A geometric construction of a class of twisted field planes by Ree-Tits unitals

We investigate the intersection pattern of Ree-Tits unitals in the split Cayley HexagonH (q) associated to Dickson's groupG ₂ (q). Using these patterns, we are able to define an incidence geometry which turns out to be a twisted field plane of order 3^2h+1, non-Desarguesian ifh 0. We also show that a general point of the underlying generalized hexagon defines an oval in .Dedicated to H. Reiner Salzmann for his sixtieth birthdayThe second author is Senior Research Associate of the National Fund for Scientific Research (Belgium)     

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).     

Wrapping polygons in polygons

This paper is developed toI ₂(2g).c-geometries, namely, point-line-plane structures where planes are generalized 2g-gons with exactly two lines on every point and any two intersecting lines belong to a unique plane.I ₂(2g).c-geometries appear in several contexts, sometimes in connection with sporadic simple groups. Many of them are homomorphic images of truncations of geometries belonging to Coxeter diagrams. TheI ₂(2g).c-geometries obtained in this way may be regarded as the standard ones. We characterize them in this paper. For everyI ₂(2g).c-geometry , we define a numberw(), which counts the number of times we need to walk around a 2g-gon contained in a plane of , building up a wall of planes around it, before closing the wall. We prove thatw()=1 if and only if is standard and we apply that result to a number of special cases.     

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     

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.     

Lax Projective Embeddings of Polar Spaces

Let be a non-degenerate polar space of rank n 3 where all of its lines have at least three points. We prove that, if admits a lax embedding e : in a projective space defined over a skewfield K, then is a classical and defined over a sub-skewfield K₀ of K. Accordingly, admits a full embedding e₀ in a K₀-projective space ₀. We also prove that, under suitable hypotheses on e and e₀, there exists an embedding such that and preserves dimensions.Received: March, 2004     

Twisted derivations and distinct sets of lines that cover the same pairs of points

A partial projective plane of ordern consists of lines andn ² +n + 1 points such that every line hasn+1 points and distinct lines meet in a unique point. Suppose that two essentially different partial projective planes and of ordern, n a perfect square, that are defined on the same set of points cover the same pairs of points. For sufficiently largen we show that this implies that and have at leastn(n+1) lines. This bound is sharp and there exist essentially two different types of examples meeting the bound.As an application, we can show that derived planes provide an example for a pair of projective planes of square order with as much structure as possible in common, that is, as many lines as possible in common. Furthermore, we present a new method (twisted derivations) to obtain planes from one another by replacing the same number of lines as in a derivation.     

Bounding the diameter of a distance regular graph by a function ofk d

Let be a distance regular graph with diameterd, and _d () the set of vertices at distanced from. is said to be thin if the induced subgraph on _d () is a union of cliques for every vertex. We show that the diameterd is bounded above by a function depending only onk _d, which is the cardinality of _d (), if is not thin. We also investigate thin distance regular graphs witha _d 0.     

Über Tuttes Cages

An undirected graph of valencyd and girth is called a (d, )-cage if its automorphism group acts transitively on the set of alls-paths in ands(+1)/2. We discuss an elementary construction of two known families of cages which allows us to prove easily some facts about their automorphism groups. We give, for example, a new proof of the fact that the automorphism group ofSp ₄(2 ⁿ ) contains elements which are not induced by Sp ₄(2 ⁿ ).     

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.     

K-theory of fine topological algebras, Chern character, and assembly

TheK-theory of the group algebra [] for a countable, discrete group is defined in terms of the simplicial ring of smooth simplices on [], where [] is given the fine topology with respect to its finite-dimensional, linear subspaces. The assembly map for this theory :K _* B K _* [] is studied and shown to be a rational injection. The proof uses the Connes-Karoubi Chern character fromK-theory of Banach algebras to cyclic homology, here generalized to any fine topological algebra, and proved to be multiplicative.     

On the boundedness of Cauchy singular operator from the spaceL p toL q,p>q>-1

It is proved that for a Cauchy type singular operator, given by equality (1), to be bounded from the Lebesgue spaceL _p () tol _q (), as = _n=1 ^Ȟ _n , _n ={z:|z|=r _n }, it is necessary and sufficient that either condition (4) or (5) be fulfilled.     

Almost Covers Of 2-Arc Transitive Graphs

Let be a G-symmetric graph whose vertex set admits a nontrivial G-invariant partition with block size v. Let be the quotient graph of relative to and [B,C] the bipartite subgraph of induced by adjacent blocks B,C of . In this paper we study such graphs for which is connected, (G, 2)-arc transitive and is almost covered by in the sense that [B,C] is a matching of v-1 2 edges. Such graphs arose as a natural extremal case in a previous study by the author with Li and Praeger. The case K _v+1 is covered by results of Gardiner and Praeger. We consider here the general case where K _v+1, and prove that, for some even integer n 4, is a near n-gonal graph with respect to a certain G-orbit on n-cycles of . Moreover, we prove that every (G, 2)-arc transitive near n-gonal graph with respect to a G-orbit on n-cycles arises as a quotient of a graph with these properties. (A near n-gonal graph is a connected graph of girth at least 4 together with a set of n-cycles of such that each 2-arc of is contained in a unique member of .)     

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.     

Circularity of Finite Groups without Fixed Points

Let be a fixed point free group given by the presentation where and are relative prime numbers, t = /s and s = gcd( – 1,), and is the order of modulo . We prove that if (1) = 2, and (2) is embeddable into the multiplicative group of some skew field, then is circular. This means that there is some additive group N on which acts fixed point freely, and |((a)+b)((c)+d)| 2 whenever a,b,c,d N, a0c, are such that (a)+b(c)+d.     

Generalized quadrangles with an ovoid that is translation with respect to opposite flags

The article [6] contains the result that if a finite generalized quadrangle of order s has an ovoid that is translation with respect to two opposite flags, but not with respect to any two non-opposite flags, then is self-polar and is the set of absolute points of a polarity. In particular, if is the classical generalized quadrangle Q(4, q) then is a Suzuki-Tits ovoid. In this article, we remove the need to assume that is Q(4, q) in order to conclude that is a Suzuki-Tits ovoid by showing that the initial assumptions in fact imply that is Q(4, q). At the same time, we also relax the requirement that have order s.Received: 14 May 2004     

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.     

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).     

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 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.