Two points in an incidence structure are called adjacent if there exists a line passing through them. The authors study maps between locally projective near $$2n$$-gons preserving adjacency. A near $$2n$$-gon $$G$$ is called locally projective if the line pencil through each point has the structure of a projective space in a natural way, all of the same dimension $$m-1\geq 1$$. In this case $$m$$ is called the rank of $$G$$.
The following theorem is proved: Let $$G$$ and $$G'$$ be locally projective near $$2n$$-gons ($$n\geq 1$$) of the same finite rank $$m\geq 2$$. Then every surjective, adjacency preserving map on the point sets is an isomorphism of the geometries. This applies in particular to generalized $$2n$$-gons ($$n>1$$) and dual polar spaces.
The theorem generalizes earlier work of W. Huang [Proc. Am. Math. Soc. 128, 2451-2455 (2000; Zbl 0955.51004)]. The authors make a case that there is not much room to further generalize the result.

### MSC:

 51A50 Polar geometry, symplectic spaces, orthogonal spaces 51E12 Generalized quadrangles and generalized polygons in finite geometry

Zbl 0955.51004
Full Text:

### References:

 [1] F. Buekenhout, P. Cameron, Projective and a ne geometry over division rings. In: Handbook of incidence geometry, 27-62, North-Holland 1995. · Zbl 0822.51001 [2] Cameron P. J., Geom. Dedicata 12 pp 75– (1982) [3] W.l. Chow, On the geometry of algebraic homogeneous spaces. Ann. of Math. (2) 50 (1949), 32-67. · Zbl 0040.22901 [4] A. M. Cohen, Point-line spaces related to buildings. In: Handbook of incidence geometry, 647-737, North-Holland 1995. · Zbl 0829.51004 [5] J. Combin. Theory Ser. 42 pp 111– (1986) [6] Proc. Amer. Math. Soc. 128 pp 2451– (2000) [7] D., Canad. J. Math. 37 pp 296– (1985) [8] J. A. Thas, Generalized polygons.In: Handbook of incidence geometry, 383-431, North-Holland 1995. [9] H. Van Maldeghem, Generalized polygons.Birkh user 1998. · Zbl 0914.51005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.