Timeline for Categorical proof subgroups of free groups are free?
Current License: CC BY-SA 3.0
9 events
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| Jan 29, 2015 at 15:40 | comment | added | Pasha Zusmanovich | @Mariano Suárez-Alvarez: No, the same is not true for Lie algebras. See a paper by Mikhalev, Umirbaev, Zolotykh (DOI:10.1070/RM1994v049n01ABEH002199) where a non-free Lie algebra of cohomological dimension 1 is constructed (in characteristic p). As far as I know, in charactersitic zero this is an open question. | |
| Jan 28, 2015 at 10:29 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| Jan 28, 2015 at 10:24 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| Jan 28, 2015 at 10:22 | comment | added | Qiaochu Yuan | @HJRW: thanks for the warning. I picked that notation because I half-remembered that similar property, but it seems I remembered it wrong. I'll pick something else. | |
| Jan 28, 2015 at 10:07 | comment | added | HJRW | I like this answer. A small comment on notation. Your choice of $F_k$ is unfortunate, since $F_k$ is usually used for the similar property of 'having a BG with finite $k$-skeleton'. So $F_1 =$ fg, $F_2 =$ fp etc. | |
| Jan 28, 2015 at 9:41 | comment | added | Mariano Suárez-Álvarez | The same is true of free Lie algebras: they are those of global dimension 1. | |
| Jan 28, 2015 at 9:37 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| Jan 28, 2015 at 9:36 | comment | added | Matthias Wendt | A reference for the last sentence: Groups of cohomological dimension one are free is a theorem of Stallings (finitely generated) and Swan (general). I support the preference of cohomological dimension over freeness. | |
| Jan 28, 2015 at 9:31 | history | answered | Qiaochu Yuan | CC BY-SA 3.0 |