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Section 2.1 Quasi isometries

Back in Section 1.5 we saw that a choice of generating set \(G = \gen A\) gave rise to a Cayley graph \(\cay A G\) which in turn endows \(G\) with a metric \(d_{\cay A G}\text{.}\) However, the subject matter in the previous sections should have reinforced the fact that there should be no expectation that such a metric be canonical for the group. A surprisingly successful strategy has been to simultaneously consider all possible metrics on \(G\) that arise from choices of finite generating sets. This motivates the following definition.

Spoiler: Quasi isometry is an unreasonably effective tool in group theory.

Definition 2.1.1. Quasi isometric embedding.
Let \((X,d_X)\) and \((Y,d_Y)\) be metric spaces. We say a function \(f:X \to Y\) is a \((K,C)\)-quasi isometric embedding where \(K,C\) are strictly positive real numbers and we have
\begin{equation} \frac{1}{K}d_y(f(x_1),f(x_2))-C \leq d_X(x_1,x_2) \leq Kd_Y(f(x_1,f(x_2)) + C,\label{eq_qi_i}\tag{2.1.1} \end{equation}
for every \(x_1,x_2 \in X\text{.}\)

We can always make the numbers bigger.

Often we will omit the bounds \((K,C)\) and simply talk of a quasi isometric embedding, i.e. we'll say that \(f:X \to Y\) is a quasi isometric embedding if there exist numbers \((K,C)\) that satisfy the requirement of Definition 2.1.1.

Definition 2.1.3. Coarsely dense.

Let \((X,d_x),(Y,d_y)\) be metric spaces and let \(f:X \to Y\) be a quasi-isometric embedding. We'll say its image is \(D\)-coarsely dense if there is some \(D>0\) such that for every \(y\in Y\) there exists some \(x_y \in X\) such that:

\begin{equation*} d_Y(y,f(x_y))\leq D. \end{equation*}

As before, we will say that the image of a quasi isometry is coarsely dense, if it is \(D\)-coarsely dense for some \(D>0\text{.}\) In this business, the actual parameters don't always matter.

Definition 2.1.4. Quasi isometry.

Let \((X,d_X),(Y,d_Y)\) be metric spaces and let \(f:X \to Y\) be a quasi isometric embedding. We say that \(f\) is a quasi-isometry if there exists a quasi isometric embedding \(g:Y \to X\) called a quasi-inverse such that there is some number \(D\) such that:

  • For all \(x \in X\text{,}\) \(d_X(x,g(f(x))\leq D\text{,}\) and
  • For all \(y \in Y\text{,}\) \(d_Y(y,f(g(y))\leq D\text{.}\)

Another way to say this is that the compositions \(g\circ f\) and \(f\circ g\) are \(D\)-close to the identity.

We will finally state two results and leave proofs and examples as exercises.

Exercises 2.1.1 Exercises

2.

Show that quasi isometry gives an equialence relation between metric spaces.

3.

The following question is stated in the context of function \(f:X\to Y\) between metric space.

By a linear growing function we mean a linear function \(\ell(x) = mx +b\) where \(m,b \in \RR\) and \(m>0\) (like in Calculus 1.) Show the following:

  1. If there is a linear growing function \(\ell_r(x)\) such that
    \begin{equation*} d_Y(f(x),f(y)) \leq \ell_r(d_X(x,y))\text{,} \end{equation*}
    then there is a linear growing function \(\ell_l(x)\) such that
    \begin{equation*} \ell_l(d_Y(f(x),f(y))) \leq d_X(x,y) \end{equation*}
  2. Show that if there are linear growing functions \(\ell_1,\ell_2\) such that for all \(x,y \in X\) we have
    \begin{equation*} d_X(x,y) \leq \ell_1(d_Y(f(x),f( y))) \end{equation*}
    and
    \begin{equation*} d_Y(f(x),f(y)) \leq \ell_2(d_X(x,y)), \end{equation*}
    then \(f:X\to Y\) is a quasi isometric embedding.
4.

Let \(\Gamma\) be a graph and let \(V(\Gamma)\) be it's vertex set. Let \(V(\Gamma)\) be equipped with the metric induced by \(\Gamma\text{.}\) Suppose every edge has length \(1\) and consider a retraction

\begin{equation*} r:\Gamma \to V(\Gamma) \end{equation*}

where each edge interior is mapped to some adjacent vertex. Show that \(r\) is a quasi-isometry.

5.

Prove Proposition 2.1.5.

Hint: Look back at Exercise 1.6.1.5

Other hint: In this case you are considering the a group \(G\) with two metrics \(d_A\) and \(d_B\) (from two finite generating sets of \(G\)) and function you want to use is the identity. Show that there are linear functions \(\ell_1,\ell_2\) such that

\begin{equation*} d_A(x,y) \leq \ell_1(d_B(x,y))\textrm{ and } d_B(x,y) \leq \ell_2(d_A(x,y)), \end{equation*}

for all \(x,y \in G.\)

6.

Prove Proposition 2.1.6.

Hint: Although the statement of Proposition 2.1.6 doesn't have any parameters, you should start by fixing parameters.

Other hint: for a quasi-inverse define \(g(y)\) to be some \(x\in X\) such that \(d_Y(f(x),y)\) is minimal, or at least unifomly bounded for all \(y\) (this is where \(D\)-coarsely dense becomes important.)

7.

Prove that with the standard Euclidean metric that the mapping

\begin{align*} \RR \amp \to \RR^2\\ x \amp \mapsto (x,0) \end{align*}

Is a quasi-isometric embeding, but not a quasi-isometry.

8.

Consider the graph \(\Gamma= \cay{\{1\}} \ZZ\text{,}\) i.e. the real line with vertices at integer points. Let's conisider what happens when we change the lengths of edges.

  1. Fix \(0\lt a\lt b\in \RR \text{.}\) Show that if for each edge \(e\) of \(\Gamma\) we randomly change its length to some number \(a\lt\ell_e\lt b\) then the resulting metric space \(\Gamma'\) is quasi-isometric to \(\Gamma\text{.}\)
  2. Show that if we let the length of edges be any \(0 \lt l_e \leq 1\text{,}\) then the resulting graph may not be quasi-isometric to \(\Gamma\)

Hint: if edges schrink too much maybe the graph will have finite diameter.