On the Oka principle in a Banach space,II

 Let X be one of the Banach spaces c ₀ , ℓ _p , 1≤p<∞; Ω⊂X pseudoconvex open, a holomorphic Banach vector bundle with a Banach Lie group G ^* for structure group. We show that a suitable Runge-type approximation hypothesis on X, G ^* (which we also prove for G ^* a solvable Lie group) implies the vanishing of the sheaf cohomology groups H ^q (Ω, 𝒪 ^E ), q≥1, with coefficients in the sheaf of germs of holomorphic sections of E. Further, letting 𝒪^Γ (𝒞^Γ) be the sheaf of germs of holomorphic (continuous) sections of a Banach Lie group bundle Γ→Ω with Banach Lie groups G, G ^* for fiber group and structure group, we show that a suitable Runge-type approximation hypothesis on X, G, G ^* (which we prove again for G, G ^* solvable Lie groups) implies the injectivity (and for X=ℓ₁ also the surjectivity) of the Grauert–Oka map H ¹ (Ω, 𝒪^Γ)→H ¹ (Ω, 𝒞^Γ) of multiplicative cohomology sets. Received: 1 March 2002 / Published online: 28 March 2003 Mathematics Subject Classification (2000): 32L20, 32L05, 46G20 RID="*" ID="*" Kedves Laci Móhan kisfiamnak. RID="*" ID="*" To my dear little Son     

Torsion in Boundary Coinvariants and K -Theory for Affine Buildings

Let (G, I, N, S) be an affine topological Tits system, and let Γ be a torsion-free cocompact lattice in G. This article studies the coinvariants H ₀(Γ; C(Ω,Z)), where Ω is the Furstenberg boundary of G. It is shown that the class [1] of the identity function in H ₀(Γ; C(Ω, Z)) has finite order, with explicit bounds for the order. A similar statement applies to the K ₀ group of the boundary crossed product C ^*-algebra C(Ω)Γ. If the Tits system has type ? ₂, exact computations are given, both for the crossed product algebra and for the reduced group C ^*-algebra.     

Preduals of semigroup algebras

For a locally compact group G, the measure convolution algebra M(G) carries a natural coproduct. In previous work, we showed that the canonical predual C ₀(G) of M(G) is the unique predual which makes both the product and the coproduct on M(G) weak^*-continuous. Given a discrete semigroup S, the convolution algebra ℓ ¹(S) also carries a coproduct. In this paper we examine preduals for ℓ ¹(S) making both the product and the coproduct weak^*-continuous. Under certain conditions on S, we show that ℓ ¹(S) has a unique such predual. Such S include the free semigroup on finitely many generators. In general, however, this need not be the case even for quite simple semigroups and we construct uncountably many such preduals on ℓ ¹(S) when S is either ℤ₊×ℤ or (ℕ,⋅).     

On the outradius of the teichmüller space

Let Γ be a fuchsian group which preserves the unit disc Δ and hence also its complement Δ^* in the Riemann sphere . The Bers embedding represents the Teichm=:uller space T(Γ) of Γ in the space (B (Δ^*, Γ) of bounded quadratic differentials for Γ in Δ^*. Then, T(Γ) is included in the closed ball centred at the origin of radius 6 inB(Δ^*, Γ) with respect to the norm employed in a paper by Nehari [The Schwarzian derivative and Schlicht functions; Bull. Amer. Math. Soc. 55 (1949), 545–551]. In other words the outradiuso(Γ) ofT(Γ) is not greater than 6. The purpose of this paper is to give a complete characterization of a fuchsian group Γ for which the outradiuso(Γ) ofT(Γ) attains this extremal value 6. The main theorem is: Let Γ be a fuchsian group preserving Δ^*. Then the outradiuso(Γ) of the Teichmüller spaceT(Γ) equals 6 if and only if for any positive numberd, either (i) there exists a hyperbolic disc of radiusd precisely invariant under the trivial subgroup, or (ii) there exists the collar of widthd about the axis of a hyperbolic element of Γ. Dedicated to Professor K?taro Oikawa on his 60th birthday     

Cusp forms

LetG andH ⊂G be two real semisimple groups defined overQ. Assume thatH is the group of points fixed by an involution ofG. Letπ ⊂L ²(H\G) be an irreducible representation ofG and letf επ be aK-finite function. Let Γ be an arithmetic subgroup ofG. The Poincaré seriesP _f(g)=Σ_H∩ΓΓ f(γ{}itg) is an automorphic form on Γ\G. We show thatP _f is cuspidal in some cases, whenH ∩Γ\H is compact. Partially supported by NSF Grant # DMS 9103608.     

