Timeline for Inductive colimits of free groups
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 27, 2017 at 20:30 | vote | accept | Hannes Thiel | ||
| Dec 27, 2017 at 20:13 | answer | added | YCor | timeline score: 5 | |
| Dec 27, 2017 at 19:43 | comment | added | Hannes Thiel | Yes, with inductive colimit I mean the colimit of a system of groups indexed over a directed set, for example a sequential system $G_1\to G_2\to G_3\to\ldots$. | |
| Dec 27, 2017 at 19:33 | comment | added | YCor | Again, I'm using filtering assumption. Indeed any amalgam of free groups is a (non-filtering) inductive limit of free groups. There are non-free such examples. | |
| Dec 27, 2017 at 19:23 | comment | added | Hannes Thiel | @YCor: Yes, you are right. I was not aware of "locally free groups". Also your argument that inductive limits of free groups are locally free seems like a perfect answer to me. | |
| Dec 27, 2017 at 19:22 | comment | added | Yonatan Harpaz | Every group that is a directed colimit of free groups has homological dimension at most 1 and cohomological dimension at most 2. In particular, $\mathbb{Z}/2$ is not a directed colimit of free groups. | |
| Dec 27, 2017 at 19:19 | comment | added | YCor | Actually I think it's true (but nontrivial) that all inductive limits of free groups are locally free. This reduces to the finitely generated case, in which case it follows from the fact that any sequence of epimorphisms between finitely generated free groups stabilizes (because the rank decreases until it stabilizes, and then we have isomorphism by Hopfianness). | |
| Dec 27, 2017 at 19:14 | comment | added | YCor | No, for instance $\mathbf{Q}$ or $\mathbf{Q}\ast\mathbf{Q}$ are not free but are inductive limits of free groups with injective homomorphisms. These are not free. Inductive limits of free groups with injective homomorphisms are called "locally free groups". Another non-free example is the $\pi_1$ of the Hawaiian earring space. | |
| Dec 27, 2017 at 19:11 | comment | added | Hannes Thiel | With injective connecting homomorphisms the question will be very different. In fact, it seems to me that an inductive limit of free groups with injective connecting homomorphisms is again a free group. | |
| Dec 27, 2017 at 19:06 | comment | added | YCor | Directed limit often refers to inductive limit of injective homomorphisms. Anyway... what do you think of a group with 2 elements, in this case too? | |
| Dec 27, 2017 at 18:46 | history | asked | Hannes Thiel | CC BY-SA 3.0 |