Structure of Chevalley groups over rings via universal localization

Abstract

In the current article we study structure of a Chevalley group $G(R)$ over a commutative ring $R$. We generalize and improve the following results: (1) standard, relative, and multi-relative commutator formulas; (2) nilpotent structure of [relative] $K_1$; (3) bounded word length of commutators. To this end we enlarge the elementary group, construct a generic element for the extended elementary group, and use localization in the universal ring. The key step is a construction of a generic element for a principle congruence subgroup, corresponding to a principle ideal.

References in corpus (5)

• The yoga of commutators
• Commutator width in Chevalley groups
• Generation of relative commutator subgroups in Chevalley groups. II
• Gauss decomposition for Chevalley groups, revisited
• Generation of relative commutator subgroups in Chevalley groups

Cited by in corpus (16)

• Commutator width in Chevalley groups
• Generation of relative commutator subgroups in Chevalley groups. II
• Commutators of relative and unrelative elementary subgroups in Chevalley groups
• Commutators of relative and unrelative elementary unitary groups
• Centrality of ${\mathrm{K}}_2$ for Chevalley groups: a pro-group approach
• Superrigidity from Chevalley groups into acylindrically hyperbolic groups via quasi-cocycles
• Elementary bounded generation for ${\mathrm{SL}}_n$ for global function fields and $n\ge 3$
• Overgroups of subsystem subgroups in exceptional groups: nonideal levels
• Relative and unrelative elementary groups, revisited
• Overgroups of subsystem subgroups in exceptional groups: 2A1-proof
• Locally isotropic elementary groups
• Inclusion among commutators of elementary subgroups
• Multiple commutators of elementary subgroups: end of the line
• Another presentation for symplectic Steinberg groups
• Verbal width in arithmetic Chevalley groups
• Commutators of elementary subgroups: curiouser and curiouser