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.

29 pages, 5 figures, 8 tables