Cocompact lattices of minimal covolume in rank 2 Kac-Moody groups, Part I: Edge-transitive lattices

Inna (Korchagina) Capdeboscq and Anne Thomas

Abstract

Let $G$ be a topological Kac-Moody group of rank 2 with symmetric Cartan matrix, defined over a finite field. An example is $G=\mathrm{SL}(2,K)$, where $K$ is the field of formal Laurent series over $F_q$. The group $G$ acts on its Bruhat-Tits building $X$, a regular tree, with quotient a single edge. We classify the cocompact lattices in $G$ which act transitively on the edges of $X$. Using this, for many such $G$ we find the minimum covolume among cocompact lattices in $G$, by proving that the lattice which realises this minimum is edge-transitive. Our proofs use covering theory for graphs of groups, the dynamics of the $G$-action on $X$, the Levi decomposition for the parabolic subgroups of $G$, and finite group theory.