PreprintCocompact lattices of minimal covolume in rank 2 KacMoody groups, Part I: Edgetransitive latticesInna (Korchagina) Capdeboscq and Anne ThomasAbstractLet \(G\) be a topological KacMoody 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 BruhatTits 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 edgetransitive. 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. This paper is available as a pdf (552kB) file.
