The ten most fundamental topics in geometric group theory

Question

What are the ten most fundamental topics in geometric group theory?

This is a pedagogical question prompted by the fact that I am teaching geometric group theory to undergraduates. They are expected to know already know “group theory, euclidean and hyperbolic geometry, fundamental group and covering spaces”.

The words "ten most" are there to make the question appealing, and to give it a bit of structure. If you think that there are only three "most fundamental" topics (or, conversely, 15 topics whose relative importance are impossible to measure) then that's fine with me - I am very interested to hear your opinion. [Since there is no "one correct answer" I've asked the mods to make this question CW.]

The most relevant posts here on MO are perhaps this list of books and this discussion of how to learn about $\mathrm{Out}(F_n)$.

Answer 1

Here is my take. Unlike Andy, I would not structure such a course around big theorems. In part, this is because your students simply do not have enough background to handle any "big theorems." Instead, I would try to emphasize "small things" and connections of geometric group theory with various areas of mathematics. Item 2 is not, strictly speaking, geometric, but, IMHO, belongs to any geometric group theory course.

1. Free groups and group presentations (likely covered in a basic algebraic topology class). Dehn problems in combinatorial group theory and undecidability results (without proofs).

2. Residual finiteness in general; residual finiteness of finitely generated subgroups of $SL(n,\mathbb Z)$ and statement (without a proof) for finitely generated general matrix groups. Application to decidability of the WP. Mikhailova example of undecidability of the membership problem for subgroups of $SL(4,\mathbb Z)$.

3. General mantra of "groups as geometric objects": Cayley graphs, Cayley complexes, quasi-isometries, MS Lemma. Surface groups and hyperbolic plane, abelian groups and Euclidean spaces.

4. Generalities of hyperbolic groups, isoperimetric inequalities, Dehn algorithm and decidability of the WP. Morse Lemma and quasi-isometry invariance of hyperbolicity.

5. Basics of small cancellation theory and why/when small cancellation implies hyperbolicity.

6. Quasiconvex subgroups and decidability of the membership problem.

7. Group actions on simplicial trees, amalgams of groups and relation to the Seifert - Van Kampen Theorem.

8. Ends of spaces as topological invariants and ends of groups as coarse geometric invariants. Statement of the Stallings Theorem: It is unlikely that you will have time for a full proof but you can give a sketch using group actions on trees.

Now (maybe even earlier!), you probably are out of time. If not:

9. Probabilistic aspects (assuming that your students took a basic discrete probability class). Gromov's density model and at least a statement of hyperbolicity of random groups with a sketch of a proof using small cancellation theory.

10. Instead of 9: Statement of the Mostow Rigidity Theorem and an outline of a proof using "zooming in" argument, blackboxing the required analytical details (do not even try to explain what quasiconformal maps are, just state the needed properties).

Answer 2

Geometric group theory is a huge subject, and a course that really tried to cover all of it would be too disjointed to be useful. If I were teaching such a course, I would choose a few major theorems and spend time developing all the tools you would need to prove and appreciate them. For instance, I think you could easily spend a semester focusing on:

1. Mostow rigidity, perhaps proved using Gromov's approach using the Gromov norm (which is probably more useful to the typical GGT student than the analytical tools needed for the original proof).

2. Gromov's theorem on groups of polynomial growth, which would also give you a nice opportunity to talk about large-scale geometry in general.

3. Stallings's theorem on ends of groups, together with some applications (my favorite would be the fact that groups of cohomological dimension one are free, but that might be too algebro-topological for you). This would require also developing Bass-Serre theory.

Three other topics that would make sense to include are hyperbolic groups, CAT(0) geometry, and the theory of cube complexes. But that would almost be a separate course, and would not have much overlap with the three big theorems above.