthis theory can be found in Oh (2005) and McDuff and Salamon (2004).

Let (M2m, ω) be a closed symplectic manifold.1 Consider the space Λ of allsmooth contractible loops x: S 1 = R/Z → M. Denote by F the space of allsmooth Hamiltonian functions F : M × S 1 → R which satisfy the followingnormalization condition:

∫

MF(·, t)ωm = 0 for any t ∈ S 1. Define the action

functional

AF(x, u) :=∫

S 1F(x(t), t) dt −

∫

uω,

where x ∈ Λ and u is a disc spanning x in M. In general, the action functionalis a multi-valued function on Λ — it depends on the homotopy class with fixedboundary of the disc u. However, it becomes single-valued on a suitable coveringΛ of Λ, and its differential dAF is a well-defined closed 1-form on Λ itself. A

1 There is a general belief that the picture presented below is valid for all closed symplectic man-ifolds. At the moment, it is confirmed under some additional assumptions on (M, ω), see (McDuff

and Salamon, 2004).

424 L. POLTEROVICH

crucial fact is that the critical points of this 1-form correspond to contractible1-periodic orbits of the Hamiltonian flow ft generated by F.

Floer homology is Morse – Novikov homology of the 1-form dAF . To developMorse theory in this context start with a loop Jt, t ∈ S 1, of ω-compatible almostcomplex structures and define a Riemannian metric on Λ by

(ξ1, ξ2) =

∫ 1

0ω(ξ1(t), Jtξ2(t)

)dt,

where ξ1, ξ2 ∈ TΛ. Lift this metric to Λ and consider the negative gradient flow ofthe action functional AF . For a generic choice of the Hamiltonian F and the loopJt the count of isolated gradient trajectories connecting critical points of AF

gives rise in a standard way to the Morse complex of AF on Λ. Each connectingtrajectory is a path in the space of loops and hence forms a 2-dimensional cylin-der. An important feature of the gradient flow is that the connecting trajectoriesare solutions of a deformed Cauchy – Riemann equation, and hence they can bestudied with the Gromov theory of pseudo-holomorphic curves. Floer homology isdefined as the homology of the above-mentioned Morse complex with coefficientsin an appropriate Novikov ring. This homology is independent of the HamiltonianF and is canonically isomorphic to the quantum homology QH(M, ω) — a suitabledeformation of the homology ring of M (see Piunikhin et al., 1996).

Let φF be the natural lift of the time-1-map f1 = φF of the flow to the universalcover Ham(M, ω) of the group of Hamiltonian diffeomorphisms of M. It turns outthat the Floer homologies of the sublevel sets AF < α ⊂ Λ depend only onthe element φF but not on the specific choice of a generating Hamiltonian F.Hence they give rise to a bunch of invariants of φF . These homologies form a richalgebraic object canonically associated to the group Ham(M, ω), and carry someinteresting information about the group.

The natural inclusion of sublevel sets

AF < α → AF < +∞ = Λ

induces a morphism in the corresponding Floer homologies:

iα: HFloer(AF < α) → HFloer(Λ) = QH(M, ω).

For a quantum homology class a ∈ QH(M, ω) set

c(a, φF) = infα ∈ R | a ∈ Image iα. (2)

This number is called a spectral invariant of the (lifted) Hamiltonian dif-feomorphism φF ∈ Ham(M, ω). Intuitively speaking, spectral invariants arehomologically essential critical values of the action functional AF . They play animportant role in symplectic dynamics.


Suppose now that π2(M) = 0. In this case the action AF is a single-valuedfunctional on Λ. Introduce the width of the Hamiltonian diffeomorphism φF asthe maximal difference between the critical values of AF . Schwarz (2000) provedthat this quantity does not depend on the specific choice of a Hamiltonian functiongenerating φF , and therefore gives rise to an invariant of the Hamiltonian diffeo-morphism φF . It is not hard to see that width(φF) is always finite. We will needthe following deep result of Floer theory:

THEOREM 2.1 (Schwarz, 2000). Assume that φF 1. Then the action functionalAF has at least two distinct critical values. In particular,

width(φF) > 0.

On the other hand, width(1) = 0. Another, this time very simple, property ofthe width is its nice behavior under iterations of a Hamiltonian diffeomorphism:

width( f n) ≥ n · width( f ) for all f ∈ Ham(M, ω). (3)

2.2. WIDTH AND TORSION

As an immediate application of the width, we prove the No-Torsion Theorem.

Proof of Theorem 1.2. Take any f ∈ Ham(M, ω) \ 1. Then width( f n) > 0 forall n ∈ N by Theorem 2.1 and formula (3). Thus f n 1, so f is of infinite order.

