Flat Lax and Weak Lax Embeddings of Finite Generalized Hexagons

In this paper we study laxly embedded generalized hexagons in finite projective spaces (a generalized hexagon is laxly embedded inPG(d,q) if it is a spanning subgeometry of the natural point-line geometry associated toPG(d,q)), satisfying the following additional assumption: for any pointxof the hexagon, the set of points collinear in the hexagon withxis contained in some plane ofPG(d,q). In particular, we show thatd ≤ 7, and ifd = 7, we completely classify all such embeddings. A classification is also carried out ford = 5, 6 under some additional hypotheses. Finally, laxly embedded generalized hexagons satisfying other additional assumptions are considered, and classifications are also obtained.