Exotic buildings, lattices and non-residual finiteness

Abstract

We provide the first examples of lattices on irreducible buildings that are not residually finite. Assuming that the normal subgroup property holds for them, which is expected, they are virtually simple. Our examples fall into five quasi-isometry classes, corresponding to their actions on five distinct exotic ~C2-buildings. Each class contains a smallest lattice, which has no proper finite index subgroups. Consequently, if the normal subgroup property holds, these are in fact simple. As exotic lattices, our examples are CAT(0)-groups, enjoy Kazhdan’s property (T), and are quasi-isometrically rigid. If the smallest lattices turn out to be simple, then, to the author’s knowledge, they constitute the first infinite examples of quasi-isometrically rigid simple groups, as well as the first infinite examples of simple CAT(0)-groups with Kazhdan’s property (T). The lattices were found through a computer search, and the methods developed proved to be fruitful; as a direct application, we classify type-preserving, vertex-regular ~A2-lattices acting on buildings of thickness three.