Motivation for Hyperbolic Groups - Soft Question

Question

I took a Geometric Group Theory course this semester. A very big part was hyperbolic groups. What I felt was a little bit lacking in this course was - why do I need hyperbolic groups? What is the motivation?

Answer 1

Here is an abbreviated answer:

1. Why do I need hyperbolic groups?

There is a good chance, you do not. For instance, if you are planning to work in commutative algebra or PDEs, most likely you will never see this concept used anywhere.

Nevertheless, the notion of hyperbolicity not just of groups but also of graphs and more general metric spaces turned out to be surprisingly useful in several area of mathematics, including applied math.

2. What is the motivation?

The general answer is that hyperbolic groups tend to be abundant among finitely presented groups (in particular, "random groups are hyperbolic" indeed). Hyperbolic groups are not "exotic" in any sense and are easy to construct using, for instance, small cancellation theory (this is what the last chapter of Ghys and de la Harpe explains). At the same, time, hyperbolic groups possess some desired properties that general finitely-presentable groups a lacking.

For instance, some of the first questions one asks about groups (and one sees these question quite frequently at MSE) are:

a) "How can I tell if the given word in a finitely presented group $G$ corresponds to the identity element of $G$?"

b) "How can I tell if two finitely presented groups are isomorphic?"

For general finitely-presented groups both problems (the "word problem" and the "isomorphism problem") are known to be undecidable and, hence, the answer to (a) and (b) is "you cannot!"

However, for hyperbolic groups (there are some details that I am omitting) both problems are decidable, although the first is much easier than the 2nd.

Another set of basic questions of group theory belongs to the circle of "Burnside problems:"

(i) If $G$ is finitely generated group where every nontrivial element has order $\le n$, does it follow that $G$ is finite? (ii) What about finitely presented groups?

Both are quite hard. For very small values of $n$ the answer to (i) is positive. But when $n$ is large, then the answer to (i) turns out to be negative. However, constructions are quite hard and they use some hyperbolicity ideas, even though first proofs predate the notion of hyperbolic groups.

On the other hand, if $G$ is hyperbolic then it is either finite or contains an element of infinite order. See if you can find a proof in the book you were reading.

The next question is a bit vague: Given a finitely-presented group, can I describe the entire group algorithmically?

I will not explain how to make this question precise, but hyperbolic groups (and some more general classes of groups) admit an automatic structure which answers this question affirmatively, making computations with hyperbolic groups possible.

Switching to a different field: It turns out that hyperbolic groups (and their close relatives) are quite useful for resolving some purely topological questions, e.g. some problems in 3-dimensional topology, some problems of higher dimensional topology (such as Novikov's conjecture, where the ability to attach a suitable "boundary" to a group is critical).

Here is a small example: Suppose that $M$ is a closed connected manifold, $f: M\to M$ is a degree 1 map. Is $f$ a homotopy-equivalence?

At the first glance, this sounds ridiculously unlikely. The question, going back to Hopf, is wide-open in dimensions $\ge 4$. However, if you assume that $M$ is aspherical and has hyperbolic fundamental group, then the answer is positive. You can even drop the asphericity assumption and conclude that $f$ at least induces an isomorphism of fundamental groups.

Lastly, regarding your question in a comment:

3. "What the hyperbolicity of the fundamental group of a topological space says about the space itself?"

Without further clarification, not much. But you get a lot of information if your manifold is 3-dimensional or if it is higher-dimensional, closed and aspherical. I will leave it at that.

Here is one book you may want to take a look at to understand importance of hyperbolicity in 3-dimensional topology:

Aschenbrenner, Matthias; Friedl, Stefan; Wilton, Henry, 3-manifold groups, EMS Series of Lectures in Mathematics. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-154-5/pbk). xiv, 215 p. (2015). ZBL1326.57001.