2.3. A GEOMETRY ON Ham(M, ω)

Assume that (M, ω) is closed and symplectically hyperbolic. Fix any Riemannianmetric ρ on M. Given a path ft, t ∈ [0; 1], of Hamiltonian diffeomorphisms ofM, denote by F(x, t) the corresponding Hamiltonian. By adding a time-dependentconstant const(t) to F we can achieve the following normalization:

∫

MFtω

n = 0for all t ∈ [0; 1]. Define

length ft =

∫ 1

0

(maxx∈M

| sgrad Ft(x)|ρ + maxx∈M

|Ft(x)|)

dt.

Introduce a norm ν on Ham(M, ω) by

ν( f ) = inf length ft,

where the infimum is taken over all paths ft of Hamiltonian diffeomorphismsjoining the identity with f . It is easy to check (and we leave this as an exercise)that ν satisfies the axioms of a norm presented in Section 1.3 above. Our nextresult relates this norm with the width introduced in Section 2.1.

426 L. POLTEROVICH

THEOREM 2.2 (Geometric inequality). There exists a constant C > 0 so that

ν( f ) ≥ C · width( f )

for all Hamiltonian diffeomorphisms f 1.

The proof is based on a simple argument of differential-geometric flavor, seePolterovich (2002).

2.4. WIDTH AND DISTORTION

THEOREM 2.3. Every element f ∈ Ham(M, ω) \ 1 is undistorted with respectto the norm ν.

Proof. Combining Theorem 2.2 with formula (3) we get

ν( f n) ≥ C · width( f n) ≥ Cn · width( f ).

But, crucially, Theorem 2.1 yields width( f ) > 0. Thus f is undistorted.

Now we are ready to prove the No-Distortion Theorem.

Proof of Theorem 1.8. Take any symmetric finite generating set S ⊂ Γ, and put

C = maxs∈S

ν(Φ(s)

).

Then ν(Φ(g)

) ≤ C · ‖g‖, and hence

limn→+∞

ν(Φ(g)n)n

≤ C · limn→+∞

‖gn‖n

= 0.

We get that Φ(g) is distorted with respect to ν. By Theorem 2.3, Φ(g) = 1.

2.5. MORE REMARKS ON THE ZIMMER PROGRAM

We conclude our presentation of the Hamiltonian version of the Zimmer pro-gram with some remarks and open problems. The geometric inequality given inTheorem 2.2 above can be extended in a weaker form to any closed symplecticmanifold with π2 = 0. In fact, one gets a lower bound on the growth type of thesequence ν( f n), f 1 in terms of the symplectic filling function of (M, ω), seePolterovich (2002). This function measures the “minimal growth” of a primitiveof the symplectic form on the universal cover M of M.

The lower bound on ν( f n), f 1 yields in the same way an analogue ofTheorem 1.8: one can show that if Γ is a finitely generated group, and ‖g‖n/n


decays sufficiently fast for some element g ∈ Γ, this element must lie in the kernelof any homomorphism Γ → Ham(M, ω).

Assume, for instance, that the lift of ω to M admits a primitive which growsnot faster than Rε , ε > 0, on balls of radius R. A good example is given by thestandard symplectic torus T

2n, where one can take ε = 1. Let Γ be a finitelygenerated group, and suppose that

lim inflog ‖gn‖

log n= 0 (4)

for some g ∈ Γ. Then g lies in the kernel of any homomorphism Γ → Ham(M, ω).The Lubotzky – Mozes – Raghunathan theorem (Lubotzky et al., 2000) states thatevery nonuniform lattice of rank ≥ 2 has an element of infinite order whichsatisfies equation (4). Arguing as in the proof of Theorem 1.6 one can show thatevery Hamiltonian action of an irreducible nonuniform lattice of rank ≥ 2 on sucha symplectic manifold is trivial.

Some of these results extend to the identity component of the group of allsymplectic (not necessarily Hamiltonian) diffeomorphisms of (M, ω), and in thecase of closed surfaces — to the group of all area preserving diffeomorphisms.

It is, of course, a challenging problem to handle the remaining case of uniform(that is, co-compact) lattices.

Let us mention that our method does not work for symplectic manifolds withπ2 0: The notion of width of a Hamiltonian diffeomorphism, which is crucialfor our approach in the case when π2(M) = 0, does not make sense anymore, say,for M = CPm. The reason is that the action functional becomes multi-valued onthe loop space Λ. It is an interesting problem of Floer theory to find an appropriatemodification of Theorem 2.1 and inequality (3) in the case when π2(M) 0. Myfeeling is that it should involve not only symplectic actions of closed orbits ofHamiltonian flows, but also the Conley – Zehnder indices. On the other hand, formanifolds with π2 0 the filtered Floer homology form a richer object than in thecase π2 = 0 due to the existence of “quantum effects.” Hopefully, one can use thefull strength of this rich structure to get information about actions of lattices.

Recently some restrictions of symplectic actions of lattices on the 2-spherewere found by Franks and Handel by using sophisticated tools of 2-dimensionaldynamics. For instance they proved the following extension of Theorem 1.6.

