A Family of Geometries Related to the Suzuki Tower

ABSTRACT

Michio Suzuki constructed a sequence of five simple groups G _i , with i = 0,…, 4, and five graphs Δ _i , with i = 0,…, 4, such that Δ _i appears as a subgraph of Δ _i+1 for i = 0,…, 3 and G _i is an automorphism group of Δ _i for i = 0,…, 4. The largest group G ₄ was a new sporadic group of order 448 345 497 600. It is now called the Suzuki group Suz. These groups and graphs form what Jacques Tits calls the Suzuki tower. In a previous work, we constructed a rank four geometry Γ(HJ) on which the Hall-Janko sporadic simple group acts flag-transitively and residually weakly primitively. In this article, we show that Γ(HJ) belongs to a family of five geometries in bijection with the Suzuki tower. The largest of them is a geometry of rank six, on which the Suzuki sporadic group acts flag-transitively and residually weakly primitively.