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This maybe a very general question.

If we have a group given by its presentation only, what kind of properties could be proven about it?

I know examples about non-amenability of some Burnside groups.

What kind of examples are there in literature where one proves some property "just" from a presentation?

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    $\begingroup$ The questions mathoverflow.net/questions/16532 and mathoverflow.net/questions/15957 seem relevant. $\endgroup$
    – Andy Putman
    Commented Jul 11, 2010 at 3:56
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    $\begingroup$ One remark I should make is that while in general you can't calculate much about a group from its presentation, sometimes you can get lucky. For instance, a theorem of C. Gordon says that you can't calculate H_2 of a group from a presentation, but in the special case of the mapping class group of a surface Pitsch has a beautiful paper doing exactly that. $\endgroup$
    – Andy Putman
    Commented Jul 11, 2010 at 4:05
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    $\begingroup$ One cautionary point is that a group with an elaborate presentation might well turn out to be trivial. And this may be extremely difficult to decide. It's a standard issue in combinatorial group theory. $\endgroup$
    – Jim Humphreys
    Commented Jul 11, 2010 at 13:14
  • $\begingroup$ One fact that I like is that the n dimensional representations of a fp group (wrt some ground field) is scheme. The relations turn into polynomial relations! $\endgroup$
    – Spice the Bird
    Commented Feb 7, 2012 at 4:37
  • $\begingroup$ Also related mathoverflow.net/questions/16565 $\endgroup$
    – David E Speyer
    Commented Jul 31, 2014 at 18:44

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Quite a lot, I believe (although `a lot' is subjective).

A neat example of a property which can be read immediately off of a (finite) presentation $\langle X; R \rangle$ is the Deficiency of said presentation. This is defined to be $|X|-|R|$. Now, this contain some intriguing properties. For example, every group of deficiency greater than 2 is large (it has a finite-index subgroup which contains a homomorphism onto a non-abelian free group), while every group of deficiency 1 is infinite and if a deficiency 1 presentation has a relator which is a proper power then this group too is large.

Also, the word, conjugacy, etc. problems for presentations is an intriguing topic.

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    $\begingroup$ I'm pretty sure you can't in general read the deficiency of a group from its presentation. $\endgroup$
    – HJRW
    Commented Aug 16, 2010 at 14:45
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    $\begingroup$ I suppose what you're saying here is that you can get a lower bound on the deficiency of a group from its presentation, and that can be enough to read off largeness etc. I just think it's a little dangerous in this sort of question to elide the difference between properties of groups and properties of presentations. $\endgroup$
    – HJRW
    Commented Aug 16, 2010 at 16:57
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Almost nothing can reliably be said about a group just from a presentation in finite time. (In fact, the abelianisation is just about the only thing one can reliably compute.) Most strikingly, there is no algorithm to recognise whether a given presentation represents the trivial group. More generally, one cannot in general solve 'the word problem' - ie, there is no algorithm to determine whether a given element is non-trivial. See Chuck Miller's survey article for details.

(Update. I inserted the word 'reliably' above in deference to Joel David Hamkins' fair comment. (Update 2. I then inserted the phrase 'in finite time' to be strictly correct, in an effort to head off further argument.) It is true that, in many special cases, there is information that can be read off from a specific presentation. This is more or less the topic of combinatorial group theory! But I want to emphasise that you can do nothing with an arbitrary presentation.)

On the other hand, there is a growing realisation that, surprisingly, if one is given a solution to the word problem (by an oracle, say) then one can compute quite a lot of information. Daniel Groves and I proved that, in these circumstances, one can determine whether the group in question is free. Nicholas Touikan generalised this to show that one cam compute the Grushko decomposition.

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    $\begingroup$ 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. $\endgroup$
    – Andy Putman
    Commented Aug 16, 2010 at 15:24
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    $\begingroup$ 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. $\endgroup$
    – Joel David Hamkins
    Commented Aug 16, 2010 at 15:46
  • $\begingroup$ Andy - good point! $\endgroup$
    – HJRW
    Commented Aug 16, 2010 at 16:03
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    $\begingroup$ 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. $\endgroup$
    – Joel David Hamkins
    Commented Aug 16, 2010 at 17:29
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    $\begingroup$ 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 $\endgroup$
    – HJRW
    Commented Aug 16, 2010 at 20:57
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If you have a finite presentation of a group $G$, then you can easily derive its abelianization $G^{ab}$ from that, by standard techniques: Basically, you create an integer matrix, with one column for each generator, and one row for each relation, noting in entry $a_{ij}$ how often the generator $g_j$ occurred in the relator $r_i$. Then, compute the Smith normal form of this and you can read of the isomorphism type of $G^{ab}$.

If this happens to be non-trivial, you immediately have a proof that your group is non-trivial. Of course, the converse fails.

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Given a group $\Gamma$ which you are interested in, given by a presentation, and an easy group $G$ which you understand (a dihedral group or a symmetric group, maybe), you can often use the presentation to determine whether or not there exists a surjective homomorphism $\Gamma\to G$, and maybe you can even count them. This is a strong tool for showing that two given presentations give rise to different groups, and for showing that your group must be "at least as complicated" as $G$. For instance, Tietze originally proved in 1908 that the trefoil is knotted by exhibiting a surjection from a Wirtinger presentation of the fundamental group of its complement onto the symmetric group $S_3$, while the fundamental group of the unknot is abelian and so can admit no such homomorphism.

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  • $\begingroup$ Presumably you mean surjective homomorphism? $\endgroup$
    – Qiaochu Yuan
    Commented Oct 21, 2012 at 6:11
  • $\begingroup$ Of course. Sorry. $\endgroup$
    – Daniel Moskovich
    Commented Oct 21, 2012 at 6:22
  • $\begingroup$ By counting homomorphisms to symmetric groups up to $S_n$, you can count the subgroups of index up to $n$. $\endgroup$
    – Douglas Zare
    Commented Oct 21, 2012 at 6:32
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Continuing the idea that working with bare presentations is "hard", automatic groups sometimes can give you a handle, if you're trying to study a specific group . Automatic groups are groups with finite state machines that can, essentially, solve the word problem for that group.

If you're studying a particular group and are lucky, a procedure such as Knuth-Bendix can compute an automatic structure for you. Then lots of hard computations become easy (e.g. the order of the group).

Magma has some of these algorithms implemented, see this Magma documentation page.

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