THEOREM 2.4 (Franks and Handel, 2003). Let Γ ⊂ SL(k,Z), k ≥ 3, be an in-finite normal subgroup. Then every homomorphism Φ: Γ → Ham(S 2) has finiteimage.

It would be interesting to reprove and extend their results, for instance tocomplex projective spaces of higher dimensions, by using Floer theory.

428 L. POLTEROVICH

3. The Calabi quasi-morphism and related topics

3.1. EXTENDING THE CALABI HOMOMORPHISM

Until recently, as far as I know, the only purely algebraic results on the structureof the group Ham were obtained by Banyaga (1978). They split into two cases:

C I: M . Then Ham(M, ω) is simple, that is it has no nontrivial normalsubgroups.

C II: M : ω = dλ. In this case Ham(M, ω) admits the Calabihomomorphism

CalM: Ham(M, ω) → R

whose kernel coincides with the commutator subgroup of Ham(M, ω). More-over, this commutator subgroup turns out to be a simple group. The Calabihomomorphism is defined as “the average energy required in order togenerate a Hamiltonian diffeomorphism.” More precisely,

CalM(φF) =

∫ 1

0

∫

MF(x, t)ωm dt. (5)

One can show that this map is well defined (that is, its value depends on φF

but not on a specific F) and is indeed a homomorphism.

Return now to the case when (M, ω) is a closed symplectic manifold. Cover Mby sufficiently small open discs Uα and consider the collection of homomorphismsCalUα : Ham(Uα) → R. The Calabi homomorphisms obviously agree on overlaps.Hence a natural question is whether it is possible to extend this collection to aglobal homomorphism of Ham(M, ω). The answer is of course negative sincethe group is simple by Banyaga’s above-mentioned theorem (Banyaga, 1978).It turns out, however, that for certain M’s one can perform such an extensionwith a bounded error (Entov and Polterovich, 2003). The formalism of “homo-morphisms up to a bounded error” is given by the notion of a quasi-morphismwhich originated in the works by Gromov (1982) and Brooks (1981) on boundedcohomology of groups and recently became quite popular in group theory anddynamics (see, e.g., the beautiful short survey by Kotschick, 2004). The rest ofthese notes is dedicated to an outline of this construction, its applications to thestudy of distortion with respect to Hofer’s norm on Ham(M, ω) and discussion ofsome related topics.

Another set of applications, to symplectic intersections, lies outside the scopeof the present notes, see Biran et al. (2004) and Entov and Polterovich (2005).


3.2. INTRODUCING QUASI-MORPHISMS

A quasi-morphism on a group G is a function r: G → R which satisfies thehomomorphism equation up to a bounded error: there exists C > 0 such that

|r( f g) − r( f ) − r(g)| ≤ C

for all f , g ∈ G. A quasi-morphism r is called homogeneous if r(gn) = nr(g) forall g ∈ G and n ∈ Z. Given a quasi-morphism r, one defines its homogenizationrh by

rh(g) = limn→+∞

r(gn)n

.

One can show that rh is a homogeneous quasi-morphism and the difference |rh− r|is a bounded function on G. Homogeneous quasi-morphisms are invariant underconjugations in G. They play an important role in the study of groups. We referthe reader to Bavard (1991) and Kotschick (2004) for an introduction to quasi-morphisms.

EXAMPLE 3.1 (The Barge – Ghys quasi-morphism; Barge and Ghys, 1988).Consider the isometric action of a discrete group G ⊂ PSL(2,R) on the upperhalf-plane H equipped with the hyperbolic metric. To construct quasi-morphismsof G, start with a smooth G-invariant one-form α such that |dα/Ω| ≤ C for someC > 0, where Ω is the hyperbolic area form. Given such α and a base-point z ∈ H,set

r(g) =

∫

(z,gz)α,

where (z,w) denotes the geodesic segment between two points z,w ∈ H. Take anyf , g ∈ G. Let ∆ be the hyperbolic triangle with vertices z, f z, and f gz. Clearly,

|r( f g) − r( f ) − r(g)| =∣∣∣∣∣

∫

∂∆

α

∣∣∣∣∣ =

∣∣∣∣∣

∫

∆

dα∣∣∣∣∣ ≤ Cπ

since the area of any hyperbolic triangle does not exceed π. Thus r is a quasi-morphism. Assume, for example, that G is the fundamental group of a closedoriented surface of genus ≥ 2. Applying this construction to various forms αone can show that homogeneous quasi-morphisms on G form a linear space ofdimension continuum.

