A classification of finite homogeneous semilinear spaces

Abstract. A semilinear space S is homogeneous if, whenever the semilinear structures induced on two finite subsets S1 and S2 of S are isomorphic, there is at least one automorphism of S mapping S1 onto S2. We give a complete classification of all finite homogeneous semilinear spaces. Our theorem extends a result of Ronse on graphs and a result of Devillers and Doyen on linear spaces. Key words. Semilinear space, polar space, copolar space, partial geometry, automorphism group, homogeneity. 2000 Mathematics Subject Classification. 05B25, 51E14, 20B25

Adv. Geom. 3 (2003), 105–121 Advances in Geometry ( de Gruyter 2003 Translation spreads of the split Cayley hexagon

Abstract. Some results concerning translation spreads of the classical generalized hexagon HðqÞ are given, motivated by known analogous results for translation ovoids of the generalized quadrangle Qð4; qÞ. In addition, semi-classical spreads are characterized in terms of their kernels. Finally, a new spread of HðqÞ is described which is also a new 1-system of the nondegenerate parabolic quadric in PGð6; qÞ.

( de Gruyter 2001 Near hexagons with four points on a line

Abstract. We classify, up to four open cases, all near hexagons with four points on a line and with quads through every two points at distance 2. 1 The examples A near polygon is a connected partial linear space satisfying the property that for every point x and every line L, there is a unique point on L nearest to x (distances are measured in the collinearity graph G). If d is the diameter of G, then the near polygon is called a near 2d-gon. A near polygon is said to have order …s; t † if every line is incident with s ‡ 1 points and if every point is incident with t ‡ 1 lines. Near polygons were introduced in [12]. The near quadrangles are just the generalized quadrangles, see [10] and [13] for a detailed survey of these geometries. A generalized quadrangle (GQ for short) is called nondegenerate if every point is incident with at least two lines. A near hexagon is called regular with parameters s; t; t2 if it has order …s; t † and if every two points at distance 2 have exactly t2 ‡ 1 common neighbours. In this paper we classify, up to four open cases, all near hexagons satisfying the following two properties: (i) every line is incident with 4 points …s ˆ 3†; (ii) every two point