Principal congruence subgroups in the infinite rank case

TL;DR

This paper extends classical results about finite-dimensional linear groups to the automorphism group of an infinitely generated free abelian group, providing structural classifications and analogues of known relations.

Contribution

It introduces new descriptions of normal generators, maximal normal subgroups, and principal congruence subgroups for automorphism groups of infinite rank free abelian groups.

Findings

Classified maximal normal subgroups of Aut(A)

Described normal generators of principal congruence subgroups

Established an analogue of Brenner's ladder relation

Abstract

We obtain a number of analogues of the classical results of the 1960s on the general linear groups $\mathrm{GL}_n(\mathbf Z)$ and special linear groups $\mathrm{SL}_n(\mathbf Z)$ for the automorphism group $\Gamma_A=\mathrm{Aut}(A)$ of an infinitely generated free abelian group $A.$ In particular, we obtain a description of normal generators of the group $\mathrm{Aut}(A),$ classify the maximal normal subgroups of the group $\mathrm{Aut}(A),$ describe normal generators of the principal congruence subgroups $\Gamma_{\!A}(m)$ of the group $\mathrm{Aut}(A),$ and obtain an analogue of Brenner's ladder relation for the group $\mathrm{Aut}(A).$