An introduction to combinatorial and geometric group theory

Section 2.2 Can anything be expressed by quasi isometry?

Two metric spaces are the same if they are isometric. In the previous section we saw although there is no canonical word metric on a group, if we consider metric spaces only up to quasi isometry, then we do obtain something canonical: namely the quasi isometry class of word metrics coming from finite generating sets. The issue then becomes that maybe quasi isometry is too vague and that all groups are quasi isometric. For quasi isometry to be useful, it must be able to tell groups apart.

All finite groups are quasi isometric to the metric space consisting of a single point, so this viewpoint is not useful for the study of finite groups. We also see that quasi isometry is able to distinguish finite groups from infinite groups, but that isn't particularly impressive.

In this lecture we will show that the groups \(\ZZ\) and \(\ZZ^2\) are not quasi isometric and we will prove Theorem 2.2.5, the Svarc-Milnor Lemma, which has far reaching consequences such as the fact that if \(K \leq H\) is a finite index subgroup then \(H\) and \(K\) are quasi isometric. This gives us a limitation of how much can be determined by quasi isometry.

Let \(x\) be a point in a metric space \((X,d)\text{,}\) let \(r\gt 0\) be some number, then we define the closed ball of radius \(r\) about \(x\) to be the set

When it is clear from the context in which metric space a given ball lies, we will not bother explicitly giving the metric.

It is worth recalling from Exercise 1.5.1.4 that for any ball about the identity in the standard Cayley graph of \(\ZZ^2\text{,}\) we can find a generating set of \(\ZZ\) which gives the exact same ball about the identity. In other words, at small and medium scales word metrics will not distinguish \(\ZZ\) and \(\ZZ^2\)

The proof of the following is known as an asymptotic argument.

So at least now we have established that quasi isometry is maybe not a completely useless notion as it can tell at least two groups apart. Quasi isometry, however, does have limitations and we will now explore these.

From now on we will asssume our spaces \(X\) are path connected metric spaces, the readier at this point is free to assume that \(X\) is a connected graph, all of whose edges have a given length (usually they have unit length). A path is a continuous mapping

and we can define its speed at \(s\) as

If \(X\) is a graph, viewing edges as isometric copies of intervals of \(\RR\) we can make sense of this for almost all values of \(s \in [0,L]\text{.}\)

Integrating speed gives arc length and we say that \(p\) is arc length parameterized if \(|\dot p(s)| =1\) for all \(s \in [0,L]\text{.}\) We say that a path is a geodesic if it is a shortest path between its endpoints \(p(0)\) and \(p(L)\) and we can define the path metric to be given by the minimal length of all paths joining two points. A metric space is called a geodesic metric space if its metric coincides with the path metric.

We assume the reader knows what a left group action

on a space is. We will use the \(\cdot\) symbol to denote the action of a group element on a point. We say that \(G\) acts by isometries on \((X,d)\) if for every \(g \in G\) and every \(x,y \in X\) we have:

For our current purposes it will be sufficient to only consider our metric spaces to be path connected graphs, but the results we state will be be more general. For our purposes a compact set is a finite subgraph. An action by isometries is an action by graph automorphisms. The action of \(G\) on a graph \(X\) is proper if the stablizer of any vertex is finite. The quotient \(G \backslash X\) of an action has vertices that conist of equivalence classes

and edges

and is itself a graph. The action of \(G\) on \(X\) is cocompact if \(G\backslash X\) is finite and a coarse fundamental domain is compact subset \(K \subset X\) whose \(G\) translates cover \(X\text{,}\) i.e.

With all this terminology out of the way, we can start proving results.

We will first state a useful result without proof. It follows from a careful consideration of the definitions.

The proof above illustrates the potent combination of a proper, cocompact action by isometries. Such an action is typically called a geometric action. The lemma above gives an unexpected consequence:

Recall that the quasi isometry class of a group \(G\) is the smallest quasi isometry class which contains all word metrics on \(G\text{.}\) In particular the statement below makes complete sense without having to specify some metric on \(G\text{.}\)

First note that by hypthesis \(G\cdot p_0\) is coarsely dense in \(X\) and in fact quasi isometric to \(X\text{.}\) So we need need only show that \(G\) is quasi isometric to \(G \cdot p_0 \subset X\text{.}\) We will show that the following map

\begin{align*} f:G \amp \to p_0\\ g \amp \mapsto g\cdot p_0 \end{align*}

is a quasi-isometric embedding. Since it is surjective the result will follow.

