The yoga of commutators

TL;DR

This paper explores advanced localisation techniques in algebraic groups, demonstrating their effectiveness in deriving commutator formulas, bounded commutator widths, and nilpotent filtrations, thus enhancing computational methods in algebraic group theory.

Contribution

It introduces and analyzes new localisation methods and their applications to commutator calculus, providing improved tools for algebraic group calculations.

Findings

Established relative and universal localisation techniques.

Proved bounded width of commutators with elementary generators.

Derived nilpotent filtrations of congruence subgroups.

Abstract

In the present paper we discuss some recent versions of localisation methods for calculations in the groups of points of algebraic-like and classical-like groups. Namely, we describe relative localisation, universal localisation, and enhanced versions of localisation-completion. Apart from the general strategic description of these methods, we state some typical technical results of the conjugation calculus and the commutator calculus. Also, we state several recent results obtained therewith, such as relative standard commutator formulae, bounded width of commutators, with respect to the elementary generators, and nilpotent filtrations of congruence subgroups. Overall, this shows that localisation methods can be much more efficient, than expected.