Bi-slim Flag-Transitive Geometries of Gonality \(3\): Construction and Classification

Abstract

We consider point-line geometries having three points on every line, having three lines through every point (\(bi\)-\(slim\; geometries\)), and containing triangles. We give some (new) constructions and we prove that every flag-transitive such geometry either belongs to a certain infinite class described by Coxeter a long time ago, or is one of three well-defined sporadic ones, namely, The Möbius-Kantor geometry on \(8\) points, The Desargues geometry on \(10\) points,A unique infinite example related to the tiling of the real Euclidean plane in regular hexagons.We also classify the possible groups.