I am reading a Gromov's book " Metric Structures for Riemannian and Non-Riemannian Spaces ". Consider the following concept :
$$ distort(X)\doteq sup \frac{length\ dist|_X}{dist|_X} $$
That is for instance $X=S^1(1)$ on ${\bf R}^2$. Then $distort(X)=\frac{\pi}{2}$.
We can generalize this : If $X \subset A$, $$ distort(X,D) = sup\ dist(x_1,x_2) $$ where $dist_A(x_1,x_2) \geq D$.
So if $A = {\bf R}^2$ and $X= \{ (x,y)| \ y=|x| \}$ then $$ distort(X,D)=\sqrt{2}D$$.
Here I have a question : If $A=GL_n$, which subgroup $X$ has $distort(X,D)$ of exponential growth ? Thank you in advance.