Canonical contact forms on spherical CR manifolds

We construct the CR invariant canonical contact form can(J) on scalar positive spherical CR manifold (M,J), which is the CR analogue of canonical metric on locally conformally flat manifold constructed by Habermann and Jost. We also construct another canonical contact form on the Kleinian manifold ()/, where is a convex cocompact subgroup of Aut_CRS²ⁿ⁺¹=PU(n+1,1) and () is the discontinuity domain of . This contact form can be used to prove that ()/ is scalar positive (respectively, scalar negative, or scalar vanishing) if and only if the critical exponent ()<n (respectively, ()>n, or ()=n). This generalizes Nayatanis result for convex cocompact subgroups of SO(n+1,1). We also discuss the connected sum of spherical CR manifolds.     

On Locally Projective Graphs of Girth 5

Let be a graph and G be a 2-arc transitive automorphism group of . For a vertex x let G(x)^(x) denote the permutation group induced by the stabilizer G(x) of x in G on the set (x) of vertices adjacent to x in . Then is said to be a locally projective graph of type (n,q) if G(x)^(x) contains PSL_n(q) as a normal subgroup in its natural doubly transitive action. Suppose that is a locally projective graph of type (n,q), for some n 3, whose girth (that is, the length of a shortest cycle) is 5 and suppose that G(x) acts faithfully on (x). (The case of unfaithful action was completely settled earlier.) We show that under these conditions either n=4, q=2, has 506 vertices and , and contains the Wells graph on 32 vertices as a subgraph. In the latter case if, for a given n, at least one graph satisfying the conditions exists then there is a universal graph W(n) of which all other graphs for this n are quotients. The graph W(3) satisfies the conditions and has 2²⁰ vertices.     

Superrigidity of lattices in solvable Lie groups

Summary Let be a closed, cocompact subgroup of a simply connected, solvable Lie groupG, such that Ad _G has the same Zariski closure as AdG. If : GL _n () is any finite-dimensional representation of , we show that virtually extends to a representation ofG. (By combining this with work of Margulis on lattices in semisimple groups, we obtain a similar result for lattices in many groups that are neither solvable nor semisimple.) Furthermore, we show that if is isomorphic to a closed, cocompact subgroup of another simply connected, solvable Lie groupG, then any isomorphism from to extends to a crossed isomorphism fromG toG. In the same vein, we prove a more concrete form of Mostow's theorem that compact solvmanifolds with isomorphic fundamental groups are diffeomorphic.Oblatum 5-VII-1994 & 15-IV-1995     

The martin boundary for harmonic functions on groups of automorphisms of a homogeneous tree

Consider a closed subgroup of the automorphism group of a homogeneous treeT, and assume that acts transitively on the vertex set. Suppose that is a probability measure on which has continuous density with respect to Haar measure and whose support is compact open and generates as a closed semigroup. It is shown that the Martin boundary of with respect to the random walk with law coincides with the space of ends ofT. This extends known results for free groups and applies, for example, to the affine group over a non archimedean local field.     

Extremal quasiconformal mappings compatible with Fuchsian groups

For every torsion free Fuchsian group with Poincaré's -operator norm é=1, it is proved that there exists an extremal Beltrami differential of which is also extremal under its own boundary correspondence. It is also proved that the imbedding of the Teichmüller spaceT() into the universal Teichmüller spaceT is not a global isometry unless is an elementary group.     

Configuration of subgroups in the special linear group over a skew field with infinite center

Let T be a skew field with infinite center, let be the special linear group over T of degree 3, and let be the subgroup of diagonal matrices with unit Dieudonee determinant. It is proved that for each intermediate subgroup H, H , there exists a net of order n such that ( H N().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 175, pp. 5–12, 1989.In conclusion, the author would like to thank his instructor Z. I. Borevich, as well as N. A. Vavilov, for their assistance.     

On the removable singularities of CR functions given on a generic manifold

Summary In this paper we consider the problem of holomorphic continuation and removal of singularities of the CR functions given on K, where is a generic manifold with nondegenerate Levi form and K is a meromorphically p-convex compactum. We get some conditions on , relative to p-convexity and q-concavity, under which every integrable CR function given on K extends holomorphically in some domain \K, where is a wedge domain with edge . Our Results are local.Authors had a support of Russian Fund of Fundamental Investigations (grant 93-011-258).     

Algebraisierung Scharf 2-Fach Transitiver Permutationsmengen durch Quasibereiche

Sharply 2-transitive permutation sets (M,) with the property: every which interchanges two distinct elements of M is in and V (where := { ; ²=id }) can be characterized algebraically by quasi-domains as defined by G. Kist.Among other examples there are some which show that a sharply 2-transitive permutation set (M,) is not necessarily a group a neither if (M,) satisfies (R^*): id and (, ^–1 ) which answers a question of H. Karzel [7], nor if (M,) is symmetric (contrary to the case of a symmetric sharply 3-transitive permutation set (M,) where is always a group [8]).

---

---

Herrn Rafael Artzy zum 70. Geburtstag     

A family of multiply extended grids

We construct an infinite family{ _n}_n=5 of finite connected graphs _n that are multiple extensions of the well-known extended grid discovered in [1] (which is isomorphic to ₅). The graphs _n are locally _n–1 forn > 5, and have the following property: the automorphism groupG(n) of _n permutes transitively the maximal cliques of _n (which aren-cliques) and the stabilizer of somen-clique of _n inG(n) induces _n on the vertices of. Furthermore we show that the clique complexes of the graphs _n are simply connected.     

