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The normal subgroup structure of the Picard group. (English) Zbl 0624.20031

Unter der Picard-Gruppe \(\Gamma\) verstehen wir die Gruppe PSL(2,\({\mathbb{Z}}[i])\) der linear gebrochenen Transformationen mit ganzen Gaußschen Zahlen als Koeffizienten. Die Autoren untersuchen die Struktur von Normalteilern von \(\Gamma\). Grundlage ist dabei eine detaillierte Untersuchung der höheren Kommutatorgruppen \(\Gamma^{(n)}\). Insbesondere gilt: \(\Gamma^{(1)}\) hat Index 4, \(\Gamma^{(2)}\) hat Index 12, \(\Gamma^{(3)}\) hat Index 768 und \(\Gamma^{(n)}\), \(n\geq 4\), hat unendlichen Index in \(\Gamma\). Dies erlaubt unter anderem eine vollständige Klassifizierung der Normalteiler von \(\Gamma\) mit Index kleiner als 60. Schließlich geben die Autoren Beispiele von normalen Nicht-Kongruenzuntergruppen von \(\Gamma\).
Reviewer: G.Rosenberger

MSC:

20H05 Unimodular groups, congruence subgroups (group-theoretic aspects)
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
11F06 Structure of modular groups and generalizations; arithmetic groups
20F14 Derived series, central series, and generalizations for groups
20E15 Chains and lattices of subgroups, subnormal subgroups
20E07 Subgroup theorems; subgroup growth
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