On the congruence kernel of isotropic groups over rings

2013/04/30 by Stavrova, A.
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1305.0057

Abstract

Let R be a connected noetherian commutative ring, and let G be a simply connected reductive group over R of isotropic rank ge 2. The elementary subgroup E(R) of G(R) is the subgroup generated by the R-points U_P⁺(R) and U_P^-(R) of the unipotent radicals of two opposite parabolic subgroups P⁺ and P^- of G. Assume that 2 is invertible in R if G is of type B_n,C_n,F₄,G₂ and 3 is invertible in R if G is of type G₂. We prove that the congruence kernel of E(R), defined as the kernel of the natural homomorphism between the profinite completion of E(R) and the congruence completion of E(R) with respect to congruence subgroups of finite index, is central. In the course of the proof, we construct Steinberg groups associated to isotropic reductive groups and show that they are central extensions of E(R) if R is a local ring.

Related

• Centrality of the congruence kernel for elementary subgroups of Chevalley groups of rank >1 over noetherian rings
• Locally isotropic elementary groups
• Centralizer of the elementary subgroup of an isotropic reductive group
• Centrality of the congruence subgroup kernel of higher rank non-uniform arithmetic groups
• Isotropic reductive groups over discrete Hodge algebras