We prove a uniqueness theorem for homogeneous quasimorphisms on the universal cover of the symplectic linear grou

Salamon, Dietmar

Simon, Gabi

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ISRAEL JOURNAL OF MATHEMATICS 175 (2010), 221–224DOI: 10.1007/s11856-010-0010-4HOMOGENEOUS QUASIMORPHISMSON THE SYMPLECTIC LINEAR GROUPBYGabi Ben Simon and Dietmar A. SalamonDepartment Mathematik, ETH, HG G 37.1,Ramistrasse 101, 8092 Zurich, Switzerlande-mail: gabi.ben.simon@math.ethz.ch and salamon@math.ethz.chABSTRACTWe prove a uniqueness theorem for homogeneous quasimorphisms on theuniversal cover of the symplectic linear group.Let G be a group. A quasimorphism on G is a map ρ : G→ R satisfying|ρ(gh)− ρ(g)− ρ(h)| ≤ Cfor all g, h ∈ G and a suitable constant C. It is called homogeneous if ρ(gk) =kρ(g) for every g ∈ G and every integer k ≥ 0. LetSp(2n) :={Ψ ∈ R2n×2n |ΨJ0ΨT = J0}, J0 :=(0 −1l1l 0),denote the group of symplectic matrices and S˜p(2n) its universal cover. Thinkof an element of S˜p(2n) as a homotopy class [Ψ] (with fixed endpoints) of asmooth path Ψ : [0, 1]→ Sp(2n) satisfying Ψ(0) = 1l.Theorem 1: There is a unique homogeneous quasimorphism µ on S˜p(2n) thatdescends to the determinant homomorphism on U(n) in the sense thatdet(X + iY ) = exp (2piiµ([Ψ])) ,(X −YY X):= Ψ(1),for every [Ψ] ∈ S˜p(2n) with Ψ(1) ∈ Sp(2n) ∩O(2n) ∼= U(n).Received December 20, 2007 and in revised form May 14, 2008221222 GABI BEN SIMON AND DIETMAR A. SALAMON Isr. J. Math.The quasimomorphism of Theorem 1 plays a central role in [3] and this mo-tivated the present note. Two explicit constructions of the quasimorphism canbe found in [1] and [5]. The construction in [1] uses the unitary part in a polardecomposition and homogenization. The construction in [5] uses the eigenvaluedecomposition of a symplectic matrix (but does not mention the term quasi-morphism).Lemma 1: If ρ : G → R is a homogeneous quasimorphism, then ρ is invariantunder conjugation and ρ(g−1) = −ρ(g) for every g ∈ G.Proof of Lemma 1. Let C be the constant in the definition of quasimorphism.By homgeneity, we have ρ(1) = 0. Hence∣∣ρ(gk) + ρ(g−k)∣∣ ≤ C for every g ∈ Gand every integer k ≥ 0. By homogeneity, we obtain∣∣ρ(g) + ρ(g−1)∣∣ ≤ C/k forevery k and so ρ(g−1) = −ρ(g). Hence∣∣ρ(ghg−1)− ρ(h)∣∣ = ∣∣ρ(ghg−1)− ρ(g)− ρ(h)− ρ(g−1)∣∣ ≤ 2C.Using homogeneity again we obtain ρ(ghg−1) = ρ(h) for all g, h ∈ G.Proof of Theorem 1. Let P ⊂ Sp(2n) denote the set of symmetric positive defi-nite symplectic matrices. This space is contractible and hence there is a naturalinjection ι : P → S˜p(2n). Explicitly, the map ι assigns to a matrix P ∈ P theunique homotopy class of paths Φ : [0, 1] → P with endpoints Φ(0) = 1l andΦ(1) = P .Let µ : S˜p(2n) → R be a homogeneous quasimorphism that descends to thedeterminant homomorphism on U(n). It suffices to prove that the restrictionof µ to ι(P) is bounded. (If µ′ is another quasimorphism satisfying the require-ments of Theorem 1 and µ, µ′ are bounded on ι(P) then, by polar decompositionand the determinant assumption, their difference is bounded and so, by homo-geneity, they are equal.) We prove that µ vanishes on ι(P). For every unitarymatrix Q ∈ U(n) ⊂ Sp(2n) and every P ∈ P we have(1) µ(ι(QPQT )) = µ(ι(P )).To see this, choose two paths Φ : [0, 1] → P and Ψ : [0, 1] → U(n) such thatΦ(0) = Ψ(0) = 1 and Φ(1) = P , Ψ(1) = Q. Then µ([Φ]) = µ([ΨΦΨ−1]), byLemma 1, and so (1) follows from the fact that Ψ−1 = ΨT . Now let P ∈ P .Since P is a symmetric symplectic matrix we have PJ0P = J0 and henceµ(ι(P )) = µ(ι(J0P−1J−10)) = µ(ι(P−1)) = µ(ι(P )−1) = −µ(ι(P )).Vol. 175, 2010 HOMOGENEOUS QUASIMORPHISMS 223Here the second equation follows from (1) and the last from Lemma 1. Thisshows that µ(ι(P )) = 0 for every P ∈ P .Remark 1: Lemma 1 is well known to the experts [2]. We included a proof togive a self-contained exposition.Remark 2: Related results, obtained with different methods, are contained in [1]and [4]. Our main theorem can in fact be deduced from these results.Remark 3: The determinant homomorphism det : U(n) → S1 is uniquelydetermined by the condition that it induces an isomorphism on fundamentalgroups. Hence it follows from Theorem 1 that the homogeneous quasimorphismµ : S˜p(2n)→ R is uniquely determined by the condition that it restricts to anisomorphism of the fundamental group of Sp(2n) to the integers.Remark 4: The referee pointed out to us the following generalization.Let G be a uniformly perfect group and Z → G˜ → G be a central extension.If ρ is a homogeneous quasimorphism on G˜ that vanishes on Z then ρ ≡ 0.To see this we first observe that, since ρ vanishes on Z, we haveρ(zg) = limk→∞k−1ρ(zkgk) = limk→∞k−1ρ(gk) = ρ(g)for all z ∈ Z and g ∈ G˜. Hence ρ descends to G. Now let c > 0 be the constant inthe definition of quasimorphism. Then, by Lemma 1, we have∣∣ρ(ghg−1h−1)∣∣ =∣∣ρ(ghg−1h−1)− ρ(g)− ρ(hg−1h−1)∣∣ ≤ c for all g, h ∈ G. Since every element ofG can be expressed as a product of at most N commutators we have |ρ(g)| ≤(2N−1)c for all g ∈ G. Thus the quasimorphism is bounded and hence vanishesidentically.Theorem 1 follows from this generalization because Sp(2n) is uniformly per-fect and S˜p(2n) is a central extension of Sp(2n). However, the geometric prop-erties of the Maslov quasimorphism µ : S˜p(2n) → R derived in the proof ofTheorem 1 do not follow from the above algebraic argument.References[1] J. Barge and E. Ghys, Cocycle D’Euler et de Maslov, Mathematische Annalen 294 (1992),235–265.[2] C. Bavard, Longueur stable des commutateurs, Enseignement des Mathe´matiques 37(1991), 109–150.224 GABI BEN SIMON AND DIETMAR A. SALAMON Isr. J. Math.[3] G. Ben Simon, The geometry of partial order on contact transformations of prequan-tization manifolds, in Arithmetic and Geometry Around Quantization, (O. Ceyhan,Y. I. Manin and M. Marcolli, eds.), Progress in Mathematics, Birkha¨user, Basel,[4] M. Burger, A. Iozzi and A. Wienhard, Surface group representations with maximal Toledoinvariant, Annals of Mathematics, to appear.[5] D. A. Salamon and E. Zehnder, Floer homology, the Maslov index, and periodic solutionof Hamiltonian equations, in Analysis et cetera, (P. H. Rabinowitz and E. Zehnder, eds.),Academic Press, New York, 1990, pp. 573–600.

Homogeneous quasimorphisms on the symplectic linear group

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Homogeneous quasimorphisms on the symplectic linear group

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