On the thin regular geometries of rank four for the Janko group $J_1$

Abstract

The Janko group $J_1$ acts regularly on six rank four thin residually connected geometries. Two of them are polytopes of type $\left\{5,3,5\right\}$ and $\left\{5,6,5\right\}$. In this paper, we show that starting from the $\left\{5,3,5\right\}$ polytope, the five other thin geometries may be constructed in a simple manner.