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. This would provide a rather satisfactory conclusion: In the beginning (item 1) they learn that "nothing can be done" but at the end of the course (item 9), they learn that "everything can be done asymptotically almost surely." Just as in Hegelian "Logic."

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

I don't have 10 things for your list, but I can describe the syllabus of the Introduction to Geometric Group Theory Masters' course I have taught for the last couple of years. I hope this is worth mentioning, because it takes a different tack to the other two answers.

The idea is very much not to comprehensively cover all the main topics of the subject, nor to focus on "big theorems". Rather, the course motivates the kinds of problems that interested Dehn and his contemporaries, and shows how geometric techniques can be used to solve them.

The course consists of 24 50-minute lectures, and the syllabus is as follows.

I COMBINATORIAL GROUP THEORY

Free groups and presentations. Historical case study: Dehn's construction of infinitely many 3-dimensional homology spheres. (This motivates the question: How can we distinguish them? Dehn's examples are Seifert-fibred spaces with triangular base orbifolds.) Van Kampen diagrams.

II BASICS OF GEOMETRIC GROUP THEORY

Cayley graphs. The Schwarz--Milnor lemma. Case study: free groups. (We return to free groups and study them via free actions on trees.)

III BASS--SERRE THEORY

Amalgamated free products. HNN extensions. Graphs of groups. The Bass--Serre tree. Property FA. (Dehn's examples have FA, so Bass--Serre theory won't help us with them!)

IV FUCHSIAN GROUPS

Hyperbolic geometry. Examples of Fuchsian groups. (We prove Poincaré's polygon theorem, to see that hyperbolic triangle groups are Fuchsian.) Centres and Dehn's examples. (We apply our understanding to Dehn's examples, and conclude that they are all different. The point is that their quotients by their centres are all triangle groups with different orders of torsion.)

V HYPERBOLIC GROUPS

Hyperbolic metric spaces. The Mostow--Morse lemma. (This seems to be the correct name for what is often called the Morse lemma.) Hyperbolic groups. Local geodesics. Dehn's algorithm.

Answer 3

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.

Answer 4

Here is a version of the table of contents of Bowditch's lecture notes A course on geometric group theory.

1. Group presentations, free groups, abelianisation.

2. Cayley graphs.

3. Quasi-isometries and their invariants.

4. Fundamental groups, covering spaces.

5. Hyperbolic geometry, the plane, the space, surfaces, three-manifolds.

6. Hyperbolic spaces, trees, the four-point condition, exponential growth of distance, quasi-geodesics, Hausdorff distances, qi-invariance, hyperbolic groups and their properties.

7. Isoperimetric functions, linear bounds and hyperbolicity, qi-invariance, Dehn functions, the word problem.