Abstracts of talks
Minicourse «Reductive groups over rings» (in Russian).
Миникурс «Редуктивные группы над кольцами»:
слайды, текст с некоторыми доказательствами
(версия от 07.02.17).
We introduce ``axioms'' for the relative elementary subgroups of a reductive algebraic group.
Using these axioms only we prove all commutator formulas, including a generalization of the
nilpotency of K1. Since we use the “universal localization”, which is
a kind of the generic element method, we get a bound for the length of commutators
w.r.t. a functorial generating set. The universal localization method is used also to
prove the normal structure theorem for a Chevalley group over a ring.
Subgroup structure of Chevalley groups over rings.
We study the lattice Lof subgroups of a Chevalley group over a commutative ring, containing
a given subgroup D. Sandwich classification theorem for L. Which D can we handle?
Relations with Aschbacher clasess. Known result and conjectures. Main steps of the standard proof.
Overgroups of subring subgroups.
(Subgroups of a Chevalley group, containing the elementary subgroup over a subring)
Sandwich classification theorem for the lattice of subgroups of a
Chevalley group G(Φ,A) over a ring A, containing the elementary
subgroup E(Φ,R) over a subring R of A. Known results,
recent progress. Quasi-algebraic ring extensions. A conjecture about the final answer.
Localization techniques for Chevalley groups over rings.
Different versions of localization method: Quillen--Suslin, adding independent variables,
localization-completion, universal localization. Ideas of proofs: commutator
formulas, nilpotent structure of K1, length of commutators, normal structure.
Structure of Chevalley groups over rings.
A new version of localization techniques. Standard commutator formulas.
Universal ring for the principle congruence subgroup corresponding to a principle ideal.
Nilpotent structure of K1. Bounded word length. Normal structure.
Multiple relative commutator formula.
Ideas of the proof of the formula
Other talks are more introductory. Notion of a Chevalley group or an algebraic group scheme is
not important, one can have in mind particular examples, e.g. the special (general) linear
group or other classical groups.
Sandwich classification in linear groups.
Sandwich classification theorem for the lattice of subgroups of a group G,
containing a given subgroup D. Examples for G=GL(n,R).
Results and problems for a Chevalley group G.
Sandwich classification for subgroups of G, normalized by D.
General theorem. Examples.
Structure of Chevalley groups over commutative rings.
Chevalley group G(Φ,R) (e.g. SL(n,R) ),
elementary subgroup E(Φ,R). Normality of the elementary subgroup.
Subgroups, normalized by the elementary subgroup. Nilpotent structure of
K1(Φ,R)=G(Φ,R)/E(Φ,R). Relations on elementary
generators, the Steinberg group St(Φ,R). Centrality of
Width of linear groups. There are two versions:
1. More information, less ideas.
2. Less information, more ideas of proofs.
Width of a group G with respect to a generating set S.
Elementary generators. When elementary generators span SL(n,R)?
Width of SL(n,Z) and
SL(n,C[x]) in elementary generators.
G is called perfect if the set of all commutators C is a generating set.
Width of G=SL(R)=lim SL(n,R) with respect to C.
Obstruction for width of SL(n,R) to be finite. Width of C with
respect to elementary generators.
Generators of relative elementary subgroup.
Extended abstract (in Russian).