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     

Unique ergodicity of flows on homogeneous spaces

LetG be a unimodular Lie group, Γ a co-compact discrete subgroup ofG and ‘a’ a semisimple element ofG. LetT _a be the mapgΓ →ag Γ:G/Γ →G/Γ. The following statements are pairwise equivalent: (1) (T _a, G/Γ,θ) is weak-mixing. (2) (T _a, G/Γ) is topologically weak-mixing. (3) (G ^u, G/Γ) is uniquely ergodic. (4) (G ^u, G/Γ,θ) is ergodic. (5) (G ^u, G/Γ) is point transitive. (6) (G ^u, G/Γ) is minimal. If in additionG is semisimple with finite center and no compact factors, then the statement “(T _a, G/Γ,θ) is ergodic” may be added to the above list. The authors were partially supported by NSF grant MCS 75-05250.     

Semi-primary hereditary rings

A semi-primary hereditary ring Σ, with radicalM and residue ring Γ=Σ/M, is uniquely determined by Γ and a Γ-bimoduleA=M/M ², whenever Σ admits a splitting Σ=Γ+A+M ².     

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     

Some remarks on universal graphs

LetΓ be a class of countable graphs, and let ℱ(Γ) denote the class of all countable graphs that do not contain any subgraph isomorphic to a member ofΓ. Furthermore, letTΓ andHΓ denote the class of all subdivisions of graphs inΓ and the class of all graphs contracting to a member ofΓ, respectively. As the main result of this paper it is decided which of the classes ℱ(TK ⁿ ) and ℱ(HK ⁿ ),n≦ℵ₀, contain a universal element. In fact, for ℱ(TK ⁴)=ℱ(HK ⁴) a strongly universal graph is constructed, whereas for 5≦n≦ℵ₀ the classes ℱ(TK ⁿ ) and ℱ(HK ⁿ ) have no universal elements. Dedicated to Klaus Wagner on his 75^th birthday     

On representation of the solution of the Neumann problem in a domain with peak by the harmonic simple layer potential

The problem of finding a solution of the Neumann problem for the Laplacian in the form of a simple layer potential Vρ with unknown density ρ is known to be reducible to a boundary integral equation of the second kind to be solved for density. The Neumann problem is examined in a bounded n-dimensional domain Ω⁺ (n > 2) with a cusp of an outward isolated peak either on its boundary or in its complement Ω⁻ = R ⁿ \Ω⁺. Let Γ be the common boundary of the domains Ω^±, Tr(Γ) be the space of traces on Γ of functions with finite Dirichlet integral over R ⁿ , and Tr(Γ)* be the dual space to Tr(Γ). We show that the solution of the Neumann problem for a domain Ω⁻ with a cusp of an inward peak may be represented as Vρ⁻, where ρ⁻ ∈ Tr(Γ)* is uniquely determined for all Ψ⁻ ∈ Tr(Γ)*. If Ω⁺ is a domain with an inward peak and if Ψ⁺ ∈ Tr(Γ)*, Ψ⁺ ⊥ 1, then the solution of the Neumann problem for Ω⁺ has the representation u ⁺ = Vρ⁺ for some ρ⁺ ∈ Tr(Γ)* which is unique up to an additive constant ρ₀, ρ₀ = V ⁻¹(1). These results do not hold for domains with outward peak.     

Subspaces of<Emphasis Type="Italic">l</Emphasis><Superscript>∞</Superscript> (Γ) without quasicomplements

J. Lindenstrauss proves in [L] thatc ₀(Γ) is not quasicomplemented inl ^∞(Γ) while H. P. Rosenthal in [R] proves that subspaces, whose dual balls are weak* sequentially compact and weak* separable, are quasicomplemented inl ^∞(Γ). In this note it is proved that weak* separability of the dual is the precise condition determining whether a subspace, without isomorphic copies ofl ₁ and whose dual balls are weak* sequentially compact, is quasicomplemented or not inl ^∞(Γ). Especially spaces isomorphic tol _p(Γ), for 1<p<∞, have no quasicomplements inl ^∞(Γ) if Γ is uncountable.     

Boundedness of maximal operators and potential operators on Carleson curves in Lebesgue spaces with variable exponent

We prove the boundedness of the maximal operator Mr in the spaces L^p（·）（Г,p） with variable exponent p（t） and power weight p on an arbitrary Carleson curve under the assumption that p（t） satisfies the log-condition on Г. We prove also weighted Sobolev type L^p（·）（Г, p） → L^q（·）（Г, p）-theorem for potential operators on Carleson curves.     

Distinguishing labeling of the actions of almost simple groups

Suppose Γ is a group acting on a set X, written as (Γ,X). An r-labeling f: X→{1,2, ..., r} of X is called distinguishing for (Γ,X) if for all σ∈Γ,σ≠1, there exists an element x∈X such that f(x)≠f(x ^σ ). The distinguishing number d(Γ,X) of (Γ,X) is the minimum r for which there is a distinguishing r-labeling for (Γ,X). If Γ is the automorphism group of a graph G, then d(Γ,V (G)) is denoted by d(G), and is called the distinguishing number of the graph G. The distinguishing set of Γ-actions is defined to be D*(Γ)={d(Γ,X): Γ acts on X}, and the distinguishing set of Γ-graphs is defined to be D(Γ)={d(G): Aut(G)≅Γ}. This paper determines the distinguishing set of Γ-actions and the distinguishing set of Γ-graphs for almost simple groups Γ.     

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.     

On the surjectivity of some trace maps

LetK be a commutative ring with a unit element 1. Let Γ be a finite group acting onK via a mapt: Γ→Aut(K). For every subgroupH≤Γ define tr _H :K→K ^H by tr _h (x)=Σ_σ∈H σ(x). We proveTheorem: tr_Γ is surjective onto K ^Γ if and only if tr _P is surjective onto K ^P for every (cyclic) prime order subgroup P of Γ. This is false for certain non-commutative ringsK.     