Homogeneous quasi-morphisms, when they exist, serve as a substitute of ho-momorphisms in some interesting situations. For instance, using quasi-morphismsone can prove that certain elements are undistorted with respect to some meaning-ful metrics on the group. As an illustration, consider the commutator subgroup G′

of a group G. Every element g ∈ G′ can be written as the product of a finite

430 L. POLTEROVICH

number of simple commutators of the form aba−1b−1 with a, b ∈ G. Define thecommutator norm of g as the minimal number of simple commutators required torepresent g. It is an easy exercise to show that if r(g) > 0 for some homogeneousquasi-morphism r on G, the element g is undistorted in the sense of the commu-tator norm. Surprisingly, the converse statement is true as well. Its proof, which isdue to Bavard (1991), is nonconstructive — it uses the Hahn – Banach theorem.

3.3. QUASI-MORPHISMS ON Ham(M, ω)

Now we focus on the case when G is either the group Ham(M, ω) of Hamiltoniandiffeomorphisms of a symplectic manifold, or its universal cover Ham(M, ω).It was shown by Banyaga (1978) that, when M is closed, such a group is per-fect (that is, it coincides with its commutator subgroup) and therefore admits nonontrivial homomorphisms to R. Sometimes, however, these groups admit non-trivial homogeneous quasi-morphisms. Existence of such quasi-morphisms on thegroup of Hamiltonian diffeomorphisms and/or its universal cover is known for thefollowing classes of closed symplectic manifolds:

− orientable surfaces (Gambaudo and Ghys, 2004);

− closed manifolds whose fundamental group has trivial center and admitsnontrivial quasi-morphisms (see Section 3.7 below);

− manifolds with c1 = 0 (Barge and Ghys, 1992; Entov, 2004);

− complex projective spaces (see Givental′, 1990; Entov and Polterovich, 2003)and, more generally, spherically monotone symplectic manifolds with semi-simple even quantum homology algebra (Entov and Polterovich, 2003).

The quasi-morphism constructed in Entov and Polterovich (2003) for spheri-cally monotone symplectic manifolds with semi-simple even quantum homologyalgebra comes from Floer theory. The list of such manifolds includes for instanceCPm, S 2 × S 2 and CP2 blown up at one point. For simplicity, let us concentrateon the case when M = CPm. Our quasi-morphism is defined as follows. Take anyf ∈ Ham(M, ω). Let f be its lift to the universal cover Ham(M, ω) associated toany Hamiltonian flow generating f . Put

µ( f ) = −Volume(M) · limn→∞

c([M], f n)n

,

where c is the spectral number defined by formula (2) and [M] is the fundamentalclass of M which corresponds to the unity in the quantum homology ring. Onecan show that µ( f ) is well defined (in particular, it does not depend on the specificlift f ) and the map µ: Ham(M) → R is a quasi-morphism. In addition, it has thefollowing peculiar Calabi property: Let U ⊂ M be an open displaceable subset,which means that φ(U) ∩ Closure(U) = ∅ for some Hamiltonian diffeomorphism


φ ∈ Ham(M, ω). Assume that the restriction of the symplectic form ω to U isexact. Consider the subgroup Ham(U) ⊂ Ham(M) consisting of all Hamiltoniandiffeomorphisms generated by Hamiltonian functions vanishing outside U. Thenthe restriction of µ to Ham(U) coincides with the Calabi homomorphism CalUgiven by equation (5). Hence we fulfilled the promise given in Section 3.1. Wecall a homogeneous quasi-morphism with the Calabi property a Calabi quasi-morphism.

3.4. DISTORTION IN HOFER’S NORM ON Ham(M, ω)

Here we present a geometric application of the Calabi quasi-morphism. Let (M, ω)be a symplectic manifold. Given a path ft, t ∈ [0; 1], of Hamiltonian dif-feomorphisms of M, denote by F(x, t) = Ft(x) the corresponding Hamiltonian.Define

lengthH ft =

∫ 1

0

(maxx∈M

Ft(x) − minx∈M

Ft(x))

dt.

Define Hofer’s norm νH on Ham(M, ω) by

νH( f ) = inf length ft,

where the infimum is taken over all paths ft of Hamiltonian diffeomorphismsjoining the identity with f . This norm was introduced by Hofer (1990). A deepfact2 is that νH is a genuine norm on Ham(M, ω), that is νH( f ) > 0 for f 1. Thiswas proved in Hofer (1990) and Polterovich (1993) for some classes of symplec-tic manifolds, and by Lalonde and McDuff (1995) in full generality. The metricρH( f , g) := νH( f g−1) on Ham(M, ω) is called Hofer’s metric. It was an objectof intensive study since its discovery in 1990. We refer the reader to Polterovich(2001) for preliminaries on Hofer’s geometry.

Any compactly supported smooth function F on (M, ω) generates a oneparameter subgroup ft of Hamiltonian diffeomorphisms. Define its distortion

d(F) = limt→+∞

νH( ft)t

. (6)

We say that a one-parameter subgroup is undistorted if d(F) > 0.It was shown by Sikorav (1990) that when M = R

2n every one parametersubgroup of Ham(M, ω) remains a bounded distance from the identity. This resultwas extended in Polterovich and Siburg (2000) as follows:

THEOREM 3.2 (Dichotomy theorem). Let (M, ω) be an open surface of infinitearea. Every one parameter subgroup of Hamiltonian diffeomorphisms is eitherundistorted, or remains a bounded distance from the identity.

