Lastly, in this appendix, we aim to provide an intuition regarding closed balls in spaces of curvature smaller than zero. The hyperbolic spaces we consider in this article are hyperbolic manifolds (that is, hyperbolic in the differentiable sense). However, there are other generalisations of the notion of hyperbolicity that carry over to the setting of geodesic metric spaces (such as Gromov hyperbolicity). This theory unifies the two extremes of spaces of negative curvature, the real hyperbolic spaces  $\mathbb{H}^n$  and trees. Recall that a tree is a simple graph that does not have closed paths. Trees provide a great intuition for the metric phenomena that we should expect in spaces of negative curvature. For example, let us denote by  $G$  an infinite planar grid (observe that this graph is far from being a tree), and let us denote by  $T$  a regular tree of valency 4 (this means that every vertex in  $T$  is connected to exactly other four vertices). The 2-dimensional Euclidean space  $\mathbb{E}^2$  is quasi-isometric to  $G$  and the 2-dimensional real hyperbolic space  $\mathbb{H}^2$  is quasi-isometric (but not isometric) to  $T$ . This observation is due to Švarc and Milnor. Recall that the graphs  $G$  and  $T$  are endowed with the natural path metric. A ball of radius  $n > 0$  in  $G$  has between  $n^2$  and  $4n^2$  points (polynomial in  $n$ ), while a ball of radius  $n$  in  $T$  has exactly  $(4^n - 1)/3$  points (exponential in  $n$ ). Now let us come back to  $\mathbb{H}^3$ , where the "continuous" analogue of this principle is also true; that is, that there exists a constant  $c > 1$  such that for every  $n > 0$ , one needs at least  $c^n$  balls of radius one to cover a ball of radius  $n$  in  $\mathbb{H}^3$ . If we only want to understand the local geometry of  $\mathbb{H}^3$ , we can re-scale this principle to say the following. Let  $B_1^{\mathbb{H}^3}$  denote the ball of radius one in  $\mathbb{H}^3$ . For all positive integers  $n > 0$ , one needs at least  $c^n$  balls of radius  $1/n$  to cover  $B_1^{\mathbb{H}^3}$ . This tree-like behaviour should be compared with the opposite principle that holds in  $\mathbb{E}^3$ , being that for all  $n > 0$ , one needs at most  $8n^3$  (a polynomial bound) of balls of radius one to cover  $B_1^{\mathbb{E}^3}$ . Essentially, we should think of  $B_1^{\mathbb{H}^3}$  as a manifold that is isomorphic to  $B_1^{\mathbb{E}^3}$  which is difficult to cover by balls of smaller radius, in fact, it will be coarsely approximated by a finite tree of balls of small radius (the underlying tree being bigger as the chosen radius is smaller). Furthermore, the philosophy that travelling around  $B_1^{\mathbb{H}^3}$  becomes more and more expensive as we leave the origin is made accurate with the consideration of divergence functions: geodesics in  $\mathbb{H}^3$  witness exponential divergence. More precisely, there exists a constant  $c > 1$  such that for every  $r > 0$  and for every geodesic  $\gamma : \mathbb{R} \longrightarrow \mathbb{H}^3$  passing through the origin at time  $t = 0$  it is true that if  $\alpha$  is a path that connects  $\gamma(-r)$  and  $\gamma(r)$  and that lies entirely outside of the ball of radius  $d(0, \gamma(r))$ , then  $\alpha$  must have length at least  $c^r$ . Again, this is better appreciated when put in comparison with the Euclidean space  $\mathbb{E}^3$ . Any geodesic (any line)  $\gamma : \mathbb{R} \longrightarrow \mathbb{E}^3$  passing through the origin at time  $t = 0$  will have the property that for all  $r > 0$  there is a path of length  $\pi r + \varepsilon$ , with  $\varepsilon > 0$  arbitrarily small, that lies entirely outside of the ball of radius  $r$  with centre at the origin. The length  $\pi r + \varepsilon$  is linear in  $r$ , as opposed to the exponential length that is required in hyperbolic spaces.