Eighth Monday: Geometric Actions and a Schwarz Lemma

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 $\delta$ -hyperbolic space.

We’ve established what it means for a space to be $\delta$ -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 $d_X$ on $X$ , for all $x, y \in X$ and $g \in G$ it’s the case that

$d_X (x, y) = d_X (g \cdot x, g \cdot y).$

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

Definition. A group $G$ acts properly discontinuously on a space $X$ if for any ball of radius $r$ around a point $x \in X$ , then

$|\{g \in G \big| gB \cap B \neq \emptyset\}| < \infty$

Definition. A group $G$ acts cocompactly if the quotient $X/G$ is compact. This is a topological notion, and for the purposes of our class, it will suffice to show that every orbit $G_x$ forms a net in $X$ , 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 $G$ and a metric space $X$ , suppose $G \curvearrowright X$ geometrically. Then for all $x \in X$ , the function

$f: G \to X$

$f: g \mapsto g \cdot x$

is a quasi-isometry.

Proof. By assumption of the action of $G$ being geometric we know that $f(G)$ forms a net in $X$ , and by definition of being a net there is some $R$ such that $N_R (G \cdot x) = X$ . Define the set $S$ as

$S := \{g \in G \big| gN_{2R+1} (x) \cap N_{2R+1} (x) \neq \emptyset \}.$

We will seek to show that $\langle S \rangle = G$ . As a consequence of the action of $G$ being geometric we know the action must be properly discontinuous, and so $S$ must be finite. Consider an arbitrary $g \in G$ and $x \in X$ , and let $\gamma$ be a geodesic connecting $x$ to $g \cdot x$ . We’ll parameterize $\gamma$ into some finite number $L$ many segments $i$ , with $\gamma(i) = x_i$ . Since $f(G)$ forms a net, there’s some $R$ such that for all $i$ , $\gamma(i) \in N_R (g_i \cdot x)$ .

Significantly, we can pull from this relationship that $d(g_i \cdot x, g_{i + 1} \cdot x) \leq 2R + 1$ , because $g_i \cdot x$ and $g_{i + 1} \cdot x$ can both differ from $x$ by at most $R$ , plus 1 to represent going through $x$ . Consider the picture:

We can take this relationship and multiply by $g_{i+1}^{-1}$ for

$d(g_{i + 1}^{-1} g_i x, x) \leq 2R + 1,$

which implies that $g_{i + 1}^{-1} g_i \in S$ . Fortuitously, we may apply this process over the entirety of $\gamma$ , since $g_k = (g_k g_{k - 1}^{-1})(g_{k - 1} g_{k - 2}^{-1}) \ldots (g_1 e)$ :

$d \big( (g_{L + 1} g_L^{-1})(g_L g_{L - 1}) \ldots (g_2 g_1^{-1})(g_1 e_1^{-1}) \cdot x \big) = d(g \cdot x) \leq 2R + 1.$

Since our choice of $g$ was arbitrary, this shows that all $g$ are in $S$ , and since elements of $S$ are defined as elements of $G$ , it must be the case that $\langle S \rangle = G$ .

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 $g, h \in G$ be arbitrary . Note that by multiplying by $g^{-1}$ we have

$d(gx, hx) = d(x, g^{-1}hx).$

Great news! Since $g$ and $h$ are arbitrary, $g^{-1}h$ spans all of $G$ , so it’s equivalent to consider bounds on $d(x, g \cdot x)$ for some new, arbitrary $x \in G$ . Let $\gamma$ be a geodesic from $x$ to $g \cdot x$ . Then

$|\gamma| + 1 = d_x(x, g \cdot x) + 1 \geq d_S(e, g),$

since $G$ acts by isometries. We see from the above that it suffices for the lower bound of our purported quasi-isometry to let $K = \epsilon = 1$ . We’ll now seek an upper bound. Suppose $d_S(e, g) = k$ , and thus $g = s_1 s_2 \ldots s_k$ where $s_i$ are letters on $G$ . As per the graphic, we can subdivide the geodesic from $x$ to $g_x$ into $k$ -many pieces, each of which must have length less than or equal to $2(R + 1)$ .

The total distance of the path is $d_S (e, g) \cdot (4R + 2)$ . Since $G$ acts isometrically, this means that the inequality $d_X (x, gx) \leq d_S (e, g) \cdot (4R + 2)$ implies that the upper bound is $d_X(x, gx)$ times $4R + 2$ , which becomes our choice of $K$ .

The total distance of the path is $d_S (e, g) \cdot (4R + 2)$ . Since $G$ acts isometrically, this means that the inequality $d_X (x, gx) \leq d_S (e, g) \cdot (4R + 2)$ implies that the upper bound is $d_X(x, gx)$ times $4R + 2$ , which becomes our choice of $K$ .

This gives us our upper bound as $(4R + 2, 0)$ . Selecting the highest values of $K$ and $\epsilon$ from our lower and upper bounds, we conclude that $f: G \to X$ is a $(4R + 2, 1)$ 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 $\simeq$ to indicate quasi-isometries.

Corollary. For a group $G$ and spaces $X$ and $Y$ , if $G \curvearrowright X, Y$ geometrically, then $X \simeq Y$ .

Corollary. If groups $G, H$ act on a space $X$ geometrically, then $G \simeq H$ .

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

Corollary. If $H \leq G$ has finite index, then $H \simeq G$ .

Proof. We know $G \curvearrowright \Gamma_{G, S}$ geometrically, as does $H$ . Since the index of $H$ is finite with respect to $G$ , $H_x$ is a net.

Corollary. If $n \geq 2$ , then $F_2 \simeq F_n$ . This is unsettling!

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

Theorem. For a group $G$ and a space $X$ , the following are equivalent:

1) G is hyperbolic;

2) $G \curvearrowright X$ geometrically implies that $X$ is $\delta$ -hyperbolic;

3) $\Gamma_{G, S}$ is $\delta$ -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( $\mathbb{H}^n$ )

The following are some non-examples of hyperbolic groups:

$\mathbb{Z}^2$
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 $G = \langle G \big| R \rangle$ is a presentation such that

1) for all $r \in R$ , there exist finitely many ways to write $r = uv$ such that $|u| > |v|$ ;

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

Theorem. If $G$ is hyperbolic, then $G$ 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 $X$ , if $X$ is $\delta$ -hyperbolic and $\gamma$ is a path in such that every subpath of length $\leq 8 \delta$ is geodesic, then $\delta$ 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 $\delta$ -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.

3 Responses to Eighth Monday: Geometric Actions and a Schwarz Lemma

Osip Surdutovich says:

November 3, 2022 at 7:07 am

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 $F_2$ and $F_n$ are isomorphic and yet $F_2$ contains a copy of the other! Also really amazing how we can find out so much about a structure from a metric space being $\delta$ -hyperbolic.

Akash Ganguly says:

November 7, 2022 at 1:34 am

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!

Michaela Polley says:

November 10, 2022 at 5:25 am

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!

3 Responses to Eighth Monday: Geometric Actions and a Schwarz Lemma

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