2 In contrast to the nondegeneracy of the norm ν defined in Section 2.3 above.

432 L. POLTEROVICH

QUESTION 3.3. Does the dichotomy above remain true on any symplecticmanifold?

The answer is not known even on closed oriented surfaces. However one canstate the following weaker result.

THEOREM 3.4. Let M be a closed oriented surface equipped with an area form.The one-parameter subgroup of Ham(M) generated by a generic Hamiltonianfunction on M is undistorted with respect to Hofer’s norm.

Here by a generic Hamiltonian function we mean a function from an opendense set in C∞(M). We believe that this holds true on every closed symplecticmanifold, though at the moment such a result seems to be out of reach. In thecase of the 2-torus the proof is given in Polterovich (2001), and the same argu-ment settles the case of surfaces of higher genus. For the 2-sphere, the problemremained open until the appearance of the Calabi quasi-morphism. Let us explainthe argument in this case (see Entov and Polterovich, 2003).

First of all, it turns out that the Calabi quasi-morphism µ introduced above isLipschitz with respect to Hofer’s metric. Assume without loss of generality thatVolume(M) = 1. Then

|µ( f )| ≤ νH( f ) (7)

for every f ∈ Ham(M, ω).Let us focus on the case when M is the 2-sphere of total area 1, and F is a

Morse function on M with∫

S 2 F · ω = 0. Look at connected components of thelevel sets of F. A simple combinatorial argument shows that there exists a unique(maybe singular) component, say γ, with the following property:

S 2 \ γ = U1 ( · · · ( Uk,

where Ui are open topological discs of area ≤ 12 . Recall that φF stands for the

Hamiltonian diffeomorphism generated by a function F. The next lemma wasproved in Entov and Polterovich (2003) (see Entov and Polterovich, 2005, forvarious generalizations of this result):

LEMMA 3.5. Let µ: Ham(S 2) → R be any Calabi quasi-morphism continuousin Hofer’s metric. Then µ(φF) = −F(γ).

Proof. After a small C0-perturbation (which is a legitimate operation in viewof the continuity of µ) we can assume that F ≡ F(γ) in a neighborhood of γ. PutH := F − F(γ). We see that H decomposes as

H = H1 + · · · + Hk, support(Hi) ⊂ Ui.


Note that φH = φH1 · · · φHk and all φHi’s pair-wise commute. Now we use asimple algebraic property of any homogeneous quasi-morphism on any group: itsrestriction to an abelian subgroup is a genuine homomorphism. Hence

µ(φF) = µ(φH) =

k∑

i=1

µ(φHk ). (8)

Note that since the area of Ui is ≤ 12 , each function Hi has displaceable support.

Therefore, the Calabi property guarantees that

µ(φHi) =

∫

S 2Hi · ω

for all i = 1, . . . , k. Substituting this to equation (8) we get that

µ(φF) =

k∑

i=1

∫

S 2Hi · ω = −F(γ),

as required.

Proof of Theorem 3.4 for S 2. It suffices to prove the theorem for functions withzero mean (otherwise, add a constant to achieve this). Choose such a function, say,F to be Morse and assume in addition the following generic property: the level setF = 0 is regular and does not contain a connected component which dividesS 2 into two discs of equal areas. Therefore F(γ) 0 and hence |µ(φF)| > 0. Let ft be the one parameter subgroup generated by F. With our notation, f1 = φF .Combining the Lipschitz property of the Calabi quasi-morphism with Lemma 3.5,we get that

νH( ft) ≥ |µ( ft)| = t|µ( f1)| = t|F(γ)|,and hence

d(F) = limt→+∞

νH( ft)t

≥ |F(γ)| > 0,

as required.

3.5. EXISTENCE AND UNIQUENESS OF CALABI QUASI-MORPHISMS

Here are some basic questions which we are unable to answer at the moment.

QUESTION 3.6. Which symplectic manifolds admit a Calabi quasi-morphism?

This problem is open already for all closed surfaces of genus ≥ 1. Re-cently Ostrover (2005) extended the results of Entov and Polterovich (2003) tononmonotone symplectic forms on S 2 × S 2 and CP2 blown up at one point.

434 L. POLTEROVICH

QUESTION 3.7. Is a Calabi quasi-morphism unique?

This is unknown even for the 2-sphere. Note that Lemma 3.5 shows thatall Calabi quasi-morphisms which are continuous in Hofer’s metric must coin-cide on the set of Hamiltonian diffeomorphisms of S 2 generated by autonomousHamiltonians.

