Classification of certain types of tilde geometries (Q1326554)

The purpose of this article is to prove the following theorem: Let \(\mathcal G\) be a residually connected geometry belonging to the diagram \[ \underset {0} \circ\cong \underset {1} \circ-\underset {2} \circ\cdots \underset{n-2} \circ-\underset{n-1} \circ \] with \(n\geq 4\). Suppose \(\mathcal G\) possesses a flag-transitive automorphism group \(G\) inducing on the rank 3 residual tilde geometry of \(\mathcal G\) the group \(3^ 7\cdot\text{Sp}_ 6(2)\). Then \(\mathcal G\) is the symplectic type tilde geometry and \(G\cong 3^{{n\choose 2}_ 2}\cdot \text{Sp}_{2n}(2)\), where \({n\choose 2}_ 2= (2^ n- 1)(2^{n-1}- 1)/3\). It is convenient to get acquainted also with the paper by \textit{A. A. Ivanov} and the first author [Geom. Dedicata 45, No. 1, 1-23 (1993; Zbl 0788.51017)].