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We construct cocompact lattices Γ'0 < Γ0 in the group G = PGLd $({\mathbb{F}_q(\!(t)\!)\!})$ which are type-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 Isr. J. Math.103 (1998), 125–140. The stabiliser of each vertex in Γ'0 is a Singer cycle and the stabiliser of each vertex in Γ0 is isomorphic to the normaliser of a Singer cycle in PGLd(q). We show that the intersections of Γ'0 and Γ0 with PSLd $({\mathbb{F}_q(\!(t)\!)\!})$ are lattices in PSLd $({\mathbb{F}_q(\!(t)\!)\!})$ , and identify the pairs (d, q) such that the entire lattice Γ'0 or Γ0 is contained in PSLd $({\mathbb{F}_q(\!(t)\!)\!})$ . Finally we discuss minimality of covolumes of cocompact lattices in SL3 $({\mathbb{F}_q(\!(t)\!)\!})$ . Our proofs combine the construction of Cartwright–Steger Isr. J. Math.103 (1998), 125–140 with results about Singer cycles and their normalisers, and geometric arguments.



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