Thin geometries for the Suzuki simple group \(Sz(8)\) (Q1281145)

This article is a part of the project to classify the firm, residually connected geometries on which the Suzuki simple group Sz\((q)\) acts flag-transitively and fulfils some primitivity condition, such as \textbf{Pri, Rpri} or \textbf{Rwpri}. This classification is already finished for the rank 2 geometries in [\textit{D. Leemans}, Beitr. Algebra Geom. 39, No. 1, 97-120 (1998; Zbl 0914.51009)]. In \textit{D. Leemans} [Lond. Math. Soc., Lect. Note. Ser. 261, Vol. 2, 517-526 (1999)], all the firm residually connected geometries, on which Sz(8) acts flag-transitively and residually weakly primitively, are classified. This classification gave 147 thin geometries. The author mentions that when classifying the firm, residually connected geometries on which a group acts flag-transitively and fulfils some primitivity condition, if thin geometries arise, then generally, a lot of these thin geometries arise. This is precisely what might occur when one wants to classify the firm, residually connected geometries of rank bigger than two, on which the Suzuki simple group \(\text{Sz}(q)\) acts flag-transitively and fulfils some primitivity condition. The objective of this article is to classify all the thin geometries on which the group Sz(8) acts flag-transitively; thus obtaining a tool to guess what will happen in general in the classification of the firm, residually connected geometries, on which the Suzuki simple group \(\text{Sz}(q)\) acts flag-transitively and fulfils some primitivity condition. This work showed that there are exactly 183 such thin geometries. The results obtained were first found by looking at the dihedral subgroups of Sz(8) and seeing which of these subgroups could give rise to thin geometries, and secondly, by writing a series of MAGMA programs to classify all thin residually connected geometries on which a group \(G\) acts flag-transitively. Of all the thin geometries, their correlation groups and their rank 2 truncations are given.