The Ξ operator and its relation to Krein's spectral shift function

We explore connections between Krein's spectral shift function ζ(λ,H ₀, H) associated with the pair of self-adjoint operators (H ₀, H),H=H ₀+V, in a Hilbert spaceH and the recently introduced concept of a spectral shift operator Ξ(J+K ^*(H ₀−λ−i0)⁻¹ K) associated with the operator-valued Herglotz functionJ+K ^*(H ₀−z)⁻¹ K, Im(z)>0 inH, whereV=KJK ^* andJ=sgn(V). Our principal results include a new representation for ζ(λ,H ₀,H) in terms of an averaged index for the Fredholm pair of self-adjoint spectral projections (E J+A(λ)+tB(λ)(−∞, 0)),E _J((−∞, 0))), ℝ, whereA(λ)=Re(K ^*(H ₀−λ−i0⁻¹ K),B(λ)=Im(K ^*(H ₀−λ-i0)⁻¹ K) a.e. Moreover, introducing the new concept of a trindex for a pair of operators (A, P) inH, whereA is bounded andP is an orthogonal projection, we prove that ζ(λ,H ₀, H) coincides with the trindex associated with the pair (Ξ(J+K ^*(H ₀−λ−i0)K), Ξ(J)). In addition, we discuss a variant of the Birman-Krein formula relating the trindex of a pair of Ξ operators and the Fredholm determinant of the abstract scattering matrix. We also provide a generalization of the classical Birman—Schwinger principle, replacing the traditional eigenvalue counting functions by appropriate spectral shift functions.     

Affinely equivalent complete flat manifolds

Let E _Aff(Γ,G, m) be the set of affine equivalence classes of m-dimensional complete flat manifolds with a fixed fundamental group Γ and a fixed holonomy group G. Let n be the dimension of a closed flat manifold whose fundamental group is isomorphic to Γ. We describe E _Aff(Γ,G, m) in terms of equivalence classes of pairs (ε, ρ), consisting of epimorphisms of Γ onto G and representations of G in ℝ ^m-n . As an application we give some estimates of card E _Aff(Γ,G, m).     

On Optimal Orientations of Cartesian Products of Trees

. Let d(D) (resp., d(G)) denote the diameter and r(D) (resp., r(G)) the radius of a digraph D (resp., graph G). Let G×H denote the cartesian product of two graphs G and H. An orientation D of G is said to be (r, d)-invariant if r(D)=r(G) and d(D)=d(G). Let {T _i }, i=1,…,n, where n≥2, be a family of trees. In this paper, we show that the graph ∏ _{i =1} ⁿ T _i admits an (r, d)-invariant orientation provided that d(T ₁)≥d(T ₂)≥4 for n=2, and d(T ₁)≥5 and d(T ₂)≥4 for n≥3. Received: July 30, 1997 Final version received: April 20, 1998     

Average bending of convex pleated planes in hyperbolic three-space

If P is a pleated plane in 3-dimensional hyperbolic space H ³ and α a geodesic in its intrinsic metric we define B(P,α), the average bending of P in the direction α. We show that if P is a convex pleated plane embedded in H ³ then B(P,α)≤K for some universal K. Furthermore if P_Γ is a boundary component of the convex hull of a quasi-Fuchsian group Γ then B(P_Γ,α)=B(Γ) almost everywhere, where B(Γ) is a constant times the length of the bending lamination β_Γ of the pleated surface X _Γ=P_Γ/Γ. We use these to prove a number of results about quasi-Fuchsian groups including a universal bound on the Lipschitz constant for the map to infinity and a bound on the length of β_Γ by a constant times the Euler characteristic of X _Γ. Oblatum 10-X-1996 & 23-V-1997     