By Lemma 2.2.3, Item 3 \(S = R_{r_0}\) is a finite generating set of \(G\text{,}\) therefore(by Lemma 2.2.3, Item 3) so must be \(A = R_{3r_0}\text{.}\) We equip \(G\) with the finite generating set \(A\text{.}\) Note that for each \(a \in A\) we have

\begin{equation*} d_X(p_0,a\cdot p_0) \leq 6\cdot r_0 \end{equation*}

So if \(g = a_1\cdots a_n\) is a product of minimal length representing \(g \in G\text{,}\) i.e. \(d_A(1,g)=n\) then considering the broken path:

we have by the triangle inequality that

\begin{equation*} d_X(p_0,g\cdot p_0) \leq 6\cdot r_0 d_A(1,g), \end{equation*}

which is one inequality we need to show.

We now need to bound \(d_A(1,g)\) above by some linear function of \(d_X(p_0,g\cdot p_0)\text{.}\) This follows by noting that every point of \(X\) is a distance at most \(r_0\) from some point in \(G\cdot p_0\) ( Lemma 2.2.3, Item 1) and that every point in \(v\in G \cdot p_0 \cap B(p_1,3r_0)\) there is some \(a\) such that \(a \cdot p_0 = v\text{.}\) The picture below which is obtained by dividing the shortest path from \(P_0\) to \(g\cdot P_0\) into a minimal number of segments of length at most \(r_0\) implies that \(d_A(1,g) \leq \frac{1}{r_0}d_X(p_0,g\cdot p_0)+1\text{.}\)

At this point showing that \(f\) is a quasi isometry is straightforward.

From this we get the following, which gives us the ultimate limitiation of quasi-isometry

The result above seems trivial for the following reason. If \((X,d)\) is a metric space and have a subset \(S \subset X\) and we equip \(S\) with the subspace metric \(d|_S\text{,}\) then \(S\) is isometrically embedded into \(X\) with respect to this metric. In particualr if \(S\) is coarsely dense in \(X\text{,}\) as is the case with a finite index subgroup, then \((S,d|_S)\) will be quasi isometric to \((X,d_X)\text{.}\) Let us call this the extrinsic metric on \(S\) from the embedding \(S\subset X\text{.}\) Above corollary is stronger than that. In particular any finite index subgroup of \(H\) is finitely generated so a finite generating set \(\gen A = H\) endows \(H\) with an intrinsic quasi isometry class.

The above corollary states that if \(H\leq G\) is a finite index subgroup of of a finitely generated group. Then \(H\) and \(G\) are quasi isometric with respect to their respective intrinsic quasi isometry classes.

Now it may happen that we have an injective homomorphism \(H \into G\) where both \(H\) and \(G\) are finitely generated but given aword metric \(d\) on \(G\) the induced extrinsic metric \(d|_H\) induced on \(H\) is not quasi isometric to any (intrinsic) word metric on \(H\text{.}\) In this case we say \(H\) is a distorted subgroup. We illustrate this with an example

\(BS(1,2)\text{,}\) the Baumslag-Solitar group is a weird group. It has the following presentation:

\begin{equation*} BS(1,2) = \pres{a,t}{tat^{-1}a^{-2}}. \end{equation*}

Notice that the single relation gives the following commutation relation:

\begin{equation*} ta = a^2t \end{equation*}

which means that every element can be expressed as a word of the form:

\begin{equation*} a^nt^m; n,m \in \ZZ. \end{equation*}

So far that doesn't seem weird, but its Caley graph looks like this:

.

Consider the homomorphism \(\phi:\ZZ \into BS(1,2)\) given by \(\psi(n) = a^n\text{.}\) In the next chapter we will see that \(\phi\) is injective. For now we can see this as \(\gen a \leq BS(1,2)\) as being a copy of \(\ZZ\) sitting in \(BS(1,2)\text{.}\)

We can now compare the standard metric \(d_\ZZ\) on \(\ZZ\) to the extrinsic metric \(d|_{\gen a}\) induced by its inclusion into \(BS(1,2)\text{.}\)

On the one hand we have \(d_\ZZ(1,a^n) = |n|\text{,}\) which is straightforward. Now the relation lets us rewrite, for example,

\begin{equation*} a^2 = tat^{-1} \Rightarrow a^{2^3} = aaaaaaaa = tttat^{-1}t^{-1}t^{-1}. \end{equation*}

More generally we have \(a^{2^n} = t^nat^{-n}\text{,}\) so on the other hand we have two different metrics:

\begin{equation*} d_\ZZ(1,a^{2^n}) = 2^n, d|_{\gen a}(1,a^{2^n})=2n+1. \end{equation*}

In other words this embedding \(\phi:\ZZ \into BS(1,2)\) exponentially distorts the intrinsic metric on \(\ZZ\text{.}\) In particular this embedding is not a quasi isometric embedding, as the latter only allows linear distortion.

We will say that groups \(G\) and \(H\) are abstractly commensurable if they have finite index subgroups \(K\leq G\) and \(K'\leq H\) which are isomorphic, i.e \(K \approx K'\text{.}\) Corollary 2.2.6 tells us that commensurable groups are quasi-isometric therefore we have the limitation that quasi isometry cannot distinguish between groups in a commensurability class.