Cocompact lattices on \(\tilde{A}_n\) buildings

Inna Capdeboscq, Dmitriy Rumynin and Anne Thomas

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 \(G = \mathrm{PGL}_d(K)\) 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 \(\mathrm{PGL}_d(q)\). We then show that the intersections of \(\Gamma'_0\) and \(\Gamma_0\) with \(\mathrm{PSL}_d(K)\) are lattices in \(\mathrm{PSL}_d(K)\), and identify the pairs \((d,q)\) such that the entire lattice \(\Gamma'_0\) or \(\Gamma_0\) is contained in \(\mathrm{PSL}_d(K)\). Finally we discuss minimality of covolumes of cocompact lattices in \(\mathrm{SL}_3(K)\). Our proofs combine a construction of Cartwright and Steger with results about Singer cycles and their normalisers, and geometric arguments.