Moufang trees and generalized triangles
Richard M. Weiss · Osaka City University (Osaka City University) · 1995
for any subset {;c,jv ,z} of F(Γ). The graph Γ will be called thick if |ΓJ>3 for all we F(Γ). An apartment of Γ is a connected subgraph Δ such that |ΔJ = 2 for every u e K(Δ). When there is no danger of confusion, we will often use integers to denote vertices of Γ. A generalized w-gon (for n > 2) is a bipartite graph of diameter n and girth 2n. A generalized «-gon Γ for n>3 is called Moufang if Gι*3..,w-ι acts transitively on ΓM\{>z-l} for every (w-l)-path (1, •••,«) of Γ for some G<aut(Γ). In [6], Tits showed that thick Moufang w-gons exist only for n — 3,4,6 and 8. If Γ is a thick generalized «-gon and G<aut(Γ), then G^}n(/0 ...„= 1 for every w-path (0, •••,«) of Γ. (This is a special case of [5,(4.1.1)]; see Theorem 2 of [8].) Thus, the following (Theorem 1 of [8]) is a generalization of Tits' result: