The Margulis-Platonov conjecture for SL1,D and 2-generation of finite simple groups

Abstract

We give a new proof of the Margulis-Platonov conjecture for the groups SL 1,D which is shorter than the previous proofs and in which the use of the classification of finite simple groups is limited to the fact that every such group is generated by two elements. The argument is based on further development of the methods of [18].

Commensurated Subgroups of Arithmetic Groups, Totally Disconnected Groups and Adelic Rigidity

“…Notice that this same prediction is made by Conjecture 7.7, as follows from part 1 of Proposition 7.8 applied to any one of the almost all non-archimedean ν ∈ V where G is K νisotropic (a similar situation is expected when S is co-finite, which is consistent with the general CSP result of Prasad-Rapinchuk [39, Section 9] conditioned on (MP). See also Rapinchuk's [46] and the references therein for the latest progress on this conjecture.) Thus, Conjecture 7.7 removes the restriction in Serre's conjecture by offering a unified statement.…”

On graphs and valuations

“…Proof. When |S 0 | = 1 this is proved (in an equivalent form) in [R,Lemma 2]. The general case follows by induction.…”

Finite and Infinite Quotients of Discrete and Indiscrete Groups

“…That conjecture describes the normal subgroup structure of the group of rational points of a simply connected simple algebraic group over a number field. Although the conjecture is still open in full generality, the special case of the reduced norm 1 group SL 1 (D) of a division algebra D over a number field was settled in a series of important papers by various authors (see [SS02], [Rap06] and references therein). Roughly speaking, the normal subgroup structure of SL 1 (D) is elucidated by a similar scheme as in the proof of Theorem 3.1: infinite quotients and finite quotients are investigated separately, with completely different methods.…”

“…Roughly speaking, the normal subgroup structure of SL 1 (D) is elucidated by a similar scheme as in the proof of Theorem 3.1: infinite quotients and finite quotients are investigated separately, with completely different methods. While the treatment of infinite quotients is based on Margulis' Normal Subgroup Theorem and Strong Approximation as above, the Classification of the Finite Simple Groups (CFSG) is used in an essential way to investigate the finite quotients of SL 1 (D) for all division algebras D of degree ≥ 3 over number fields, see [RP96], [SS02] and [Rap06]. We finish this section by mentioning the following striking culmination of this direction of research, which is valid over an arbitrary ground field.…”