Non-residually finite ${\tilde{C}}_2$-lattices

Abstract

We provide the first known examples of non-residually finite lattices on irreducible buildings. They contain the first known simple CAT(0)-groups with property (T), and the first known CAT(0)-groups that are not quasi-isometric to a direct product. We also classify type-preserving vertex-regular lattices on buildings of type ${\tilde{A}}_2$ and thickness three, and discover an arithmetic example that is not commensurable to previously studied lattices.

55 pages, 10 figures; v2: include classification of ~A_2-lattices