Lattices in complete rank 2 Kac-Moody groups. (Q456822)

Let \(\Lambda\) be a minimal Kac-Moody group of rank 2 defined over the finite field \(\mathbb F_q\), where \(q=p^a\) with \(p\) prime. Let \(G\) be the topological Kac-Moody group obtained by completing \(\Lambda\). An example is \(G=\mathrm{SL}_2(K)\), where \(K\) is the field of formal Laurent series over \(\mathbb F_q\). The group \(G\) acts on its Bruhat-Tits building \(X\), a tree, with quotient a single edge. The authors construct new examples of cocompact lattices in \(G\), many of them edge-transitive. They show that if cocompact lattices in \(G\) do not contain \(p\)-elements, the constructed lattices are the only edge-transitive lattices in \(G\) and that their constructions include the cocompact lattice of minimal covolume in \(G\). They also observe that, with an additional assumption on \(p\)-elements in \(G\), the arguments of Lubotzky (1990) for the case \(G=\mathrm{SL}_2(K) \) may be generalized to show that there is a positive lower bound on the covolumes of all lattices in \(G\), and that this minimum is realised by a non-cocompact lattice, a maximal parabolic subgroup of \(\Lambda\).