**Hyperbolic Groups**

The next thrust of the course is *hyperbolic groups*. We care about hyperbolic groups for four reasons:

- In many models of “random groups,” groups are hyperbolic with probability 1. For example, under the few-relators model of random groups, where we pick relators at random given a set of generators, as the allowed length of relators goes to , the probability of the group generated being hyperbolic goes to 1.
- All free groups and finite groups are hyperbolic.
- All hyperbolic groups are automatic.
- Hyperbolic groups have an exceptionally nice solution to the word problem.

We will now give the definition of a hyperbolic group, acknowledging that there are two words that we have not previously defined.

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

From the motivating facts above, one might think that it is very difficult to find groups that are not hyperbolic! This is in fact not the case. For exception, any group which contains (including itself) is not hyperbolic. Additionally, the Baumslag–Solitar groups are also not hyperbolic. We don’t have the tools yet to show that these are not hyperbolic, they just serve as examples to show that not every group we know about falls under this new categorization.

Next, we will discuss isometries and quasi-isometries, which we will need when thinking about hyperbolic spaces.

**Isometries**

We begin by reviewing the definition of a metric space.

**Definition. **A *metric space* is a set along with a *metric* on , such that the metric satisfies three conditions:

- if and only if ,
- , and
- .

Then,

**Definition**. An *isometry* is a map from one metric space to another (or itself) that preserves distances. That is,

such that .

Additionally,

**Definition.** Two spaces, and , are *isometric* if there exists a surjective isometry from to .

Let us now consider a couple of examples. We know that ( with the Euclidean metric) is a metric space. Isometries of to itself include reflections, rotations, and translations. Non-Isometries of to itself include projection onto a line, shearing, and scaling since none of these preserve distance between points. Consider as well the map from to defined by While this map is an isometry, the two spaces are not isometric since the mapping is not surjective and no such surjective mapping exists. While it “feels” right that the real number line and the Euclidean plane are not isometric (they have different dimensions for starters), actually showing that they are not isometric is much more difficult and beyond the scope of this blog post.

Our overall goal is to define some sort of relationship so that a group would be “equivalent” to its Cayley graph. How to go about doing this is not immediately obvious, since is a group that does not depend on the choice of generating set and the Cayley graph is a graph that does depend on the choice of generating set. In order to do this we will massage the definition of an isometry and define a *quasi-isometry* and what it means for two spaces to be *quasi-isometric*. This is a common practice in math: if a definition isn’t working the way you want it to, loosen the definition slightly and hope that you aren’t loosening it so much that it is no longer useful.

**Quasi-Isometries**

Many of the spaces we will be interested in studying will be graphs, thus it will be helpful to define a metric for graphs before diving into the heart of quasi-isometries.

Let be a graph and let us induce a metric on by path length. We can think of each edge as an isomorphic copy of the interval . We want to think of the edges like this so that we can define any point between two vertices. For any given path between two vertices, we will set the starting vertex as zero and then build the path up, with each vertex we pass adding a total of one to the path and following edges only partially resulting in adding the corresponding fraction of one to the path. We then define the metric to be the infimum of the set of path lengths of all possible paths between and . Checking that is in fact a metric is straightforward and left as an exercise to the reader. Let us now formally define *path*.

**Definition.** A *path* of length is an isometric embedding of on .

The paths we will care most about are those of minimum length:

**Definition.** A *geodesic* is a path of minimum length. Additionally, a metric space is called *geodesic* if all pairs of points can be connected with a geodesic.

For example, and with the path metric are both geodesic. However, if we remove the origin from we no longer have a geodesic. Consider a point on the negative x-axis and a point on the positive x-axis. In order to connect them with a path of minimum length we would need to go through the origin. Given any path that went around the origin, we would always be able to find a shorter path that got closer to the origin and would thus be shorter. Hence, the space is not geodesic.

We have now formally established graphs as metric spaces and can consider their isometries. Recall that our goal was to find some sort of equivalence between a group and its Cayley graph (with respect to some generating set). We know that this relationship cannot simply be being isometric, since is always countable and is always uncountable (every edge is isometric to which is itself uncountable). Thus, there will be no surjective mapping from to . We will fix this problem by loosening our definition of isometry, beginning with the subjectivity requirement.

**Definition.** A *net* in a metric space is a subset such that for some and being finite.

Essentially, a net is a subset of points in the metric space which are spread out enough around the space, so that if we blur the edges of the net (take some finite neighborhood around each point in ), we get the whole set . Hence, it doesn’t make sense to talk about the net of a finite set, since we can simply pick any point that pleases us and then set to be large enough to cover the whole set .

The idea of a net allows us to define a mapping from to where is the set of vertices of the Cayley graph and thus is a net is . However, now we don’t have inverses. We fix this by loosening our definition in a slightly different way:

.

This statement says that the distance under the mapping is “close enough” to the distance in the original metric space where there is some sort of fudge factor which works for all pairs of points .

Finally, using this additive fudge factor and adding in a multiplicative fudge factor to account for variations between generating sets of groups, we get our definition of quasi-isometric and quasi-isometry.

**Definition.** If are metric spaces, is *quasi-isometric* if

.

Furthermore, if is a net in , then is a quasi-isometry.

Notice that this definition doesn’t quite mirror the definition of isometry/isometric. In fact, the distance-preserving relationship falls under quasi-isometric and isometry and the additional requirement of being surjective-(ish) falls under being isometric and being a quasi-isometry.

This achieves our goal! There is a quasi-isometry between a group and its Cayley graph, and two Cayley graphs of the same group are quasi-isometric. Next time we will show that being quasi-isometric is an equivalence relation on metric spaces.

**Sources:**

Great post Michaela! As MurphyKate mentioned, we’ve decided to skip some backgrounding on hyperbolic geometry, but since my final project topic is something to do with it, I have some fun facts and sources to share. Fun fact! There are no squares in hyperbolic geometry. There is a ‘biggest’ triangle in hyperbolic geometry!

I’ve linked a short video I thought was cool

https://www.youtube.com/watch?v=zQo_S3yNa2w

Great post Michaela, really clear! It’s intriguing how we allow ourselves to squint our eyes at all these structures in order to make the theorem work. From one hand, it seems a bit unscientific, but it also works and helps us describe very valuable ideas. I guess this is also how physics/chemistry work with classical and quantum scales embedded in the science. I’ll think about this some more.