Crystallographic Haar Wavelets

(Γ,a)-crystallographic multiwavelets are a finite set of functions Y = { yⁱ}_i=1^L\Psi= \{ \psi ^{i}\}_{i=1}^{L}, which generate an orthonormal basis, a Riesz basis or a Parseval frame for L ²(ℝ ^d ), under the action of a crystallographic group Γ, and powers of an appropriate expanding affine map a, taking the place of the translations and dilations in classical wavelets respectively. Associated crystallographic multiresolution analysis of multiplicity n ((Γ,a)-MRA) are defined in a natural way. A complete characterization of scaling function vectors which generates Haar type (Γ,a)-MRA’s in terms of (Γ,a)-multireptiles is given. Examples of (Γ,a)-MRA crystallographic wavelets of Haar type in dimension 2 and 3 are provided.     

Lower bounds at infinity of solutions of partial differential equations in the exterior of a proper cone

An extension of a classical theorem of Rellich to the exterior of a closed proper convex cone is proved: Let Γ be a closed convex proper cone inR ⁿ and −Γ′ be the antipodes of the dual cone of Γ. Let be a partial differential operator with constant coefficients inR ⁿ, whereQ(ζ)≠0 onR ⁿ−iΓ′ andP _i is an irreducible polynomial with real coefficients. Assume that the closure of each connected component of the set {ζ∈R ⁿ−iΓ′;P _j(ζ)=0, gradP _j(ζ)≠0} contains some real point on which gradP _j≠0 and gradP _j∉Γ∪(−Γ). LetC be an open cone inR ⁿ−Γ containing both normal directions at some such point, and intersecting each normal plane of every manifold contained in {ξ∈R ⁿ;P(ξ)=0}. Ifu∈ℒ′∩L _loc ² (R ⁿ−Γ) and the support ofP(−i∂/∂x)u is contained in Γ, then the condition implies that the support ofu is contained in Γ.     

Un critère de rationalité provenant de la géométrie non commutative

We prove a conjecture of A. Connes, which gives a rationality criterion for elements of the closure of ℂΓ (Γ a free group) in the space of bounded operators in l ² (Γ). We show that this criterion applies also to the ring of Malcev–Neumann series on Γ.

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Oblatum 2-VIII-1996 & 20-IX-1996     

On point to point reflection of harmonic functions across real-analytic hypersurfaces in ℝ n

Let Γ be a non-singular real-analytic hypersurface in some domainU ⊂ ℝ ⁿ and let Har₀(U, Γ) denote the linear space of harmonic functions inU that vanish on Γ. We seek a condition onx ⁰,x ¹ ∈U/Γ such that the reflection law (RL)u(x ⁰)+Ku(x ¹)=0, ∀u∈Har₀(U, Γ) holds for some constantK. This is equivalent to the class Har₀ (U, Γ) not separating the pointsx ⁰,x ¹. We find that in odd-dimensional spaces (RL)never holds unless Γ is a sphere or a hyperplane, in which case there is a well known reflection generalizing the celebrated Schwarz reflection principle in two variables. In even-dimensional spaces the situation is different. We find a necessary and sufficient condition (denoted the SSR—strong Study reflection—condition), which we described both analytically and geometrically, for (RL) to hold. This extends and complements previous work by e.g. P.R. Garabedian, H. Lewy, D. Khavinson and H. S. Shapiro.     

The subgroup growth spectrum of virtually free groups

For a finitely generated group Γ denote by μ(Γ) the growth coefficient of Γ, that is, the infimum over all real numbers d such that s _n (Γ) < n! ^d . We show that the growth coefficient of a virtually free group is always rational, and that every rational number occurs as growth coefficient of some virtually free group.     

Group Representations with Empty Residual Spectrum

Let X be a Banach space on which a discrete group Γ acts by isometries. For certain natural choices of X, every element of the group algebra, when regarded as an operator on X, has empty residual spectrum. We show, for instance, that this occurs if X is ℓ ²(Γ) or the group von Neumann algebra VN(Γ). In our approach, we introduce the notion of a surjunctive pair, and develop some of the basic properties of this construction. The cases X = ℓ ^p (Γ) for 1 ≤ p < 2 or 2 < p < ∞ are more difficult. If Γ is amenable we can obtain partial results, using a majorization result of Herz; an example of Willis shows that some condition on Γ is necessary.     

Bipartite Distance-regular Graphs, Part I

Let Γ=(X,E) denote a bipartite distance-regular graph with diameter D≥4, and fix a vertex x of Γ. The Terwilliger algebra T=T(x) is the subalgebra of Mat _X(C) generated by A, E ^* ₀, E ^* ₁,…,E ^* _D, where A denotes the adjacency matrix for Γ and E ^* _i denotes the projection onto the i ^TH subconstituent of Γ with respect to x. An irreducible T-module W is said to be thin whenever dimE ^* _i W≤1 for 0≤i≤Di. The endpoint of W is min{i|E ^* _i W≠0}. We determine the structure of the (unique) irreducible T-module of endpoint 0 in terms of the intersection numbers of Γ. We show that up to isomorphism there is a unique irreducible T-module of endpoint 1 and it is thin. We determine its structure in terms of the intersection numbers of Γ. We determine the structure of each thin irreducible T-module W of endpoint 2 in terms of the intersection numbers of Γ and an additional real parameter ψ=ψ(W), which we refer to as the type of W. We now assume each irreducible T-module of endpoint 2 is thin and obtain the following two-fold result. First, we show that the intersection numbers of Γ are determined by the diameter D of Γ and the set of ordered pairs