The Spectral Theory of Amenable Actions and Invariants of Discrete Groups

Let G denote a semisimple group, a discrete subgroup, B=G/P the Poisson boundary. Regarding invariants of discrete subgroups we prove, in particular, the following:(1) For any -quasi-invariant measure on B, and any probablity measure on , the norm of the operator () on L ²(B,) is equal to (), where is the unitary representation in L ²(X,), and is the regular representation of .(2) In particular this estimate holds when is Lebesgue measure on B, a Patterson–Sullivan measure, or a -stationary measure, and implies explicit lower bounds for the displacement and Margulis number of (w.r.t. a finite generating set), the dimension of the conformal density, the -entropy of the measure, and Lyapunov exponents of .(3) In particular, when G=PSL₂() and is free, the new lower bound of the displacement is somewhat smaller than the Culler–Shalen bound (which requires an additional assumption) and is greater than the standard ball-packing bound.We also prove that ()=_G() for any amenable action of G and L ¹(G), and conversely, give a spectral criterion for amenability of an action of G under certain natural dynamical conditions. In addition, we establish a uniform lower bound for the -entropy of any measure quasi-invariant under the action of a group with property T, and use this fact to construct an interesting class of actions of such groups, related to 'virtual' maximal parabolic subgroups. Most of the results hold in fact in greater generality, and apply for instance when G is any semi-simple algebraic group, or when is any word-hyperbolic group, acting on their Poisson boundary, for example.     

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 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).     

Invariant subrings of ℂ[X,Y,Z] which are complete intersections

It is known that for every finite subgroup G of SL(2,), the invariant subring [X,Y]^G is a hyper-surface. In this note we treat finite subgroups of SL(3,) and give complete classification of the finite subgroups of SL(3,) whose invariant subrings are complete intersections.     

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.     

Continuous Quotients for Lattice Actions on Compact Spaces

Let < SL _n ( ) be a subgroup of finite index, where n 5. Suppose acts continuously on a manifold M, where ₁(M) = ⁿ , preserving a measure that is positive on open sets. Further assume that the induced action on H ¹(M) is non-trivial. We show there exists a finite index subgroup < and a equivariant continuous map : M ⁿ that induces an isomorphism on fundamental group. We prove more general results providing continuous quotients in cases where ₁(M) surjects onto a finitely generated torsion free nilpotent group. We also give some new examples of manifolds with actions.     

Integral points on strictly convex closed curves

A negative answer is given to Swinnerton-Dyer's question: Is it true that for any > 0 there exists a positive integer n such that for any planar closed strictly convex n-times differentiable curve , when it is blown up a sufficiently large number of times, the number of integral points on the resultant curve will be less than . An example has been constructed when this number for an infinite number is not less than ^1/2, while is infinitely differentiable.Translated from Matematicheskie Zametki, Vol. 21, No. 6, pp. 799–805, June, 1977.The author thanks S. B. Stechkin for attention to the work.     

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]).

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Herrn Rafael Artzy zum 70. Geburtstag     

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.     

Linear Versus Nonlinear Rules for Mixture Normal Priors

The problem under consideration is the -minimax estimation, under L₂loss, of a multivariate normal mean when the covariance matrix is known. The family of priors is induced by mixing zero mean multivariate normals with covariance matrix I by nonnegative random variables , whose distributions belong to a suitable family G. For a fixed family G, the linear (affine) -minimax rule is compared with the usual -minimax rule in terms of corresponding -minimax risks. It is shown that the linear rule is "good", i.e., the ratio of the risks is close to 1, irrespective of the dimension of the model. We also generalize the above model to the case of nonidentity covariance matrices and show that independence of the dimensionality is lost in this case. Several examples illustrate the behavior of the linear -minimax rule.     

Embedding a non-embeddable stable plane

In [4], K. Strambach describes a 2-dimensional stable plane admitting =SL₂ as a group of automorphisms such that there exists no -equivarient embedding into a 2-dimensional projective plane. R. Löwen [3] has given a 4-dimensional analogue , admitting =SL₂. He posed the question whether there are embeddings of Strambach's plane into . We show that such embeddings exist, in fact we determine all -equivariant embeddings of 2-dimensional stable planes admitting as atransitive group of automorphisms.     

