Singer quadrangles

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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 $\mathsf{W}(3,q)$. For $q=p^f$ with $f\geq 2$ we obtain at least two nonisomorphic groups when $p\geq 5$ 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.Comment: some minor changes (including to title) after referee's comment

A question of Frohardt on 22-groups, and skew translation quadrangles of even order

We solve a fundamental question posed in Frohardt's 1988 paper [Fro] on finite $2$-groups with Kantor familes, by showing that finite groups with a Kantor family $(\mathcal{F},\mathcal{F}^*)$ having distinct members $A, B \in \mathcal{F}$ such that $A^* \cap B^*$ is a central subgroup of $H$ and the quotient $H/(A^* \cap B^*)$ is abelian cannot exist if the center of $H$ has exponent $4$ and the members of $\mathcal{F}$ are elementary abelian. In a similar way, we solve another old problem dating back to the 1970s by showing that finite skew translation quadrangles of even order $(t,t)$ are always translation generalized quadrangles.Comment: 10 pages; submitted (February 2018

Central aspects of skew translation quadrangles, I

Except for the Hermitian buildings $\mathcal{H}(4,q^2)$, up to a combination of duality, translation duality or Payne integration, every known finite building of type $\mathbb{B}_2$ satisfies a set of general synthetic properties, usually put together in the term "skew translation generalized quadrangle" (STGQ). In this series of papers, we classify finite skew translation generalized quadrangles. In the first installment of the series, as corollaries of the machinery we develop in the present paper, (a) we obtain the surprising result that any skew translation quadrangle of odd order $(s,s)$ is a symplectic quadrangle; (b) we determine all skew translation quadrangles with distinct elation groups (a problem posed by Payne in a less general setting); (c) we develop a structure theory for root-elations of skew translation quadrangles which will also be used in further parts, and which essentially tells us that a very general class of skew translation quadrangles admits the theoretical maximal number of root-elations for each member, and hence all members are "central" (the main property needed to control STGQs, as which will be shown throughout); (d) we solve the Main Parameter Conjecture for a class of STGQs containing the class of the previous item, and which conjecturally coincides with the class of all STGQs.Comment: 66 pages; submitted (December 2013

Classification of skew translation generalized quadrangles, I

We describe new classification results in the theory of generalized quadrangles (= Tits-buildings of rank 2 and type B-2), more precisely in the (large) subtheory of skew translation generalized quadrangles ("STGQs"). Some of these involve, and solve, long-standing open problems

A sixteen-relator presentation of an infinite hyperbolic Kazhdan group

We provide an explicit presentation of an infinite hyperbolic Kazhdan group with $4$ generators and $16$ relators of length at most $73$. That group acts properly and cocompactly on a hyperbolic triangle building of type $(3,4,4)$. We also point out a variation of the construction that yields examples of lattices in $\tilde A_2$-buildings admitting non-Desarguesian residues of arbitrary prime power order.Comment: 9 pages, 1 figur

Central aspects of skew translation quadrangles, 1

Modulo a combination of duality, translation duality or Payne integration, every known finite generalized quadrangle except for the Hermitian quadrangles H(4, q2), is an elation generalized quadrangle for which the elation point is a center of symmetry-that is, is a "skew translation generalized quadrangle" (STGQ). In this series of papers, we classify and characterize STGQs. In the first installment of the series, (1) we obtain the rather surprising result that any skew translation quadrangle of finite odd order (s, s) is a symplectic quadrangle; (2) we determine all finite skew translation quadrangles with distinct elation groups (a problem posed by Payne in a less general setting); (3) we develop a structure theory for root elations of skew translation quadrangles which will also be used in further parts, and which essentially tells us that a very general class of skew translation quadrangles admits the theoretical maximal number of root elations for each member, and hence, all members are "central" (the main property needed to control STGQs, as which will be shown throughout); and (4) we show that finite "generic STGQs," a class of STGQs which generalizes the class of the previous item (but does not contain it by definition), have the expected parameters. We conjecture that the classes of (3) and (4) contain all STGQs

On finite generalized quadrangles of even order

In this paper, we establish the following two results: (1) a skew translation generalized quadrangle of even order is a translation generalized quadrangle, (2) a generalized quadrangle of even order does not admit a point regular automorphism group. The first result confirms a conjecture of Payne (1975) based on earlier work of Ott (2021), and the second result confirms a conjecture of Ghinelli (1992).Comment: revised version, incorporated referees' comments, 20 page