On graphs and Lie rings

From a finite oriented graph , finite-dimensional graded nilpotent Lie rings () and () are naturally constructed; these rings are related to subtrees and connected subgraphs of , respectively. Diverse versions of these constructions are also suggested. Moreover, an embedding of Lie rings of the form () in the adjoint Lie rings of finite-dimensional associative rings (also determined by the graph ) is indicated.Translated from Matematicheskie Zametki, vol. 77, no. 3, 2005, pp. 449–459.Original Russian Text Copyright © 2005 by Yu. S. Sernenov.This revised version was published online in April 2005 with a corrected issue number.     

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 Skorohod oblique reflection problem in domains with corners and application to stochastic differential equations

Summary The Skorohod oblique reflection problem for (D, , w) (D a general domain in ^d , (x),xD, a convex cone of directions of reflection,w a function inD(⁺, ^d )) is considered. It is first proved, under a condition on (D, ), corresponding to (x) not being simultaneously too large and too much skewed with respect to D, that given a sequence {w ⁿ} of functions converging in the Skorohod topology tow, any sequence {(x ⁿ, ⁿ)} of solutions to the Skorohod problem for (D, , w ⁿ) is relatively compact and any of its limit points is a solution to the Skorohod problem for (D, , w). Next it is shown that if (D, ) satisfies the uniform exterior sphere condition and another requirement, then solutions to the Skorohod problem for (D, , w) exist for everywD(⁺, ^d ) with small enough jump size. The requirement is met in the case when D is piecewiseC _b ¹ , is generated by continuous vector fields on the faces ofD and (x) makes and angle (in a suitable sense) of less than /2 with the cone of inward normals atD, for everyxD. Existence of obliquely reflecting Brownian motion and of weak solutions to stochastic differential equations with oblique reflection boundary conditions is derived.     

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.     

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.     

Connections defined in bundles by the action of infinite Lie groups

Given a principal bundle P=X×V with an arbitrary connection, whose base space X is diffeomorphic to the group manifold of some Lie group, the author constructs an infinite Lie group ^H whose action defines the geometrical structure of P. The structure equation of P is a necessary condition for the existence of a group ^H defining the given connection in P.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 12, pp. 1599–1603, December, 1991.     

An Arithmetic Reduction of Finite Rank 3 Geometries with Linear Spaces as Plane Residues and with Dual Linear Spaces as Point Residues

Let be a rank three incidence geometry of points, lines and planes whose planes are linear spaces and whose point residues are dual linear spaces (notice that we do not require anything on the line residues). We assume that the residual linear spaces of belong to a natural class of finite linear spaces, namely those linear spaces whose full automorphism group acts flag-transitively and whose orders are polynomial functions of some prime number. This class consists of six families of linear spaces. In the amalgamation of two such linear spaces imposes an equality on their orders leading, in particular, to a series of diophantine equations, the solutions of which provide a reduction theorem on the possible amalgams of linear spaces that can occur in .We prove that one of the following holds (up to a permutation of the words point and plane):A) the planes of and the dual of the point residues belong to the same family and have the same orders,B) the diagram of is in one of six families,C) the diagram of belongs to a list of seven sporadic cases.Finally, we consider the particular case where the line residues of are generalized digons.     

The cohomology of invariant variational bicomplexes

Let: E M be a fiber bundle and let be an infinitesimal Lie transformation group acting onE. We announce various new results concerning the cohomology of the invariant variational bicomplex ( ^*,* (J(E)), d_H, d_V) and the associated invariant Euler-Lagrange complex. As one application of our general theory, we completely solve the local invariant inverse problem of the calculus of variations for finite-dimensional infinitesimal Lie transformation groups.     

Optimal boundary regularity for minimal surfaces with a free boundary

Let x(w), w=u+iv B, be a minimal surface in ³ which is bounded by a configuration , S consisting of an arc and of a surface S with boundary. Suppose also that x(w) is area minimizing with respect to , S. Under appropriate regularity assumptions on and S, we can prove that the first derivatives of x(u, v) are Hölder continuous with the exponent =1/2 up to the free part of B which is mapped by x(w) into S. An example shows that this regularity result is optimal.     

On the Self-Decomposability of Euler's Gamma Function

Let be Euler's Gamma function. We prove that, for all 0, > 0, > 0, > 0, the function (( + iz)/() ^{i z)} , z R ¹, is a self-decomposable characteristic function from the Thorin class and derive its explicit canonical form. Similarly to [1], we also describe several classes of Lévy-type stochastic processes related to .     

Completeness of the plans of sequential estimation for Wiener spaces with a drift and some uniqueness theorems

One considers a family of n-dimensional Wiener processess x(t) =(t) +t, depending on a drift parameter , where is the standard Wiener process. Let be a closed subset of the space of trajectories ⁿ×₊ (plan) and assume that the measure , defined by the first occurrence of the trajectory of the process x in the set , is a probability measure. One gives conditions which the plan has to satisfy in order that from the equality for any there should follow that f 0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 69–72, 1983.The authors are grateful to V. P.Khavin for discussions on certain questions of potential theory.     

Free Algebras of Automorphic Forms on the Upper Half-Plane

We find all pairs (,a) consisting of a cocompact Fuchsian group of genus zero and an automorphy factor a of for which the graded algebra of a--automorphic forms is free.     

On the dimensions of automorphism groups of 8-dimensional ternary fields I

Let be an eight-dimensional, connected, locally compact ternary field and let denote a connected closed subgroup of its automorphism group which is taken with the compact-open topology. It is proved that is either isomorphic to the compact exceptional Lie group G₂, or the (covering) dimension of is at most 11. This bound can be decreased to 10, if the ternary fixed fieldF of is connected.     

Orbit counting for some discrete groups acting on simply connected manifolds with negative curvature

Summary In this article we develop a method of deriving asymptotic formulae for the orbital counting function for the action of certain discrete groups of isometries of simply connected negatively curved manifolds. We consider the particular case of normal subgroups ₀ of a co-compact group ₀ for which the quotient ₀/ ^k . Even in the special case of manifolds ofconstant negative curvature, this leads to new results. In particular, we have asymptotic estimates for some groups which arenot geometrically finite.Oblatum III-1993 & 18-VIII-1993The first author was supported by The Royal Society through a University Research Fellowship. The second author was supported by the UK SERC under grant number GR/G51930     

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     

L'injectivité de l'application de Dixmier pour les algèbres de Lie résolubles

Résumé Soient g une algèbre de Lie résoluble sur un corps algébriquement clos de caractéristique 0, g^* l'espace dual de g et de groupe algébrique adjoint. Nous montrons que l'application de Dixmier de g^*/ dans l'espace des idéaux primitifs de l'algèbre enveloppante de g est injective.     

The Displacement and Split Decompositions for a Q-Polynomial Distance-regular Graph

Let denote a Q-polynomial distance-regular graph with diameter at least three and standard module V. We introduce two direct sum decompositions of V. We call these the displacement decomposition for and the split decomposition for . We describe how these decompositions are related.     

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.