Principal congruence subgroups in the infinite rank case

TL;DR

This work extends Brenner's finite-rrank analysis to the automorphism group $Aut(A)$ of an infinitely generated free abelian group, introducing principal congruence subgroups $\Gamma_{A}(m)$ and their normal closures. It characterizes normal generators via moietous direct summands and constructs a canonical normal generator $\tau$ with explicit action, while showing that non-generators must lie in $\Lambda_{A}(m)$ for some $m \ge 2$. The maximal normal subgroups are described through $\Lambda_{A}(p)$ and ultrafilter-based subgroups $\Lambda_{A}(\mathcal{F})$, with a dichotomy depending on the cofinality of $\operatorname{rank}(A)$; in many cases, these relate to almost-radiations $aR(A)$. A central technical achievement is proving that the normal closure of $\boldsymbol{\tau}^{m}$ equals $\Gamma_{A}(m)$, and that Brenner-style ladder relations govern the landscape of normal subgroups, highlighting both parallels and novelties in the infinite-rank setting.

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 $Γ_A=\mathrm{Aut}(A)$ 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 $Γ_{\!A}(m)$ of the group $\mathrm{Aut}(A),$ and obtain an analogue of Brenner's ladder relation for the group $\mathrm{Aut}(A).$

Theorems & Definitions (43)

• Proposition 1.1
• proof
• Corollary 1.2
• proof
• Proposition 1.3
• proof
• Claim 1
• Claim 2
• Theorem 1.4
• proof