The situation changes when one considers groups of compactly supportedHamiltonian diffeomorphisms of open symplectic manifolds. It is proved in Biranet al. (2004) that for the standard symplectic ball the Calabi quasi-morphismswhich are continuous in Hofer’s metric form an affine space of dimension atleast continuum. In fact, one can obtain a family of continuum cardinality of lin-early independent Calabi quasi-morphisms on Ham(B2n) by considering suitableconformally symplectic embeddings B2n → CPn and pulling back the Calabiquasi-morphism µ defined above.

3.6. “HYPERBOLIC” FEATURES OF Ham(M, ω)?

The above-mentioned ampleness of the space of quasi-morphisms was knownearlier for certain discrete subgroups of hyperbolic isometries (and more generalGromov-hyperbolic groups), see Example 3.1 above.

The Dichotomy Theorem 3.2 has the following counterpart for discrete sub-groups G of the Mobius group PSL(2,R) endowed with the commutator norm:given an element f ∈ [G,G] of infinite order, either f is undistorted or the cyclicsubgroup generated by f remains a bounded distance from the identity. This resultreadily follows from the fact (Polterovich and Rudnick, 2004; Epstein and Fuji-wara, 1997; Bestvina and Fujiwara, 2002) that either there exists a homogeneousquasi-morphism of G which does not vanish on f , or f is conjugate to its inverse inG. (The proof of this fact given in Polterovich and Rudnick, 2004 is based on theBarge-Ghys construction in Example 3.1 above combined with some elementaryhyperbolic geometry.) In the first case f is undistorted. If f is conjugate to itsinverse we have the following:

f = g f −1g−1 =⇒ f n = g f −ng−1 =⇒ f 2n = [ f n, g].

Thus, writing ‖ ‖ for the commutator norm we have that ‖ f 2n‖ = 1 and

‖ f 2n+1‖ ≤ ‖ f 2n‖ + ‖ f ‖ ≤ 1 + ‖ f ‖

for all n ∈ N. This proves the desired dichotomy.The list of common features of the symplectic and hyperbolic worlds can be

continued. I do not know whether it is a superficial coincidence, or there existssome deeper reason for that.


3.7. FROM π1(M) TO Diff0(M,Ω)

We complete these lectures with the remark that quasi-morphisms on the fun-damental group π1(M) can be, under certain assumptions, canonically lifted toquasi-morphisms on the group of volume-preserving diffeomorphisms of M. Forinstance, for closed symplectic manifolds with hyperbolic fundamental group(e.g., for surfaces of higher genus) this enables us to produce plenty of quasi-morphisms on Ham (compare with the last paragraph of the previous section.)The construction below is implicitly contained in Gambaudo and Ghys (2004)which presents various beautiful quasi-morphisms on groups of area-preservingdiffeomorphisms of surfaces.

Let M be a closed connected manifold endowed with a volume form Ω. LetG = Diff0(M,Ω) be the identity component of the group of volume-preservingdiffeomorphisms of M. Suppose for simplicity that Γ := π1(M, z) has no center(for instance, M is a closed oriented surface of higher genus).

For every x ∈ M choose a path ax between x and z. The paths are assumedto be “short,” that is their lengths are bounded in some Riemannian metric onM. Moreover, we assume that the path ax depends on x continuously outside aclosed set of measure 0 on M. One can easily produce such a system of paths asfollows: Triangulate M by sufficiently small simplices and choose ax to dependcontinuously on x in the interior of each simplex.

Choose any auxiliary Riemannian metric on M. Take any isotopy ft withf0 = id and f1 = f . For x ∈ M consider the loop ( ft, x) ⊂ M based at z,consisting of 3 pieces:

− the path ax oriented from z to x;

− the trajectory of x under ft;

− a f (x) oriented from f (x) to z.

Denote by σ( f , x) ∈ Γ the homotopy class of this loop. We claim that σ( f , x) doesnot depend on the choice of the isotopy ft. Indeed, any two such isotopies differby a loop, say, ht, t ∈ [0; 1] of diffeomorphisms with f0 = f1 = 1. It suffices toshow that the loop D := ht xt∈[0;1] is contractible. Take any element γ ∈ π1(M, x)and represent it by a closed curve C ⊂ M. Look at the torus

⋃t ht(C). We see

that the loops D and C lie on the torus and hence represent commuting homotopyclasses. The conclusion is that D belongs to the center of π1(M, x), and hence iscontractible by our assumption. The claim follows.

Observe now that σ: G × M → Γ is a cocycle, that is

σ( f g, x) = σ( f , gx) · σ(g, x)

436 L. POLTEROVICH

for all f , g ∈ G and x ∈ M. Let r: Γ → R be any quasi-morphism. Defineσ∗r: G →R by

σ∗r( f ) =

∫

Mr(σ( f , x)

) ·Ω.

