Triangle geometries
type: Item
instance of: scholarly article
title: Triangle geometries
zbMATH Open document ID: 0568.51002
DOI: 10.1016/0097-3165(84)90051-7
published in: Journal of Combinatorial Theory. Series A
publication date: 1984-01-01T00:00:00Z
review text: The aim of this paper is to study triangle geometries. A triangle geometry is a rank 3 incidence geometry or, using the recent concept of a chamber system, a rank 3 chamber system whose rank 2 residues are projective planes. (By a theorem of Tits, every triangle geometry arises as quotient of a suitable affine building). The author investigates two special classes of triangle geometries: the trivalent ones (whose rank 1 residues have three chambers) and the tight ones (containing exactly three rank 2 residues). For the first class he shows that a regular automorphism group of a trivalent geometry can only be of four types and gives an example for three of these types. For the second class the author gives a method of construction using projective planes; in particular he proves that any finite desarguesian plane gives rise to a tight triangle geometry which admits the full Singer group of the plane, together with its multipliers. Special types of tight triangle geometries are studied, too.
Mathematics Subject Classification ID: 51A05
Mathematics Subject Classification ID: 51D15
zbMATH DE Number: 3907238
zbMATH Keywords: trivalent triangle geometries
zbMATH Keywords: tight triangle geometries
zbMATH Keywords: triangle geometry
zbMATH Keywords: affine building
author: Mark A. Ronan
MaRDI profile type: Publication
full work available at URL: 0097-3165(84)90051-7
OpenAlex ID: W4212783338
cites work: Reductive groups over a local field
cites work: On Certain Groups with Parabolic-Type Subgroups over Z 2
cites work: Cyclic projective planes
cites work: Geometric ABA-groups
cites work: Q5683869
cites work: Q4145882
cites work: BN-pairs and finite groups with parabolic-type subgroups
cites work: Coverings and automorphisms of chamber systems
cites work: Q5637658
cites work: Q5522742
cites work: Tits geometries and parabolic systems in finitely generated groups. I
cites work: Buildings of spherical type and finite BN-pairs
cites work: Q3961400
description: scientific article; zbMATH DE number 3907238