Moufang Polygons

In [7], Tits classified spherical buildings of rank at least three. In the addenda of [7], he introduced the Moufang property for buildings and observed that with only the results of Chapter 4 of [7] one could show that an irreducible spherical building of rank at least three and all of its irreducible residues of rank at least two must satisfy the Moufang property. Tits and the author have recently completed the classification of irreducible spherical buildings of rank two having the Moufang property [11]. We give a brief overview of this classification. An irreducible spherical building of rank two is the same thing as a generalized polygon. A generalized polygon is simply a bipartite graph whose diameter is half the length of a shortest circuit. To avoid certain trivialities, we assume as well that every vertex has at least three (but possibly infinitely many) neighbors and that the diameter is at least three. (The diameter is not allowed to be infinite.) A generalized n-gon is a generalized polygon of diameter n. Let Γ be a generalized n-gon for some n ≥ 3 and let G = Aut(G). A circuit of length 2n in Γ is called an apartment. A root of Γ is an undirected path of length n. For each vertex x of Γ, let Γx denote the set of neighbors of x. For each root α = (x0, x1, . . . , xn) of Γ, we denote by Uα the pointwise stabilizer in G of the set Γx1 ∪ · · ·∪Γxn−1 . The group Uα is called the root group associated with α. Definition. A generalized n-gon satisfies the Moufang property if for each root α of Γ, the root group Uα acts transitively on the set of apartments containing α. A Moufang n-gon is a generalized n-gon satisfying the Moufang property. A generalized 3-gon (or triangle) is the same thing as the incidence graph of a projective plane. The notion of a Moufang generalized n-gon generalizes the notion of a Moufang projective plane first introduced in [3]. We now assume that Γ is a Moufang n-gon for some n ≥ 3. We choose an apartment Σ and label the vertices of Σ by the integers modulo 2n so that i is adjacent to i+1 and different from i+2 for all i. Let Ui be the root group corresponding to the root (i, i+1, . . . , i+n) for all i and let U+ denote the subgroup of G generated by the subgroups U1, U2, . . . , Un. The (n + 1)-tuple (U+, U1, U2, . . . , Un) is called the root group sequence associated with