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arXiv · 2609.25125

Margulis-Soifer theorem for one-relator groups

Azer Akhmedov·Atabay Mahmudov
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Abstract

We establish the Margulis-Soifer dichotomy for one-relator groups: every one-relator group is either virtually solvable or has a maximal subgroup of infinite index. We also present examples of one-relator groups with and without free maximal subgroups of infinite index, as well as examples that possess both free and non-free infinite index maximal subgroups. Triviality of the Frattini subgroup is also shown for all non-solvable one-relator groups. We close the paper with a short list of questions.

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Azer Akhmedov, Atabay Mahmudov. 2026-09-23. Margulis-Soifer theorem for one-relator groups. https://arxiv.org/abs/2609.25125

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