contradicts the choice of $\epsilon_0$.

We have proved that $\gamma_p(\epsilon_0) = \gamma_q(\epsilon_0)$ and $\gamma_p'(\epsilon_0) = -\gamma_q'(\epsilon_0)$. Therefore $\gamma_p$ is a geodesic such that $\gamma_p(0) = p \in K$ and $\gamma_p(2 \epsilon_0) = q \in K$, and so the image of $\gamma_p$ is contained in $K$ by total convexity, a contradiction. Thus $E$ is a bijective local diffeomorphism, and consequently $E\colon [0,\infty) {\rightarrow}\overline{{M \backslash K}}$ is a diffeomorphism. This means $K$ is an essential subset of $M$.

The topological compactification {#sec:tracey}
================================

\[sec:topcomp\]

In this section we explain how to compactify $M$ given a choice of essential subset $K$ by extending the exponential map to take values in ${M^{*}}\backslash K = ({M \backslash K}) \cup {M(\infty)}$. Fix an essential subset $K$ and define an extension ${\overline{E}}\colon Y \times (0,1] {\rightarrow}{M^{*}}\backslash K$ by

$${\overline{E}}(p,s) = \begin{cases} {E}(p, 2 \tanh^{-1}{s}) & \text{ if } s \in (0,1), \\
                                            [ \; {E}( p, t): t \geq 0\; ] & \text{ if } s = 1. \end{cases}$$

\[ebarref\]

Recall that the notation $[\gamma]$ above represents the equivalence class of the geodesic ray $\gamma$ under the asymptotic equivalence relation, see page . We have also collapsed the normal component using a diffeomorphism. We now verify that 