Transitive bislim geometries of gonality 3, part I: The geometrically homogeneous cases

We consider point–line geometries having three points on every line, having three lines through every point (bislim geometries), and containing triangles. We classify such geometries under the hypothesis of the existence of a collineation group acting transitively on the point set. In the first part of this work, we introduce the local structure at a point and consider some cases where this local space already determines the geometry.