An infinite family of simply connected flag-transitive tilde geometries

We consider tilde-geometries (or T-geometries), which are geometries belonging to diagrams of the following shape:[Figure not available: see fulltext.] Here the rightmost edge stands for the famous triple cover of the classical generalized quadrangle related to the group Sp₄(2). The automorphism group of the cover is the nonsplit extension 3·Sp₄(2) - 3 ·S₆. Five examples of flag-transitive T-geometries were known. These are rank 3 geometries related to the groups M₂₄ (the Mathieu group), He (the Held group) and and 3⁷·Sp₆(2) (a nonsplit extension); a rank 4 geometry related to the Conway group Co₁ and a rank 5 geometry related to the Fischer-Griess Monster group F₁. In the present paper we construct an infinite family of flag-transitive T-geometries and prove that all the new geometries are simply connected. The automorphism group of the rank n geometry in the family is a nonsplit extension of a 3-group by the symplectic group Sp_{2 n}(2). The rank of the 3-group is equal to the number of 2-dimensional subspaces in an n-dimensional vector space over GF(2).