Timeline for What can be said about a group from its presentation?
Current License: CC BY-SA 2.5
15 events
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| Aug 16, 2010 at 20:57 | comment | added | HJRW | Joel, you were right to encourage me to clarify, and no doubt my answer could still be clearer. To be honest, on re-reading the question, I fear that the OP really does want a list of theorems of combinatorial group theory - I don't know how else to interpret 'non-amenability of Burnside groups'. Unless the question is edited to make clear exactly what he wants, we may never know | |
| Aug 16, 2010 at 20:19 | comment | added | Joel David Hamkins | My point was that the procedure always accepts trivial groups correctly, not just sometimes, and it does so in finite time (for those presentations it does accept). But I apologize for pestering you in your answer. | |
| Aug 16, 2010 at 18:53 | history | edited | HJRW | CC BY-SA 2.5 |
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| Aug 16, 2010 at 18:41 | comment | added | HJRW | After all, suppose I have in mind the fact that Seifert-Weber dodecahedral space is virtually Haken (to take an absurd example). There's a procedure that accepts precisely the presentations of the fundamental group of Seifert-Weber dodecahedral space, and if it accepts a given presentation then I can conclude that said group has a finite-index subgroup that splits non-trivially! But I don't think that sort of thing answers the question. | |
| Aug 16, 2010 at 18:33 | comment | added | HJRW | Joel, OK, maybe 'reliable' is the wrong word. (Indeed, I suppose 'in finite time' is exactly what I want to say!) But if the point of the question is merely to list some things that one can sometimes prove from a presentation, then the question boils down to 'List theorems of combinatorial group theory'. | |
| Aug 16, 2010 at 17:29 | comment | added | Joel David Hamkins | Henry, I appreciate your edit. But I'm not sure that reliability is the issue. After all, there is a computable procedure that accepts all and only the finite presentations of the trivial group. And another that accepts all and only the presentations of abelian groups, and similarly for many other properties. These procedures are completely reliable and work with arbitrary finite presentations. The issue for me rather is the difference between deciding a question yes-or-no as opposed to just recognizing positive instances. This is the difference between decidable and computably enumerable. | |
| Aug 16, 2010 at 16:36 | history | edited | HJRW | CC BY-SA 2.5 |
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| Aug 16, 2010 at 16:13 | vote | accept | Mustafa Gokhan Benli | ||
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| Aug 16, 2010 at 16:10 | comment | added | HJRW | Joel, I take your point. I suppose it depends how you interpret the question. But I wanted to emphasise that it's pretty hard to do anything with just a presentation. | |
| Aug 16, 2010 at 16:03 | comment | added | HJRW | Andy - good point! | |
| Aug 16, 2010 at 15:46 | comment | added | Joel David Hamkins | Henry, isn't your first paragraph over-stated? Although many questions about groups are undecidable, they are often nevertheless c.e., and so we often can compute information from a finite presentation. For example, if a finite group presentation does represent the trivial group, then we can computably recognize this in finite time---just enumerate all ways of applying the relations until you see that the generators are trivial. What you can't do is recognize non-triviality in finite time. Similarly, we can recognize whether the group is abelian in finite time, etc. | |
| Aug 16, 2010 at 15:24 | comment | added | Andy Putman | Using the Fox calculus, you can compute all the successive quotients in the lower central series (not just the first, which gives the abelianization). Of course, your broader point still stands. | |
| Aug 16, 2010 at 15:08 | history | edited | HJRW | CC BY-SA 2.5 |
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| Aug 16, 2010 at 14:53 | history | edited | HJRW | CC BY-SA 2.5 |
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| Aug 16, 2010 at 14:46 | history | answered | HJRW | CC BY-SA 2.5 |