Cocompact lattices on \tilde{A}_n buildings

Let K be the field of formal Laurent series over the finite field of order q. We construct cocompact lattices Γ'_0 < Γ_0 in the group G = PGL_d(K) which are type-preserving and act transitively on the set of vertices of each type in the building associated to G. The stabiliser of each vertex in Γ'_0 is a Singer cycle and the stabiliser of each vertex in Γ_0 is isomorphic to the normaliser of a Singer cycle in PGL_d(q). We then show that the intersections of Γ'_0 and Γ_0 with PSL_d(K) are lattices in PSL_d(K), and identify the pairs (d,q) such that the entire lattice Γ'_0 or Γ_0 is contained in PSL_d(K). Finally we discuss minimality of covolumes of cocompact lattices in SL_3(K). Our proofs combine a construction of Cartwright and Steger with results about Singer cycles and their normalisers, and geometric arguments.

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