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.