Cocompact lattices on \tilde{A}_n buildings

TL;DR

This paper constructs specific cocompact lattices in p-adic groups associated with ilde{A}_n buildings, analyzes their properties, and discusses minimal covolumes, combining algebraic and geometric methods.

Contribution

It introduces new cocompact lattices in PGL_d(K) with transitive vertex actions, and characterizes their intersections with PSL_d(K), extending previous constructions.

Findings

Constructed cocompact lattices with transitive vertex actions

Identified conditions for lattices to lie in PSL_d(K)

Discussed minimal covolumes of lattices in SL_3(K)

Abstract

Let K be the field of formal Laurent series over the finite field of order q. We construct cocompact lattices \Gamma'_0 < \Gamma_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 \Gamma'_0 is a Singer cycle and the stabiliser of each vertex in \Gamma_0 is isomorphic to the normaliser of a Singer cycle in PGL_d(q). We then show that the intersections of \Gamma'_0 and \Gamma_0 with PSL_d(K) are lattices in PSL_d(K), and identify the pairs (d,q) such that the entire lattice \Gamma'_0 or \Gamma_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.