Moufang trees and generalized triangles (Q1917006)

Let \(V(\Gamma)\) denote the vertex set of an undirected graph \(\Gamma\) and let \(G\) be a subgroup of \(\text{aut}(\Gamma)\). For \(x\in V(\Gamma)\), let \(\Gamma_x\) denote the set of vertices adjacent to \(x\). The graph \(\Gamma\) is called thick if \(|\Gamma_u|\geq 3\) for all \(u\in V(\Gamma)\). Let \(G^{[i]}_x\) denote the pointwise stabilizer in \(G_x\) of the set of vertices at distance at most \(i\) from \(x\). Set \[ G^{[i]}_{x, y,\dots, z}= G^{[i]}_x\cap G^{[i]}_y\cap\cdots\cap G^{[i]}_z. \] A thick, connected graph \(\Gamma\) is called \((G, n)\)-Moufang if for some \(G\subset \text{aut}(\Gamma)\) and some \(n\geq 3\), \(G^{[1]}_{x_1,\dots, x_{n- 1}}\) acts transitively on \(\Gamma_{x_n}\backslash \{x_{n- 1}\}\) and \(G^{[1]}_{x_0, x_1}\cap G_{x_0,\dots, x_n}= 1\) for every \(n\)-path \((x_0,\dots, x_n)\) in \(\Gamma\). Theorem 1 in the author's paper [The nonexistence of certain Moufang polygons, Inventiones Math. 51, 261-266 (1979; Zbl 0409.05033)] which generalizes a result of \textit{J. Tits} [Moufang octagons and the Ree groups of type \(^2F_4\), Am. J. Math. 105, 539-595 (1983; Zbl 0521.20016)] says that if \(\Gamma\) is \((G, n)\)-Moufang then \(n= 3,4, 6\), or 8. The author treats the case \(n= 3\) in this paper which, together with the three papers [Moufang trees and generalized quadrangles, Forum Math. 8, No. 4, 485-500 (1996; Zbl 0848.20022)], [Moufang trees and generalized hexagons, Duke Math. J. 79, No. 1, 219-233 (1995; Zbl 0833.05041)], and [Moufang trees and generalized octagons, submitted], completes the classification of \((G, n)\)-Moufang trees.