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 ${\mathrm{\Gamma }}_0^{\prime }<{\mathrm{\Gamma }}_0$ 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 ${\mathrm{\Gamma }}_0^{\prime }$ is a Singer cycle and the stabiliser of each vertex in ${\mathrm{\Gamma }}_0$ is isomorphic to the normaliser of a Singer cycle in ${\mathrm{PGL}}_d(q)$. We then show that the intersections of ${\mathrm{\Gamma }}_0^{\prime }$ and ${\mathrm{\Gamma }}_0$ with ${\mathrm{PSL}}_d(K)$ are lattices in ${\mathrm{PSL}}_d(K)$, and identify the pairs $(d,q)$ such that the entire lattice ${\mathrm{\Gamma }}_0^{\prime }$ or ${\mathrm{\Gamma }}_0$ 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.