Flag transitive geometries with trialities and no dualities coming from Suzuki groups

Abstract

Recently, Leemans and Stokes constructed an infinite family of incidence geometries admitting trialities but no dualities from the groups $PSL(2,q)$ (where $q=p^{3n}$ with p a prime and $n>0$ a positive integer). Unfortunately, these geometries are not flag transitive. In this paper, we work with the Suzuki groups $Sz\left(q\right)$, where $q=2^{2e+1}$ with e a positive integer and $2e+1$ is divisible by 3. For any odd integer m dividing $q-1$, $q+\sqrt{2q}+1$ or $q-\sqrt{2q}+1$ (i.e.: m is the order of some non-involutive element of $Sz\left(q\right)$), we construct geometries of type $(m,m,m)$ that admit trialities but no dualities. We then prove that they are flag transitive when $m=5$, no matter the value of q. These geometries form the first infinite family of incidence geometries of rank 3 that are flag transitive and have trialities but no dualities. They are constructed using chamber systems and the trialities come from field automorphisms. These same geometries can also be considered as regular hypermaps with automorphism group $Sz\left(q\right)$.