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

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 construct the first infinite family of incidence geometries of rank three that are flag transitive and have trialities but no dualities. These geometries are constructed using chamber systems of Suzuki groups Sz(q) (where $q=2^{2e+1}$ with $e$ a positive integer and $2e+1$ is divisible by 3) and the trialities come from field automorphisms. We also construct an infinite family of regular hypermaps with automorphism group Sz(q) that admit trialities but no dualities.