Topics in Combinatorial Group Theory (for a short talk).

Combinatorial group theory is a large subject, and is somewhat ill-defined (it is thus called because it uses combinatorics heavily, or because it utilises a combination of techniques? I have heard both argued...). For the purposes of this answer, I shall take it to mean "the study of abstract, discrete infinite groups and their presentations". By discrete, I simply mean we are not considering any topology on the groups. Since the 1980s the more trendy subject of "geometric group theory" has been leading the way, but these subjects are broadly similar and the symmetric difference of the practitioners is very small (by my own observations - please feel free to disagree!).

Dislaimer: I should say that I cite quite a few of my own posts in this post. That is because I am talking about stuff I find interesting so have posted quite a lot of stuff on Math.SE on this!

When working on a talk it is often useful to know the history of the subject you are working on. I stumbled across a wonderful article on the pre-70s history of combinatorial group theory (and really pre-70 is what you are after!) called Some groups I have known by B.H.Neumann. (It is worth pointing out that the "Hanna" he makes reference to is his late wife. Both B.H. and Hanna were formidable group theorists (they are the N and N in HNN-extensions), and two of their sons, Walter and Peter Neumann, have justifiably similar reputations.)

Some Classical problems you may find interesting

Broadly speaking, there were four questions which drove combinatorial group theory in the early days. The first is called Burnside's problem, the last three are Dehn's problems (the word, conjugacy and isomorphism problems).

The Burnside problem A group is called torsion if every element has finite order. Burnside's problem asks if it is possible for a finitely generated infinite torsion group to exist. Note that without the "finitely generated" the question has an obvious answer: just take the direct sum of infinitely many finite groups. The wikipedia article is pretty good on this, and I am not sure that I have anything to add. I would like to point out that 1) a fields medal has been awarded for work on the modified version of this problem 2) the current prize for "best example" goes to Ol'shanskii, and 3) Novikov and Adian (pronounced "Adyyyyyyaaan") play important roles in the story of both this problem and Dehn's problems. Finally, all known examples are infinitely presented. Find a finitely presented infinite torsion group and people will know your name...

Dehn's problems I wrote a long post on Dehn's problems here.

Some topics which I would speak on if I was in your place

If I wanted to give a talk on combinatorial group theory, but didn't just want to point out that free groups have soluble word problem (say what now?!?), I would give an introduction to one of the following areas. But I am not you. These are just some ideas for you to ponder...

Non-Hopfian groups A group is Hopfian if every surjective endomorphism is an isomorphism, so a non-Hopfian group is a group which has a surjective endomorphism which has a non-trivial kernel. For example, consider the infinite (countable) free group and then kill a generator. The resulting group is still infinite (countable) free, so this group is non-Hopfian. A more subtle example is the so-called Baumslag-Solitar group $BS(2, 3)=\langle a, b; b^{-1}a^2b=a^3\rangle$. I wrote a lengthy post on these groups here, and an excellent, and very old, article which talks about their early history is A two-generator group isomorphic to a proper factor group by B.H.Neumann (this is pre-Baumslag-Solitar's example).

van Kampen diagrams van Kampen diagrams give a way of visualising the word problem. If all diagrams satisfy certain conditions (called the small cancellation conditions) then your group has soluble word and (if the conditions are strong enough) conjugacy problem. The classical reference is the last chapter of Lyndon and Schupp's book Combinatorial group theory. See also this post of mine.

Fundamental groups of graphs The fundamental group of a graph is a free group. If you are comfortable with covering spaces and so other stuff from topology then you can prove some remarkable results about free groups by thinking of them as fundamental groups of graphs. You would want to look up something called "Stallings' foldings". The original paper is The topology of finite graphs by J.Stallings, and you will also find Allan Hatcher's book Algebraic Topology helpful. See also this post of HJRW.

Bass-Serre theory Bass-Serre theory is the theory of groups acting on trees. HNN-extensions and free products with amalgamations all act on trees, and if you have ever heard of a "graph of groups" then it is an object from this theory. Bass-Serre theory is extremely powerful, and a wonderful introduction to group actions. The classical text is the book Trees by Serre, although there is a modern book called Groups, graphs and trees by John Meier which is much more readable.

One-relator groups A one-relator group is a group with a presentation of the form $G=\langle X; R^n\rangle$, $n\geq 1$ and $R$ not a proper power of any element. It has torsion if and only if $n>1$, has soluble word problem, every torsion element is conjugate to a power of $R$, and if it is freely indecomposable then every proper subset of $X$ generates a free group. All of these wonderful results can be proven using a standard method, called Magnus' method. The idea is to induct on the length of the word $R^n$. If you have Magnus, Karrass and Solitar's book then the proofs are tricky, employing groups with "staggered" presentations. A chap called Moldovanskii realised that you could use the same idea, but instead embed $G$ into an HNN-extension of a group with a shorter defining relation (this is substantially neater). The best references I have found are an article of Fine and Rosenberger The Freiheitssatz and its extensions in a book called The mathematical legacy of Wilhelm Magnus, and a paper of McCool and Schupp On one-relator groups and HNN-extensions.

The word problem for the fundamental groups of closed, orientable surfaces When Dehn posed his three fundamental problems of group theory he proves that the fundamental groups of closed, orientable surfaces of genus at least two have word problem soluble by "Dehn's algorithm" (not that he called it that, mind!). His proof led to the development of small cancellation theory (mentioned in the "van Kampen diagram" paragraph, above) and ultimately to the development of hyperbolic groups and all that they lead to (I talk more about this in my post on Dehn's problems which I mentioned nearer the top). Because of this pedigree, it would be interesting to go through these proofs and see if you could present them in your talk. I do not know if this is possible, but certainly if I was going to do a talk next week this would be my subject of choice: explain Dehn's original proof! (Or give a modern proof of it. Or something.)

Anyway, I hope you found this all useful. You really should speak to your professor about this, but also you want to think about it by yourself. You want to find something which you find interesting, not just something your professor thinks is "at about your level..."!

I would add to the list some of the problems from identities among relations. (A good article to consult is

J.-L. Loday, 2000, Homotopical Syzygies, in Une dégustation topologique: Homotopy theory in the Swiss Alps, volume 265 of Contemporary Mathematics, 99 – 127, AMS.

The visual element is good so it makes for a nice presentation. The idea is very simple but then its development requires a bit of care, so again a nice topic to handle. The subtopic of Pictures is also neat.