Timeline for Some questions on a paper of Baumslag and Solitar
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 20, 2021 at 21:28 | vote | accept | user101010 | ||
| May 20, 2021 at 15:11 | answer | added | HJRW | timeline score: 1 | |
| May 19, 2021 at 16:00 | answer | added | Carl-Fredrik Nyberg Brodda | timeline score: 1 | |
| May 17, 2021 at 20:52 | comment | added | Carl-Fredrik Nyberg Brodda | @HJRW I don't have much to add to what you write, other than saying that I certainly second learning from Serre! (Bogopolski is also very nice) :-) | |
| May 17, 2021 at 20:30 | comment | added | user101010 | Thank you both for the comments - I have enjoyed looking at some of those classic references in the past and I really appreciate pointers on how they fit together with our current understanding. I'll start looking at Serre tomorrow :-). | |
| May 17, 2021 at 15:57 | comment | added | HJRW | ... Likewise, Bass--Serre theory makes trivial some things that seem much more difficult from the "classical" point of view. (The Kurosh subgroup theorem is an outstanding example.) So anyone who wants to contribute to modern combinatorial or geometric group theory should learn Bass--Serre theory as soon as possible. I agree that Lyndon and Schupp is a difficult source to learn from, and I wouldn't recommend it. In fact, I usually recommend Serre's book, which is fine. | |
| May 17, 2021 at 15:55 | comment | added | HJRW | @Carl-FredrikNybergBrodda: since the OP mentions "folding-style" techniques they don't seem to want an old-fashioned combinatorial group theory approach, though perhaps they can tell us themselves! :) While it's true that none of Baumslag, Solitar, Magnus, Karrass etc would have spoken explicitly of Bass--Serre theory, it is merely a conceptual simplification and clarification of techinques that they knew quite well, so there's not really anything ahistorical about using it.... | |
| May 17, 2021 at 15:47 | comment | added | Carl-Fredrik Nyberg Brodda | @HJRW My reading of the first question includes an implicit understanding of how Baumslag and Solitar would have known it, which definitely was not via Bass-Serre theory. In any case I feel it's relevant. I find the order in MKS far easier to understand and follow than e.g. Lyndon-Schupp, especially if one wishes to avoid the machinery of geometric group theory. But HNN-extensions (as a combinatorial tool, not the later geometric interpretation) do make the proof less cumbersome to write down than the amalgamations used by Magnus (and later MKS, which predates Bass-Serre theory by a decade). | |
| May 17, 2021 at 15:32 | comment | added | HJRW | @Carl-FredrikNybergBrodda: The "Magnus breakdown procedure" just involves developing the relevant bit of Bass--Serre theory in a special case. I see no reason to avoid the full theory, which is not really any harder than the special case. It's a very unfortunate feature of combinatorial group theory that some standard textbooks (eg Rotman) develop ideas according to historial, rather than conceptual, order. I haven't checked M--K--S, but I fear it's the same. | |
| May 17, 2021 at 14:53 | comment | added | Carl-Fredrik Nyberg Brodda | The usual way (back then) to show that an element in any one-relator group is non-trivial is to just use the Magnus breakdown procedure, i.e. to solve the word problem. This is not very hard. This is spelled out in e.g. Magnus-Karrass-Solitar (and uses no geometric group theory). This is even easier if one uses HNN-extensions as HJRW suggests, but the original solution does not. | |
| May 17, 2021 at 14:42 | history | edited | YCor | CC BY-SA 4.0 |
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| May 17, 2021 at 14:23 | comment | added | HJRW | The modern answer to your first question uses Bass—Serre theory. Baumslag—Solitar groups are HNN extensions and, with respect to that structure, the given element is in normal form. This implies it’s non-trivial. | |
| May 17, 2021 at 13:27 | history | asked | user101010 | CC BY-SA 4.0 |