The conditions of selfadjointness of the operator of singular integration

It is proved that the operator of singular integration along a bounded rectifiable curve is self-adjoint in the weighted spaceL ₂(, ) if and only if is a circle and (t)=const.     

Subgroups of the general symplectic group containing the group of diagonal matrices

There are described the subgroups of the general symplectic group =GSp(2n, R) over a commutative semilocal ring R, containing the group of symplectic diagonal matrices. For each such subgroup P there is uniquely defined a symplectic D-net a such that ()pN(), where () is the net subgroup in corresponding to (cf. RZhMat, 1977, 5A288), and N() is its normalizer. The quotient group N × ()/() is calculated. There are also considered subgroups in Sp(2n, R). Analogous results for subgroups of the general linear group were obtained earlier in RZhMat, 1978, 9A237.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 103, pp. 31–47, 1980.     

On the approximation of singular integral equations by equations with smooth kernels

Singular integral equations with Cauchy kernel and piecewise-continuous matrix coefficients on open and closed smooth curves are replaced by integral equations with smooth kernels of the form(t–)[(t–) ²–n ² (t) ²]^–1,0, wheren(t), t , is a continuous field of unit vectors non-tangential to . we give necessary and sufficient conditions under which the approximating equations have unique solutions and these solutions converge to the solution of the original equation. For the scalar case and the spaceL ₂() these conditions coincide with the strong ellipticity of the given equation.This work was fulfilled during the first author's visit to the Weierstrass Institute for Applied Analysis and Stochastics, Berlin in October 1993.     

Covering the Hecke Triangle Surfaces

In the mid-1980s an equivalence was established between the simple closed geodesics on the Riemann surfaces obtained as quotients of the upper half plane H by any of the following subgroups of the modular group (1) : , (3), and ³. An axis of a hyperbolic element of (1) projects to a simple closed geodesic on one of these surfaces if and only if it does so on the other two.This equivalence was used to obtain a variety of Diophantine and geometric results. In subsequent related investigations, the role of (1) was assumed by the Hecke triangle group G_q for q 3. (For q = 3, we have (1) = G₃.) These works employed the analog of ³, denoted _q.In the context of the G_q, the present paper gives the analog of , which we denote _q. As in the case q = 3, we have [_q:_q] = 2. A rather full discussion of geometry of _q\ H is given. In particular, we demonstrate that the equivalence of simple closed geodesics on _q\ H and _q\ H does not hold for q 7.As of this writing, we have not been able to obtain an appropriate analog of (3).     

Stability results for an inverse problem in potential theory

Sunto Nel presente lavoro consideriamo un problema inverso, connesso a questioni di elettrocardiologia, relativo a un potenziale di doppio strato obliquo logaritmico, u, portato da una curva piana . Il problema consiste nel determinare conoscendo u sulla frontiera di un dominio T contenente . Il problema non è ben posto secondo Hadamard. Introduciamo delle limitazioni a priori su e dimostriamo alcuni risultati di stabilità.     

The lower algebraicK-theory of virtually infinite cyclic groups

This paper gives a proof of a conjecture of W.-C. Hsiang for the negativeK-theory of integral grouprings , when the group is a subgroup of a uniform lattice in a Lie group. The authors' earlier paper reduced this result to the very special cases where either is finite or is virtually infinite cyclic. The finite case was done much earlier by Carter extending results of Bass and Murthy. The major work of the present paper consists of proving the conjecture when is virtually infinite cyclic.Both authors were supported in part by the National Science Foundation.     

Die Automorphismengruppen hyperelliptischer Kurven

Let be a hyperelliptic curve over an algebraically closed field of characteristic O and G a group of automorphisms of that contains the canonical involution of . We describe the group-theoretical structure of G by means of the arithmetical structure of the morphism ^G.     

On groups of polynomial subgroup growth

Summary Let be a finitely generated group anda _n ()=the number of its subgroups of indexn. We prove that, assuming is residually nilpotent (e.g., linear), thena _n () grows polynomially if and only if is solvable of finite rank. This answers a question of Segal. The proof uses a new characterization ofp-adic analytic groups, the theory of algebraic groups and the Prime Number Theorem. The method can be applied also to groups of polynomial word growth.Oblatum 1-VII-1989 & 7-VI-1990     

Critical points of essential norms of singular integral operators in weighted spaces

We show that for any simple piecewise Ljapunov contour there exists a power weight such that the essential norm |S | in the spaceL ₂(, ) does not depend on the angles of the contour and it is given by formula (2). All such weights are described. For the union =₁₂ of two simple piecewise Lyapunov curves we prove that the essential norm |S | inL ₂() is minimal if both ₁ and ₂ are smooth in some neighborhoods of the common points. It is the case when the norm |S | in the spaceL ₂() as well as inL ₂(, ) does not depend on the values of the angles and it can be calculated by formula (5).     

The Trace Conjecture – A Counterexample

In the paper Geometric K-Theory for Lie Groups and Foliations, Baum and Connes conjecture in a remark following Corollary 2 of their famous Isomorphism conjecture that for a finitely generated group with torsion, the trace map tr: K₀(C^*)R maps K₀(C^*) onto the additive subgroup of Q generated by all rational numbers of the form 1/n where n is the order of a finite subgroup of . We construct a counterexample to this conjecture.