On Chamber-regular $\tilde C_2$-Lattices

Abstract

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 type-preserving, 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 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 are Bruhat-Tits buildings and all our lattices are exotic building lattices.