Compact Three-Dimensional Elation Quadrangles

We show that each compact three-dimensional EGQ is isomorphic either to the real symplectic quadrangle, or to a translation quadrangle of Tits type. In particular, the elation group is one of the two classical elation groups, and thus a simply connected nilpotent Lie group.     

Point regular groups of automorphisms of generalised quadrangles

We study the point regular groups of automorphisms of some of the known generalised quadrangles. In particular we determine all point regular groups of automorphisms of the thick classical generalised quadrangles. We also construct point regular groups of automorphisms of the generalised quadrangle of order (q−1,q+1) obtained by Payne derivation from the classical symplectic quadrangle W(3,q). For q=pf with f?2 we obtain at least two nonisomorphic groups when p?5 and at least three nonisomorphic groups when p=2 or 3. Our groups include nonabelian 2-groups, groups of exponent 9 and nonspecial p-groups. We also enumerate all point regular groups of automorphisms of some small generalised quadrangles.     

Automorphism groups of compact projective planes

Compact connected projective planes have been investigated extensively in the last 30 years, mostly by studying their automorphism groups. It is our aim here to remove the connectedness assumption in some general results of Salzmann [31] and Hähl [14] on automorphism groups of compact projective planes. We show that the continuous collineations of every compact projective plane form a locally compact transformation group (Theorem 1), and that the continuous collineations fixing a quadrangle in a compact translation plane form a compact group (Corollary to Theorem 3). Furthermore we construct a metric for the topology of a quasifield belonging to a compact projective translation plane, using the modular function of its additive group (Theorem 2).     

Tight sets and <Emphasis Type="Italic">m</Emphasis>-ovoids of generalised quadrangles

The concept of a tight set of points of a generalised quadrangle was introduced by S. E. Payne in 1987, and that of an m-ovoid of a generalised quadrangle was introduced by J. A. Thas in 1989, and we unify these two concepts by defining intriguing sets of points. We prove that every intriguing set of points in a generalised quadrangle is an m-ovoid or a tight set, and we state an intersection result concerning these objects. In the classical generalised quadrangles, we construct new m-ovoids and tight sets. In particular, we construct m-ovoids of W(3,q), q odd, for all even m; we construct (q+1)/2-ovoids of W(3,q) for q odd; and we give a lower bound on m for m-ovoids of H(4,q ²).     

Weighted intriguing sets of finite generalised quadrangles

We construct and analyse interesting integer valued functions on the points of a generalised quadrangle which lie in the orthogonal complement of a principal eigenspace of the collinearity relation. These functions generalise the intriguing sets introduced by Bamberg et al. (Combinatorica 29(1):1?C17, 2009), and they provide the extra machinery to give new proofs of old results and to establish new insight into the existence of certain configurations of generalised quadrangles. In particular, we give a geometric characterisation of Payne??s tight sets, we give a new proof of Thas?? result that an m-ovoid of a generalised quadrangle of order (s,s ²) is a hemisystem, and we give a bound on the values of m for which it is possible for an m-ovoid of the four dimensional Hermitian variety to exist.     

A hemisystem of a nonclassical generalised quadrangle

The concept of a hemisystem of a generalised quadrangle has its roots in the work of B. Segre, and this term is used here to denote a set of points such that every line ℓ meets in half of the points of ℓ. If one takes the point-line geometry on the points of the hemisystem, then one obtains a partial quadrangle and hence a strongly regular point graph. The only previously known hemisystems of generalised quadrangles of order (q, q ²) were those of the elliptic quadric , q odd. We show in this paper that there exists a hemisystem of the Fisher–Thas–Walker–Kantor generalised quadrangle of order (5, 5²), which leads to a new partial quadrangle. Moreover, we can construct from our hemisystem the 3· A ₇-hemisystem of , first constructed by Cossidente and Penttila.      

Neighbour-transitive codes and partial spreads in generalised quadrangles

Designs, Codes and Cryptography - A code C in a generalised quadrangle $${\mathcal {Q}}$$ is defined to be a subset of the vertex set of the point-line incidence graph $${\Gamma }$$ of $${\mathcal...     

Linearly small elation quadrangles

We show that each elation generalized quadrangle with parameters (p, p), where p is a prime, is isomorphic to the symplectic quadrangle W(p) or its dual Q(4, p). Our results cover the more general case of linearly small elation generalized quadrangles. In particular, we obtain a characterization of the symplectic quadrangle over the field of complex numbers among compact connected quadrangles. We prove that every root elation quadrangle (Q, c, H _F ) is a skew translation quadrangle.     

Correction: Generalized Polygons with Highly Transitive Collineation Groups

In Geom Dedicata 58 (1995), 91–100, the author tried to classify generalized quadrangles with a collineation group acting transitively on ordered pentagons. Unfortunately, this paper contains several mistakes. The main result is affected such that there now is an additional compact connected quadrangle (and its dual) with a 5-gon transitive group. I am indebted to Martin Wolfrom who pointed out the error and offered a correction.     

Generalized quadrangles constructed from topological Laguerre planes

The Lie geometry of a finite-dimensional locally compact connected Laguerre plane is a topological generalized quadrangle.     

Partial circle geometries and antiregular quadrangles

A new and rather general definition of circle geometries is given. This definition is such that circle planes and chain spaces are circle geometries. Also the geometry of points and traces of an antiregular quadrangle is a partial circle geometry. Orthogonal quadrangles can then be characterised as those antiregular generalised quadrangles where in the associated partial circle geometry the Miquel condition is satisfied.     