A Computer-Assisted Proof of the Uniqueness of the Perkel Graph

The Perkel graph is a distance-regular graph of order 57, degree 6 and diameter 3, with intersection array (6, 5, 2; 1, 1, 3). We describe a computer assisted proof that every graph with this intersection array is isomorphic to the Perkel graph. The computer proof relies heavily on the fact that the minimal idempotents for , and their submatrices, are positive semidefinite.To minimize the risk of computer errors we have used two different methods to establish the same theorem and as an added precaution large parts of the corresponding programs were written by different authors.The first method generates plausible subgraphs induced by all vertices at distance 3 from a fixed vertex of and then tries to extend each of the generated graphs to a full graph with the given intersection array.The second method generates possible neighborhoods for a pentagon in . It turns out that every such pentagon can be extended to a Petersen graph in . We then prove mathematically that there is, up to isomorphism, only a single graph with this property.     

K-theory of reduced C*-algebras and rapidly decreasing functions on groups

We associate to any length function L on a group a space of rapidly decreasing functions on (in the l ² sense), denoted by H _L (). When H _L () is contained in the reduced C*-algebra C _r ^* () of (), then it is a dense *-subalgebra of C _r ^* () and we prove a theorem of A. Connes which asserts that under this hypothesis H _L () has the same K-theory as C _r ^* (). We introduce another space of rapidly decreasing functions on (in the l ¹ sense), denoted by H _L ^1, (), which is always a dense *-subalgebra of the Banach algebra l ¹(), and we show that H _L ^1, () has the same K-theory as l ¹().     

Ergodic Properties of Function Groups

Let PSL(2,) be a torsion free function group. We discuss some aspects of the asymptotic relation between the geometry of the hyperbolic manifold H³/ and the geometry of the boundary of its convex core. This leads to a characterization of Fuchsian function groups by dynamical invariants.     

Bases for Splines on a Subdivided Domain

Let S^r() be the module of all splines of smoothness r on the rectilinear partition which subdivides some domain D. Further, let S^r() be the module of all splines of smoothness r on which also subdivides D, where is a finer subdivision of . We study the relationship between a generating set of S^r() and a generating set for S^r(). This paper gives an algorithm for extending a generating set for S^r() to one for for S^r(). This method is built on algebraic properties of splines and the Gröbner Basis Algorithm.     

On the stochastic independence properties of hard-core distributions

A probability measurep on the set of matchings in a graph (or, more generally 2-bounded hypergraph) ishard-core if for some : [0,), the probabilityp(M) ofM is proportional to . We show that such distributions enjoy substantial approximate stochastic independence properties. This is based on showing that, withM chosen according to the hard-core distributionp, MP () the matching polytope of , and >0, if the vector ofmarginals, (Pr(AM):A an edge of ), is in (1–) MP (), then the weights (A) are bounded by someA(). This eventually implies, for example, that under the same assumption, with fixed, as the distance betweenA, B tends to infinity.Thought to be of independent interest, our results have already been applied in the resolutions of several questions involving asymptotic behaviour of graphs and hypergraphs (see [14, 16], [11]–[13]).Supported in part by NSFThis work forms part of the author's doctoral dissertation [16]; see also [17]. The author gratefully acknowledges NSERC for partial support in the form of a 1967 Science and Engineering Scholarship.     

GAMMA－MINIMAX ESTIMATORS FOR THE MEAN OF A MULTIVARIATE NORMAL DISTRIBUTION WITH PARTIALLY UNKNOWN COVARIANCE MATRIX

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On Embeddings of the Flag Geometries of Projective Planes in Finite Projective Spaces

The flag geometry =( ) of a finite projective plane of order s is the generalized hexagon of order (s, 1) obtained from by putting equal to the set of all flags of , by putting equal to the set of all points and lines of and where I is the natural incidence relation (inverse containment), i.e., is the dual of the double of in the sense of Van Maldeghem Mal:98. Then we say that is fully and weakly embedded in the finite projective space PG(d, q) if is a subgeometry of the natural point-line geometry associated with PG(d, q), if s = q, if the set of points of generates PG(d, q), and if the set of points of not opposite any given point of does not generate PG(d, q). Preparing the classification of all such embeddings, we construct in this paper the classical examples, prove some generalities and show that the dimension d of the projective space belongs to {6,7,8}.     

The Laplacian with Wentzell-Robin Boundary Conditions on Spaces of Continuous Functions

We investigate the Laplacian on a smooth bounded open set Rn with Wentzell-Robin boundary condition $\beta u+\frac{\partial u}{\partial \nu} + \Delta u=0$ on the boundary . Under the assumption $\memb$ C() with $\geq$ 0 , we prove that generates a differentiable positive contraction semigroup on $C(\bar{\Omega})$ and study some monotonicity properties and the asymptotic behaviour.