Characterizations of the Suzuki tower near polygons

TL;DR

This paper characterizes certain near polygons in the Suzuki tower as unique structures containing embedded smaller polygons, extending previous characterizations and classifying near hexagons of order (2, 2).

Contribution

It provides new characterizations of near polygons in the Suzuki tower based on embedded substructures, without relying on regularity assumptions.

Findings

Characterization of near polygons as unique containing embedded smaller polygons.

Complete classification of near hexagons of order (2, 2).

Extension of the Hall-Janko near octagon characterization.

Abstract

In recent work, we constructed a new near octagon $\mathcal{G}$ from certain involutions of the finite simple group $G_2(4)$ and showed a correspondence between the Suzuki tower of finite simple groups, $L_3(2) < U_3(3) < J_2 < G_2(4) < Suz$, and the tower of near polygons, $\mathrm{H}(2,1) \subset \mathrm{H}(2)^D \subset \mathsf{HJ} \subset \mathcal{G}$. Here we characterize each of these near polygons (except for the first one) as the unique near polygon of the given order and diameter containing an isometrically embedded copy of the previous near polygon of the tower. In particular, our characterization of the Hall-Janko near octagon $\mathsf{HJ}$ is similar to an earlier characterization due to Cohen and Tits who proved that it is the unique regular near octagon with parameters $(2, 4; 0, 3)$, but instead of regularity we assume existence of an isometrically embedded dual split Cayley hexagon, $\mathrm{H}(2)^D$. We also give a complete classification of near hexagons of order $(2, 2)$ and use it to prove the uniqueness result for $\mathrm{H}(2)^D$.