A characterisation of the triality locally projective graph

The paper contributes to the classification of locally projective graphs and their locally projective groups of automorphisms. This project aimed to merge sporadic and classical simple groups in a uniform setting. The list of known examples of locally projective groups of automorphisms includes the classical groups$L_n\left(2\right),S{p}_{2n}\left(2\right),O_{2n}^{+}\left(2\right),{\Omega }_8^{+}\left(2\right):S_3,G_2\left(3\right)$ as well as the sporadic simple groups$M_{22},M_{23},M_{24},He,C{o}_2,C{o}_1,J_4,BM,M,$ where M is the Monster sporadic simple group, the largest and most famous sporadic simple group. The locally projective graph for the Monster gives an important insight in the structure of 2-local subgroups in the Monster. The list also includes some remarkable non-split extensions which probably would not be discovered otherwise:$3^7\cdot S{p}_6\left(2\right),3^{23}\cdot C{o}_2,3^{4371}\cdot BM.$ This article focuses on the locally projective graph constructed by Giudici, Li and Praeger from the triality of the $D_4$-geometry over $GF\left(2\right)$. We call it the triality graph and prove that (up to coverings and quotients) it is the unique thick locally projective graph in dimension 3 where (a) the stabiliser of a plane realises a completion of the Goldschmidt amalgam$G_2^4=\{D_8\times S_3,S_4\times 2\},$ and (b) the vertex-wise stabiliser of the ball of radius 2 centred at a vertex in the collinearity graph has order 8. In the triality graph itself the completion of $G_2^4$ is the wreath product $S_3\wr S_3$.