Fourier Analysis of Semistable Distributions

We consider the generalized convolution powers G _α ^*u (x) of an arbitrary semistable distribution function G _α (x) of exponent α∈(0,2), and prove that for all j, k∈{0,1,2,…} and u>0 the derivatives G _α ^(k,j)(x;u)=∂ ^k+j G _α ^*u (x)/∂ x ^k ∂ u ^j , x∈ℝ, are of bounded variation on the whole real line ℝ. The proof, along with an integral recursion in j, is new even in the special case of stable laws, and the result provides a framework for possible asymptotic expansions in merge theorems from the domain of geometric partial attraction of semistable laws. An erratum to this article can be found at     

Recurrence in unipotent groups and ergodic nonabelian group extensions

LetT be a measure-preserving and ergodic transformation of a standard probability space (X,S, μ) and letf:X → SUT _d (ℝ) be a Borel map into the group of unipotent upper triangulard ×d matrices. We modify an argument in [12] to obtain a sufficient condition for the recurrence of the random walk defined byf, in terms of the asymptotic behaviour of the distributions of the suitably scaled mapsf(n,x)=(fT ⁿ⁻¹·fT ⁿ⁻²…fT·f). We give examples of recurrent cocycles with values in the continuous Heisenberg group H₁(ℝ)=SUT₃(ℝ), and we use a recurrent cocycle to construct an ergodic skew-product extension of an irrational rotation by the discrete Heisenberg group H₁(ℤ)=SUT₃(ℤ). The author was partially supported by the FWF research project P16004-MAT.     

The chromatic number of the product of two ℵ1-chromatic graphs can be countable

We prove (in ZFC) that for every infinite cardinal ϰ there are two graphsG ₀,G ₁ with χ(G ₀)=χ(G ₁)=ϰ⁺ and χ(G ₀×G ₁)=ϰ. We also prove a result from the other direction. If χ(G ₀)≧≧ℵ₀ and χ(G ₁)=k<ω, then χ(G ₀×G ₁)=k.     

On Projections of Semi-Algebraic Sets Defined by Few Quadratic Inequalities

Let S⊂ℝ ^k+m be a compact semi-algebraic set defined by P ₁≥0,…,P _ℓ ≥0, where P _i ∈ℝ[X ₁,…,X _k ,Y ₁,…,Y _m ], and deg (P _i )≤2, 1≤i≤ℓ. Let π denote the standard projection from ℝ ^k+m onto ℝ ^m . We prove that for any q>0, the sum of the first q Betti numbers of π(S) is bounded by (k+m) ^{O(q ℓ)}. We also present an algorithm for computing the first q Betti numbers of π(S), whose complexity is . For fixed q and ℓ, both the bounds are polynomial in k+m. The author was supported in part by an NSF Career Award 0133597 and a Sloan Foundation Fellowship.     

The non-commutative Weil algebra

For any compact Lie group G, together with an invariant inner product on its Lie algebra ?, we define the non-commutative Weil algebra ? _G as a tensor product of the universal enveloping algebra U(?) and the Clifford algebra Cl(?). Just like the usual Weil algebra W _G =S(?^*)⊗∧?^*, ? _G carries the structure of an acyclic, locally free G-differential algebra and can be used to define equivariant cohomology ℋ _G (B) for any G-differential algebra B. We construct an explicit isomorphism ?: W _G →? _G of the two Weil algebras as G-differential spaces, and prove that their multiplication maps are G-chain homotopic. This implies that the map in cohomology H _G (B)→ℋ _G (B) induced by ? is a ring isomorphism. For the trivial G-differential algebra B=ℝ, this reduces to the Duflo isomorphism S(?) ^G ≅U(?) ^G between the ring of invariant polynomials and the ring of Casimir elements. Oblatum 13-III-1999 & 27-V-1999 / Published online: 22 September 1999     

