Principal congruence subgroups in the infinite rank case

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 $\mathrm{Aut}(A).$

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