Properties unique to free groups

Asked Apr 13 '20 at 20:55

Active Apr 13 '20 at 21:21

Viewed 38 times

I was just working on a problem for my Algebra class about proving that a group was free. I was able to figure that out, but it got me wondering: are there any properties that are unique to free groups that are useful in proving that a group is free? Or a few properties that are only all satisfied in a free group?

That may have been a little vague, and I am sorry about that. I know that this kind of vagueness is not desirable on MSE, but I am really just curious about this :)

EDIT: I am interested in finitely generated free groups

edited Apr 13 '20 at 21:21

asked Apr 13 '20 at 20:55

Anonymous

Which definition of a free group are you using? – Shaun Apr 13 '20 at 20:59
@Shaun I am talking about non-abelian free groups – Anonymous Apr 13 '20 at 21:12
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Still, free groups have more than one definition. They are, of course, equivalent, but which one you use would determine what unique properties - properties in alternative definitions - they have. – Shaun Apr 13 '20 at 21:14
@Shaun I am considering $F_n = \langle a_1, a_2, \dots, a_n \rangle$ where there are no relations between the generators. – Anonymous Apr 13 '20 at 21:16
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So, do they have to be finitely generated? – Shaun Apr 13 '20 at 21:19
@Shaun Yes, I am interested in finitely generated free groups. I will update the question! – Anonymous Apr 13 '20 at 21:20
Related. – Shaun Apr 13 '20 at 21:23
@Shaun this is probably completely wrong, but I heard once that Hopfian, imperfect, and finitely generated implies free. Is this true? – Anonymous Apr 13 '20 at 21:27
The closest thing I can think of in that regard is that every finitely generated free group is Hopfian. This is Proposition 3.5 of Lyndon & Schupp. – Shaun Apr 13 '20 at 21:30
@Shaun not super related, but is it known exactly which families of groups are Hopfian? – Anonymous Apr 13 '20 at 21:36
I don't know (yet), @Anonymous. I'll look into it. – Shaun Apr 13 '20 at 21:40
"Hopfian, imperfect, and finitely generated implies free" is very much false. For instance, finite rank abelian groups, fundamental groups of surfaces of genus $\ge 1$, etc, all share these properties. – Moishe Kohan Apr 14 '20 at 03:36

Properties unique to free groups

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