Group measure space decomposition of II1 factors and W*-superrigidity

We prove a “unique crossed product decomposition” result for group measure space II₁ factors L ^∞(X)⋊Γ arising from arbitrary free ergodic probability measure preserving (p.m.p.) actions of groups Γ in a fairly large family G\mathcal{G}, which contains all free products of a Kazhdan group and a non-trivial group, as well as certain amalgamated free products over an amenable subgroup. We deduce that if T _n denotes the group of upper triangular matrices in PSL (n,ℤ), then any free, mixing p.m.p. action of G = \operatornamePSL(n,\mathbbZ)*_Tn\operatornamePSL(n,\mathbbZ)\Gamma=\operatorname{PSL}(n,\mathbb{Z})*_{T_{n}}\operatorname{PSL}(n,\mathbb{Z}) is W^∗-superrigid, i.e. any isomorphism between L ^∞(X)⋊Γ and an arbitrary group measure space factor L ^∞(Y)⋊Λ, comes from a conjugacy of the actions. We also prove that for many groups Γ in the family G\mathcal{G}, the Bernoulli actions of Γ are W^∗-superrigid.     

EQUIVARIANT SELF EQUIVALENCES OF PRINCIPAL FIBRE BUNDLES

Let E be a compact Lie group, G a closed subgroup of E, and H a closed normal sub-group of G. For principal fibre bundle (E,p, E,/G;G) tmd (E/H,p‘,E/G;G/H), the relation between auta(E) (resp. autce (E)) and autG/H(E/H) (resp. autGe/H(E/H)) is investigated by using bundle map theory and transformation group theory. It will enable us to compute the group JG(E) (resp. SG(E)) while the group J G/u(E/H) is known.     

Metric properties of the tropical Abel–Jacobi map

Let Γ be a tropical curve (or metric graph), and fix a base point p∈Γ. We define the Jacobian group J(G) of a finite weighted graph G, and show that the Jacobian J(Γ) is canonically isomorphic to the direct limit of J(G) over all weighted graph models G for Γ. This result is useful for reducing certain questions about the Abel–Jacobi map Φ _p :Γ→J(Γ), defined by Mikhalkin and Zharkov, to purely combinatorial questions about weighted graphs. We prove that J(G) is finite if and only if the edges in each 2-connected component of G are commensurable over ℚ. As an application of our direct limit theorem, we derive some local comparison formulas between ρ and \varPhi_p^*(r){\varPhi}_{p}^{*}(\rho) for three different natural “metrics” ρ on J(Γ). One of these formulas implies that Φ _p is a tropical isometry when Γ is 2-edge-connected. Another shows that the canonical measure μ _Zh on a metric graph Γ, defined by S. Zhang, measures lengths on Φ _p (Γ) with respect to the “sup-norm” on J(Γ).     

Quantum cohomology of <Emphasis Type="Italic">G/P</Emphasis> and homology of affine Grassmannian

Let G be a simple and simply-connected complex algebraic group, P ⊂ G a parabolic subgroup. We prove an unpublished result of D. Peterson which states that the quantum cohomology QH ^*(G/P) of a flag variety is, up to localization, a quotient of the homology H _*(Gr _G ) of the affine Grassmannian Gr _G of G. As a consequence, all three-point genus-zero Gromov–Witten invariants of G/P are identified with homology Schubert structure constants of H _*(Gr _G ), establishing the equivalence of the quantum and homology affine Schubert calculi.     

Deformations of complex structures on Γ/SL 2(C)

LetG be a connected complex semisimple Lie group. Let Γ be a cocompact lattice inG. In this paper, we show that whenG isSL ₂(C), nontrivial deformations of the canonical complex structure onX exist if and only if the first Betti number of the lattice Γ is non-zero. It may be remarked that for a wide class of arithmetic groups Γ, one can find a subgroup Γ′ of finite index in Γ, such that Γ′/[Γ′,Γ′] is finite (it is a conjecture of Thurston that this is true for all cocompact lattices inSL(2, C)). We also show thatG acts trivially on the coherent cohomology groupsH ⁱ(Γ/G, O) for anyi≥0.