# Geodesic Quadrangle in a Hyperbolic Space

I'm trying to follow a proof of the fact that if $g$ is an element of a hyperbolic group $G$ with infinite order, then $\langle g \rangle$ is an undistorted subgroup of $G$. The proof relies on the following lemma:

If $[p,q] \cup [q,r] \cup [r,s] \cup [s, p]$ is a geodesic quadrangle in a $\delta$-hyperbolic space $X$, then for any pair of points $x \in [p, q]$, $y \in [r, s]$ with $d(p, x) = d(s, y)$ we have $$d(x, y) \leq \max \{d(p, s), d(q, r) \} + 10\delta.$$

The problem is that the proof of this lemma is left as an exercise. I have now spent over 3 hours trying to prove this lemma, but I seem to make no progress whatsoever. This leads me to believe that I missed something elementary and I therefore wanted to ask if anyone could point me to a proof/give me an idea how to prove it.

Thanks!

• I personally think that it is easier to approach the asstertion that $\mathbb{Z}$-subgroups of an hyperbolic groups are undistorted via the notion of quasiconvexity. – M.U. Jan 2 '17 at 12:37