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arXiv · 2505.12924

Principal congruence subgroups in the infinite rank case

Vladimir A. Tolstykh
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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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Vladimir A. Tolstykh. 2025-05-19. Principal congruence subgroups in the infinite rank case. https://doi.org/10.1112/jlms.12250

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