Note that for a fixed isotopy between 1 and f , the loops ( ft, x) have uniformly(with respect to x ∈ M) bounded length. Hence the function σ( f , ·): M → Γ hasfinite image. By our assumption on the system of paths ax, the function σ( f , x)is locally constant on an open set of full measure. Thus the integral above exists.Further, by the cocycle property,

σ∗r( f g) =

∫

Mr(σ( f , gx) ·σ(g, x)

) ·Ω =

∫

Mr(σ( f , gx)

) ·Ω+

∫

Mr(σ(g, x)

) ·Ω+ Q,

where|Q| ≤ K := Volume(M) · sup

u,v∈Γ

|r(uv) − r(u) − r(v)|.

Making the change of variable y = gx and using that g is volume-preserving weget that ∫

Mr(σ( f , gx)

) ·Ω =

∫

Mr(σ( f , y)

) ·Ω.

Thus σ∗r is a quasi-morphism of G.It is easy to see that the quasi-morphisms σ∗r obtained in this way are

nontrivial.

Acknowledgements

I owe a lot to Marc Burger for his generous help with my paper (Polterovich,2002) on the Zimmer program. I am grateful to Paul Biran and Michael Entov forsharing with me their insight on various aspects of the Calabi quasi-morphism.I thank Yaron Ostrover and Felix Schlenk for carefully reading the manuscriptand pointing out numerous mistakes. Finally, I thank Octav Cornea and FrancoisLalonde for their warm hospitality in Montreal where these lectures have beendelivered.

References

Banyaga, A. (1978) Sur la structure du groupe des diffeomorphismes qui preservent une formesymplectique, Comment. Math. Helv. 53, 174 – 227.

Barge, J. and Ghys, E. (1988) Surfaces et cohomologie bornee, Invent. Math. 92, 509 – 526.Barge, J. and Ghys, E. (1992) Cocycles d’Euler et de Maslov, Math. Ann. 294, 235–265.Bavard, C. (1991) Longueur stable des commutateurs, Enseign. Math. (2) 37, 109 – 150.


Bestvina, M. and Fujiwara, K. (2002) Bounded cohomology of subgroups of mapping class groups,Geom. Topol. 6, 69 – 89.

Biran, P., Entov, M., and Polterovich, L. (2004) Calabi quasimorphisms for the symplectic ball,Commun.Contemp. Math. 6, 793 – 802.

Brooks, R. (1981) Some remarks on bounded cohomology, In I. Kra and B. Maskit (eds.), RiemannSurfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference, Vol. 97 of Ann.of Math. Stud., Stony Brook, NY, 1978, pp. 53–63, Princeton, NJ, Princeton Univ. Press.

Burger, M. and Monod, N. (1999) Bounded cohomology of lattices in higher rank Lie groups, J.Eur. Math. Soc. (JEMS) 1, 199 – 235.

Burger, M. and Monod, N. (2002) Continuous bounded cohomology and applications to rigiditytheory, Geom. Funct. Anal. 12, 219 – 280.

de la Harpe, P. (2000) Topics in Geometric Group Theory, Chicago Lectures in Math., Chicago, IL,Univ. Chicago Press.

Entov, M. (2004) Commutator length of symplectomorphisms, Comment. Math. Helv. 79, 58 – 104.Entov, M. and Polterovich, L. (2003) Calabi quasimorphism and quantum homology, Int. Math.

Res. Not. 30, 1635 – 1676.Entov, M. and Polterovich, L. (2005) Quasi-states and symplectic intersections, Comment. Math.

Helv., to appear; arXiv:math.SG/0410338.Epstein, D. B. A. and Fujiwara, K. (1997) The second bounded cohomology of word-hyperbolic

groups, Topology 36, 1275 – 1289.Farb, B. and Shalen, P. (1999) Real-analytic actions of lattices, Invent. Math. 135, 273 – 296.Feres, R. (1998) Dynamical Systems and Semisimple Groups: An Introduction, Vol. 126 of

Cambridge Tracts in Math., Cambridge, Cambridge Univ. Press.Fisher, D. (2003) Nonlinear representation theory of finitely generated groups, available at http:

//comet.lehman.cuny.edu/fisher/.Floer, A. (1988) Morse theory for Lagrangian intersections, J. Differential Geom. 28, 513 – 547.Floer, A. (1989)a Cuplength estimates on Lagrangian intersections, Comm. Pure Appl. Math. 42,

335 – 356.Floer, A. (1989)b Symplectic fixed points and holomorphic spheres, Comm. Math. Phys. 120,

575 – 611.Franks, J. and Handel, M. (2003) Area preserving group actions on surfaces, Geom. Topol. 7,

757 – 771.Franks, J. and Handel, M. (2004) Distortion elements in group actions on surfaces, arXiv:math.DS/

0404532.Gambaudo, J.-M. and Ghys, E. (2004) Commutators and diffeomorphisms of surfaces, Ergodic

Theory Dynam. Systems 24, 1591 – 1617.Ghys, E. (1999) Actions de reseaux sur le cercle, Invent. Math. 137, 199 – 231.Givental′, A. B. (1990) Nonlinear generalization of the Maslov index, In Theory of Singularities

and its Applications, Vol. 1 of Adv. Soviet Math., pp. 71–103, Providence, RI, Amer. Math. Soc.Gromov, M. (1982) Volume and bounded cohomology, Inst. Hautes Etudes Sci. Publ. Math. 56,

5–99.Hofer, H. (1990) On the topological properties of symplectic maps, Proc. Roy. Soc. Edinburgh Sect.

