Inna Capdeboscq

COCOMPACT LATTICES ON Ã n BUILDINGS

We construct cocompact lattices Γ’0< Γ0 in the group G = PGLd(Fq((t))) which are ntype-preserving and act transitively on the set of vertices of each type in the building Δ associated to G. These lattices are commensurable with the lattices of Cartwright [Steger [CS]. The stabiliser of each vertex in Γ’0 is a Singer cycle and the stabiliser of each vertex in Γ0 is isomorphic to nthe normaliser of a Singer cycle in PGLd(q). We show that the intersections of Γ’0 and Γ0 with nPSLd(Fq((t))) are lattices in PSLd(Fq((t))), and identify the pairs (d; q) such that the entire lattice Γ’0 or Γ0 is contained in PSLd(Fq((t))). Finally we discuss minimality of covolumes of cocompact nlattices in SL3(Fq((t))). Our proofs combine the construction of Cartwright{Steger [CS] with results nabout Singer cycles and their normalisers, and geometric arguments.