On Chamber-regular ${\tilde{C}}_2$-Lattices

We construct the first examples of chamber-regular lattices on ${\tilde{C}}_2$-buildings. Assuming a conjecture of Kantor our list of examples becomes a classification for chamber-regular ${\tilde{C}}_2$-lattices on locally-finite ${\tilde{C}}_2$-buildings. The links of special vertices in the buildings we construct, are all isomorphic to (the incidence graph of) the unique generalized quadrangle $Q$ of order (3,5). In particular our constructions involve chamber-regular actions on $Q$. These actions on $Q$ are the first (and if Kantor’s conjecture holds the only) chamber-regular actions on a finite generalized quadrangle and therefore interesting in their own right. Moreover $Q$ is not Moufang and therefore none of our examples is a Bruhat-Tits building and all our lattices are exotic building lattices.