On this fine Halloween, we discussed what it means to act geometrically, vindicating our work on hyperbolicity with a discussion of the Schwarz Lemma. First, recall the definition from last week:

**Definition.** A group *G* is *hyperbolic* if it acts geometrically on a -hyperbolic space.

We’ve established what it means for a space to be -hyperbolic, but what is a geometric action?

**Definition.** A group *G* acts on a metric space *X* *geometrically* if the following are true:

1) *G* acts by isometries;

2) *G* acts properly discontinuously;

3) *G* acts cocompactly.

This is all well and good, but what do these conditions mean? Fortunately, we have the tools to answer these questions.

**Definition.** A group *G* *acts by isometries* on a metric space *X* if given a metric on , for all and it’s the case that

We see that this isn’t too unfamiliar a notion; we’re basically just asserting that an action of corresponds to an isometry on the space.

**Definition. **A group acts *properly discontinuously* on a space if for any ball of radius around a point , then

**Definition.** A group *acts cocompactly* if the quotient is compact. This is a topological notion, and for the purposes of our class, it will suffice to show that every orbit forms a net in , although this is technically not the same.

All of three conditions of what it means for an action to be geometric seem to enforce some sort of “nice behavior.” But what can we pull out from these assumptions? As it happens, quite a bit.

**Theorem.** (Schwarz’ Lemma.) For some finitely generated group and a metric space , suppose geometrically. Then for all , the function

is a quasi-isometry.

**Proof.** By assumption of the action of being geometric we know that forms a net in , and by definition of being a net there is some such that . Define the set as

We will seek to show that . As a consequence of the action of being geometric we know the action must be properly discontinuous, and so must be finite. Consider an arbitrary and , and let be a geodesic connecting to . We’ll parameterize into some finite number many segments , with . Since forms a net, there’s some such that for all , .

Significantly, we can pull from this relationship that , because and can both differ from by at most , plus 1 to represent going through . Consider the picture:

We can take this relationship and multiply by for

which implies that . Fortuitously, we may apply this process over the entirety of , since :

Since our choice of was arbitrary, this shows that all are in , and since elements of are defined as elements of , it must be the case that .

This is good! We are now ready to put a bound on subsets of our geodesic. The strategy we’re going to employ won’t necessarily work for non-hyperbolic spaces, but here we’ll be able to turn this statement about subpaths into a conclusion about least distances between arbitrary points.

Let be arbitrary . Note that by multiplying by we have

Great news! Since and are arbitrary, spans all of , so it’s equivalent to consider bounds on for some new, arbitrary . Let be a geodesic from to . Then

since acts by isometries. We see from the above that it suffices for the lower bound of our purported quasi-isometry to let . We’ll now seek an upper bound. Suppose , and thus where are letters on . As per the graphic, we can subdivide the geodesic from to into -many pieces, each of which must have length less than or equal to .

This gives us our upper bound as . Selecting the highest values of and from our lower and upper bounds, we conclude that is a quasi-isometric map. ⬛

This result is one of the big reasons we spent a week pushing through definitions for isometry and hyperbolicity. Having made it to the peak of this conceptual mountain, we may gaze on a valley full of beautiful corollaries. On this voyage through some implications, we will use to indicate quasi-isometries.

**Corollary.** For a group and spaces and , if geometrically, then .

**Corollary.** If groups act on a space geometrically, then .

These results follows from the fact that, as proven previously, quasi-isometry is an equivalence relation, and thus transitive.

**Corollary.** If has finite index, then .

**Proof.** We know geometrically, as does . Since the index of is finite with respect to , is a net.

**Corollary.** If , then . This is unsettling!

Next we stated another theorem that helps tie together the definitions we’ve been considering.

**Theorem.** For a group and a space , the following are equivalent:

1) *G* is hyperbolic;

2) geometrically implies that is -hyperbolic;

3) is -hyperbolic.

The following are examples of hyperbolic groups:

- every finite group
- free groups
- virtually free groups
- virtually hyperbolic groups
- subgroups of the isometry group of hyperbolic space Isom()

The following are some non-examples of hyperbolic groups:

- most Baumslag-Solitar groups

There are far fewer non-examples. As previously discussed, that’s because even though it took us a lot of machinery to see what we were looking at, hyperbolic groups are quite common, for fairly rigorous notions of “common.” As discussed on previous blog posts, “random” groups are almost always hyperbolic!

Last, we tied things up with some discussion of Dehn presentations.

**Definition.** A *Dehn presentation* for a finitely presented group is a presentation such that

1) for all , there exist finitely many ways to write such that ;

2) if is a word on representing the identity, then either isn’t reduced, or contains as a subword.

**Theorem.** If is hyperbolic, then admits a Dehn presentation.

This is quite a claim; we didn’t finish a proof. But to get the juices flowing, we did consider some thought-provoking examples, and set up a lemma.

**Lemma.** For a space , if is -hyperbolic and is a path in such that every subpath of length is geodesic, then has no loops.

Much like how our algorithm in the proof of the Schwarz lemma doesn’t always yield the *shortest* expression of a word, instead yielding a path which satisfies -slimness, the Dehn presentation isn’t the only or necessarily the most efficient way to present a given group. However, what these results suggest is that there’s a deep relationship between hyperbolicity — that is, with having upper bounds on the distance between points on paths — and the ability to bound the number of letters needed to express an arbitrary word on generators and relators.

Awesome post John! It’s really cool how we can relate hyperbolic groups back to our beloved free groups. And I agree with you that it is quite unsettling that and are isomorphic and yet contains a copy of the other! Also really amazing how we can find out so much about a structure from a metric space being -hyperbolic.

Great post as always John! The pictures really helped a lot. As I’m doing my final project draft, I came across some equivalent definitions of a properly discontinuous action. Thought I’d add them here!

1) The G orbit of any point is locally finite.

2) The G orbit of any point is discrete and the stabilizer of that point is

finite.

3) For any point, there is a neighborhood of that point, V , for which

only finitely many T ∈ G satisfy T(V ) ∩ V = ∅

I also think that a direct comparison with the euclidean plane at times reminds me how crazy some of these properties. For instance, the idea that every segment in delta hyperbolic space is somewhat close to a geodesic segment is a completely wild idea in euclidean space!

Great post, John! It was really fun to finally define and understand hyperbolic groups, having spent so much time setting up machinery. I also liked that we simply need to show that the Cayley Graph is hyperbolic and that that is enough to show that the group is hyperbolic!