Density and maximality of arithmetic subgroups.
A. Borel
1966
Citations
148
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Journal
Journal für die reine und angewandte Mathematik (Crelles Journal)
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Let G be a connected semi-simple group defined over the field Q of rational numbers and H an arithmetic subgroup of G. By definition, then, H is a subgroup of GQ and if £ :G->GL W is an injective Jg-morphism, then ρ(Η) is commensurable with Q(G)Z (see [4]). We show that if G fulfills an obviously necessary condition, H is Zariski-dense in G (Thm 1), and is contained in only finitely many discrete subgroups of GR (Gor. to Thm 4). Theorem 2 describes the group C(H) of elements g£Gc such that g · H · g~ is commensurable with H. In particular, C(H) is equal to GQ if G has no normal Jg-subgroup N Φ {e} whose group NR of real points is compact. Similar results hold for groups defined over a number field k of finite degree over Q (Thms 3, 5). In § 7 we consider the case where G "splits over A" and H is the group of units of certain lattices, and prove that H is then discrete maximal if either k = Q or k has class number one and G has no center (Thm 7). Theorem l is closely related to the density theorem of [1]; s a matter of fact, it can easily be derived from it and from the finiteness of the volume of GRIH. Theorem 2 was stated in [2], For both, the proofs given here are different from the original ones and were obtained jointly with J.-P. Serre. Theorem 4 generalizes a result of Ramanathan [11] and the proof is in part patterned after bis. Theorem 7 applies in particular to 5/>(2n, Z), S L (n, Z) or the group of units of the Standard quadratic form of maximal index in 2n variables; in those eases, it is proved in [11] (and goes back to Hecke for 5X(2, Z)). Theorem 7 was suggested by these results. For algebraic groups, we follow the notation and conventions of [5]. In particular: -group or "algebraic group defined over a field A" are used synonymously; the identity component of a /c-group G is denoted G°; a connected semi-simple /c-group is almost simple over k if it has no normal connected A-subgroup N φ {e}. Often, °A Stands for g ' A ·g~\ Throughout this paper, k is a number field of finite degree over Q, k an algebraic closure of A, o the ring of integers of k, G a connected semi-simple k-group, and H an arithmetic subgroup of G.