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 work with the Suzuki groups Sz(q), 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+root 2q +1 or q - root 2q + 1 (i.e.: m is the order of some non-involutive element of Sz(q)), 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(q). (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).