Multiplicative Loops of 2-Dimensional Topological Quasifields

We determine the algebraic structure of the multiplicative loops for locally compact 2-dimensional topological connected quasifields. In particular, our attention turns to multiplicative loops which have either a normal subloop of positive dimension or which contain a 1-dimensional compact subgroup. In the last section, we determine explicitly the quasifields which coordinatize locally compact translation planes of dimension 4 admitting an at least 7-dimensional Lie group as collineation group.     

Coverings and dimensions in infinite profinite groups

Answering a question of Miklós Abért, we prove that an infinite profinite group cannot be the union of less than continuum many translates of a compact subset of box dimension less than 1. Furthermore, we show that it is consistent with the axioms of set theory that in any infinite profinite group there exists a compact subset of Hausdorff dimension 0 such that one can cover the group by less than continuum many translates of it.     

Compact abelian group extensions of discrete dynamical systems

Summary This paper introduces the notion of a free G extension of a dynamical system where G is a compact abelian group. The concept is closely allied to that of generalised discrete spectrum (which includes Abramov's quasi-discrete spectrum as a special case). We give necessary and sufficient conditions for a G extension of a minimal (uniquely ergodic) dynamical system to be minimal (uniquely ergodic) and show that in a certain sense a general G extension lifts these properties. Stable G-extensions always lift these properties if the underlying space is connected. This fact is then used to characterise all uniquely ergodic and minimal affine transformations of a certain three dimensional nilmanifold. The rest of the paper is devoted to the exhibition of group invariants for systems with generalised discrete spectrum. In particular it is shown that such systems always have a compact abelian group as underlying space. A lemma which facilitates this result gives necessary and sufficient conditions for a connected G-extension of a compact abelian group to be a compact abelian group.     

Compact Ovoids in Quadrangles I: Geometric Constructions

This paper is about ovoids in infinite generalized quadrangles. Using the axiom of choice, Cameron showed that infinite quadrangles contain many ovoids. Therefore, we consider mainly closed ovoids in compact quadrangles. After deriving some basic properties of compact ovoids, we consider ovoids which arise from full imbeddings. This leads to restrictions for the topological parameters (m,m). For example, if there is a regular pair of lines or a full closed subquadrangle, then mm. The existence of full subquadrangles implies the nonexistence of ideal subquadrangles, so finite-dimensional quadrangles are either point-minimal or line-minimal. Another result is that (up to duality) such a quadrangle is spanned by the set of points on an ordinary quadrangle. This is useful for studying orbits of automorphism groups. Finally we prove general nonexistence results for ovoids in quadrangles with low-dimensional line pencils. As one consequence we show that the symplectic quadrangle over an algebraically closed field of characteristic 0 has no Zariski-closed ovoids or spreads.     

Generalised sk-Spline Interpolation on Compact Abelian Groups

The notion of sk-spline is generalised to arbitrary compact Abelian groups. A class of conditionally positive definite kernels on the group is identified, and a subclass corresponding to the generalised sk-spline is used for constructing interpolants, on scattered data, to continuous functions on the group. The special case ofd-dimensional torus is considered and convergence rates are proved when the kernel is a product of one-dimensional kernels, and the data are gridded.     

The automorphism group of Payne derived generalized quadrangles

We solve a long-standing open problem by proving that the automorphism group of any thick Payne derived generalized quadrangle with ambient quadrangle S a thick generalized quadrangle of order s, s?5 and odd, with a center of symmetry, is induced by the automorphism group of S.     

Tight sets and m-ovoids of finite polar spaces

An intriguing set of points of a generalised quadrangle was introduced in [J. Bamberg, M. Law, T. Penttila, Tight sets and m-ovoids of generalised quadrangles, Combinatorica, in press] as a unification of the pre-existing notions of tight set and m-ovoid. It was shown in [J. Bamberg, M. Law, T. Penttila, Tight sets and m-ovoids of generalised quadrangles, Combinatorica, in press] that every intriguing set of points in a finite generalised quadrangle is a tight set or an m-ovoid (for some m). Moreover, it was shown that an m-ovoid and an i-tight set of a common generalised quadrangle intersect in mi points. These results yielded new proofs of old results, and in this paper, we study the natural analogue of intriguing sets in finite polar spaces of higher rank. In particular, we use the techniques developed in this paper to give an alternative proof of a result of Thas [J.A. Thas, Ovoids and spreads of finite classical polar spaces, Geom. Dedicata 10 (1-4) (1981) 135-143] that there are no ovoids of H(2r,q²), Q⁻(2r+1,q), and W(2r−1,q) for r>2. We also strengthen a result of Drudge on the non-existence of tight sets in W(2r−1,q), H(2r+1,q²), and Q⁺(2r+1,q), and we give a new proof of a result of De Winter, Luyckx, and Thas [S. De Winter, J.A. Thas, SPG-reguli satisfying the polar property and a new semipartial geometry, Des. Codes Cryptogr. 32 (1-3) (2004) 153-166; D. Luyckx, m-Systems of finite classical polar spaces, PhD thesis, The University of Ghent, 2002] that an m-system of W(4m+3,q) or Q⁻(4m+3,q) is a pseudo-ovoid of the ambient projective space.     

The Automorphism Group of a Compact Generalized Polygon has Finite Dimension

 For every compact generalized polygon whose point space has finite dimension, we show that the automorphism group has finite dimension as well. This opens the field for the application of Lie theoretic methods. Received 12 November 1997; in revised form 10 March 1998     

Constructions of intriguing sets of polar spaces from field reduction and derivation

The concepts of a tight set of points and an m-ovoid of a generalised quadrangle were unified recently by Bamberg, Law and Penttila under the title of intriguing sets. This unification was subsequently extended to polar spaces of arbitrary rank. The first part of this paper deals with a method of constructing intriguing sets of one polar space from those of another via field reduction. In the second part of this paper, we generalise an ovoid derivation of Payne and Thas to a derivation of intriguing sets.