While I’m aware (and as it was pointed out in the comments) it is generally undecidable whether a given presentation represents a free group, I’m interested in criteria that nevertheless ensure that the represented group is free (the most obvious one being, that there are no relations for example). In other words: Given a presentation of some group, what are nice criteria to conclude that the presentation describes a free group?

Even stronger is there a subclass of presentations (where one easily checks whether the presentation is in that class), for which it is algorithmically decidable whether a presentation yields a free group?

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