A 115, 25 – 28.Hofer, H. and Zehnder, E. (1994) Symplectic Invariants and Hamiltonian Dynamics, Birkhauser

Adv. Texts Basler Lehrbucher, Basel, Birkhauser.Kotschick, D. (2004) What is . . . a quasi-morphism?, Notices Amer. Math. Soc. 51, 208 – 209.Lalonde, F. and McDuff, D. (1995) The geometry of symplectic energy, Ann. of Math. (2) 141,

349–371.

438 L. POLTEROVICH

Lubotzky, A., Mozes, S., and Raghunathan, M. S. (2000) The word and Riemannian metrics onlattices in semisimple groups, Inst. Hautes Etudes Sci. Publ. Math. 91, 5 – 53.

Margulis, G. A. (1991) Discrete Subgroups of Semisimple Lie Groups, Vol. 17 of Ergeb. Math.Grenzgeb. (3), Berlin, Springer.

McDuff, D. and Salamon, D. (1995) Introduction to Symplectic Topology, Oxford Math. Monogr.,Oxford, Oxford Univ. Press.

McDuff, D. and Salamon, D. (2004) J-Holomorphic Curves and Symplectic Topology, Vol. 52 ofAmer. Math. Soc. Colloq. Publ, Providence, RI, Amer. Math. Soc.

Newman, M. (1972) Integral Matrices, Vol. 45 of Pure Appl. Math., New York, Academic Press.Oh, Y.-G. (1997) Symplectic topology as the geometry of action functional. I. Relative Floer theory

on the cotangent bundle, J. Differential Geom. 46, 499 – 577.Oh, Y.-G. (1999) Symplectic topology as the geometry of action functional. II. Pants product and

cohomological invariants, Comm. Anal. Geom. 7, 1 – 54.Oh, Y.-G. (2002)a Chain level Floer theory and Hofer’s geometry of the Hamiltonian diffeomor-

phism group, Asian J. Math. 6, 579–624.Oh, Y.-G. (2002)b Mini-max theory, spectral invariants and geometry of the Hamiltonian diffeo-

morphism group, arXiv:math.SG/0206092.Oh, Y.-G. (2005) Lectures on Floer theory and spectral invariants of Hamiltonian flows, in this

volume.Ostrover, Y. (2005), Ph.D. thesis, Tel Aviv University, in preparation.Piunikhin, S., Salamon, D., and Schwarz, M. (1996) Symplectic Floer – Donaldson theory and

quantum cohomology, In Contact and Symplectic Geometry, Vol. 8 of Publ. Newton Inst.,pp. 171–200, Cambridge, Cambridge Univ. Press.

Polterovich, L. (1993) Symplectic displacement energy for Lagrangian submanifolds, ErgodicTheory Dynam. Systems 13, 357 – 367.

Polterovich, L. (2001) The Geometry of the Group of Symplectic Diffeomorphisms, Lectures Math.ETH Zurich, Basel, Birkhauser.

Polterovich, L. (2002) Growth of maps, distortion in groups and symplectic geometry, Invent. Math.150, 655–686.

Polterovich, L. and Rudnick, Z. (2004) Stable mixing for cat maps and quasi-morphisms of themodular group, Ergodic Theory Dynam. Systems 24, 609 – 619.

Polterovich, L. and Siburg, K. F. (2000) On the asymptotic geometry of area-preserving maps,Math. Res. Lett. 7, 233–243.

Schwarz, M. (2000) On the action spectrum for closed symplectically aspherical manifolds, PacificJ. Math. 193, 419–461.

Sikorav, J.-C. (1990) Systemes hamiltoniens et topologie symplectique, Quaderni del Dottorato,Pisa, Universita di Pisa.

Viterbo, C. (1992) Symplectic topology as the geometry of generating functions, Math. Ann. 292,685 – 710.

Witte, D. (2001) Introduction to arithmetic groups, arXiv:math.DG/0106063.Zimmer, R. (1984) Ergodic Theory and Semisimple Groups, Vol. 81 of Monogr. Math., Basel,

Birkhauser.Zimmer, R. (1987) Actions of semisimple groups and discrete subgroups, In Proceedings of the In-

ternational Congress of Mathematicians, Vol. 1, 2, Berkeley, 1986, pp. 1247–1258, Providence,RI, Amer. Math. Soc.

floer homology, dynamics and groups leonid

Documents