Lattices in Kac - Moody Groups

Initially, we set out to construct non-uniform ‘arithmetic’ lattices in Kac-Moody groups of rank 2 over finite fields, as constructed by Tits ([Ti1], [Ti2]) using the BruhatTits tree of a Tits system for such groups. This attempt succeeded, and in fact, the construction we used can be applied to higher rank Kac-Moody groups over sufficiently large finite fields, and their buildings (Theorem 1.7 below). After completing this work, we learned that B. Remy has obtained an equivalent result for the more general class of almost split Kac-Moody groups ([R1], [R2]). We have also constructed an uncountably infinite family of non-uniform lattices in the rank 2 Kac-Moody case, that is, we have succeeded in carrying over A. Lubotzky’s construction of non-uniform lattices in SL2 over a Laurent series field (Theorem 2.9 below). The basic tool for this extension is a (new) spherical Tits system (Theorems 2.2 and 2.7 below). It remains to determine whether, as in the case of SL2, we have constructed uncountably many distinct conjugacy classes of non-uniform lattices within the Kac-Moody group. In further analogy with Lubotzky’s construction of lattices in SL2, we have constructed an uncountably infinite family of cocompact lattice subgroups of rank 2 Kac-Moody groups. Once again, it remains to determine if there are uncountably many distinct conjugacy classes of these lattices. In rank 2, the Kac-Moody groups and their lattice subgroups fail to have property T (Proposition 4.1 below). In contrast to this, in the higher rank case, a result of Dymara and Januszkiewicz ([DJ]) implies that certain ‘hyperbolic’ Kac-Moody groups do have property T . Hence, the lattices that we construct in these cases are finitely generated and have finite commutator quotients. Detailed proofs of the results mentioned above will appear elsewhere. The authors would like to thank B. Remy and T. Januszkiewicz for their correspondence, and for informing us of their results. Thanks to A. Lubotzky for encouraging us to undertake this work and for explaining his constructions to us, and to H. Bass for many illuminating conversations.