Congruence Subgroups of Lattices in Rank 2 Kac–Moody Groups over Finite Fields. Issue 3 (3rd March 2016)

• Record Type: Journal Article
• Authors: Ali, Abid; Carbone, Lisa
• Is Part Of: Communications in algebra. Volume 44:Issue 3(2016)
• Page Start: 1236
• Page End: 1264
• Publication Date: 2016-03-03
• Subjects: Congruence subgroup -- Kac–Moody group -- Lattice; 20E42 -- 22E40 -- 22F50; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2015.1012673
• Languages: English
• ISSNs: 0092-7872

Abstract

Abstract : Let G be a rank 2 complete affine or hyperbolic simply-connected Kac–Moody group over a finite field k . Then G is locally compact and totally disconnected. Let B − = HU − be the negative minimal parabolic subgroup of G, where H is the analog of a diagonal subgroup and U − is generated by all negative real root groups. Let w 1 and w 2 be the generators of the Weyl group. Let be the negative standard parabolic subgroup of G corresponding to w 1 . It is known that the subgroups U −, B − and, are nonuniform lattice subgroups of G . Here we construct an infinite sequence of congruence subgroups of as natural generalizations of the corresponding notions for lattices in Lie groups. We also show that the group U − contains analogous congruence subgroups. Our technique involves determining graphs of groups presentations for U −, B −, and with the fundamental apartment of the Bruhat–Tits tree X a quotient graph for U − and for B − on X . When k = 𝔽 q and q = 2 s, the graph of groups for has the the positive half of the fundamental apartment as quotient graph. We explicitly construct the graphs of groups for the principal (level 1) congruence subgroup of and the analogous subgroups of U − giving generalized amalgam presentations for them.