[line width=0.5pt, dashed] (4.5,9) – (4.75,7.25);
at (4.5,7.35) [label=below:$\pi_{\beta_0}(p)$];
[thick](0.75,6.5) – (5.5,5.5);
[thick] (5.5,5.5) – (7.75,7.5);
at  (11.25,8.75)[label=right:$\xi(\bfb)$];
[vertex] at  (7.75,8)[label=above:$(\beta_0)_r$];
[vertex] at  (7.75,7.5)[label=below:$q'$];
[thick] (7.75,7.5) – (12,5.75);
at (12,5.75)[label=right:$\alpha \in \xi(\bfa)$];
(12,5.9) to[bend left=70,](11.25,8.7);
at (10.8, 7) [label=right:$\ell$];

A transition point $(\beta_0)_r$ separates the point $p$ and any geodesic line that connects $\xi(\bfb)$ and $\xi(\bfa)$.

§.§ Case II
Suppose otherwise that $\bfb$ is not transient. By Lemma <ref>
there exists a unique set $A \in \calA$ such that $\xi(\bfb) = *p_A$.  Let $\beta_0$ be the central element of $\bfb$. Let

\[
r \gg  2\big(\Norm{\go_A}+ \Norm{p} \big).
\]

Let $\bfa \in \calU(\bfb, r)$ and let $\alpha_0$ be the central element of $\bfa$. Then $\alpha_0$ can be
$f_\bfb(1,0)$–redirect to $\beta_0$ at radius $r$. Let $\Ge \in A$ be the point near where $\alpha_0$
leaves the $M_0$–neighborhood of $A$.

Consider any geodesic segment or ray $\ell$ in $K$ connecting $\xi(\bfa)$ to $*p_A$. By  <cit.>,
$\ell$ enters $N_{\tau(f_\bfb(\qq))}(A)$ at a point that is boundedly close to $\Ge$. Since $*p_A$ is the final point
in $\ell$, $*p_A$ does not appear in interior of $\ell$ and hence, for any other vertex $x$ in $\ell$, we have
$\Norm{x} \geq \Norm{\Ge} - D(1,0)$. This implies $\Norm{x} \gg \Norm{p}$ and hence $\ell$ does not pass through $p$.

\[
\bfa \in m(\xi(\bfb), p)
\]

and hence $\calU(\bfb, r) \subset m(\xi(\bfb), p)$.

For any $\bfb \in \partial G$ and $r>0$, there exists a finite set of vertices $W \subset V(K)$ such that
for every $\bfa$

\[ \xi(\bfa) \in m(\xi(\bfb), W) \qquad \Longrightarrow \qquad \bfa \in \calU(\bfb, r).\]

Therefore, $\xi^{-1}$ is continuous.

Let the geodesic ray $\beta_0$ be the central element of $\bfb$.

§.§ Case I
Suppose that $\beta_0$ is
a transient geodesic ray. Given any $r$, let $r'$ be as in Lem:Inclusion-Criterion and
let $(\beta_0)_{r_1}$ be a transition point of $\beta_0$ where $r_1 \geq 2 r'$. Let $\delta_0$ be the constant from Lemma <ref> and let $\delta_1$ be the constants of Proposition <ref> accordingly. Let

\begin{equation}\label{r1}
r_2 > \delta_0+ \delta_1.
\end{equation}

\[
W = B((\beta_0)_{r_1}, r_2) \subset \Cay(G) \subset K
\]

be the ball of radius $r_2$ in $\Cay(G)$ centered at the vertex $\beta_{r_1}$. However, we consider
$W$ as a subset of $K$. Note that Since $\Cay(G)$ is proper, $|B((\beta_0)_{r_1}, r_2) | < \infty$.

Consider any element $\bfa \in \partial_B G$ such that $\xi(\bfa) \in m(\xi(\bfb), W)$. That is, there exists a
bi-infinite geodesic line $\ell$ in $K$ from $\xi(\bfa)$ to $\xi(\bfb)$ avoiding $W$.  Since $\bfb$ is transient $|\ell| \geq 3$,
by Lemma <ref> there exists  $\overline{\ell}_0$ whose projection to $K$ is boundedly close to $\ell$. If $\alpha_0$
is transient, then the geodesics $\alpha_0$ (the central element of $\bfa$) $\beta_0$ and $\overline{\ell}_0$ form an ideal
geodesic triangle in $\Cay(G)$. If $\alpha_0$ is not transient, then there exists $A_\bfa$ such that $\alpha_0$ eventually stays
in $A_\bfa$. Then by Lemma <ref> we can chose the starting point of $\overline{\ell}_0$ to be any vertex in
$\alpha_0 \cap N_{M_0}(A_\bfa)$, and we have that $\alpha_0$, $\beta_0$ and $\overline{\ell}_0$ also form a semi-ideal
geodesic triangle. Prop:thin implies that $(\beta_0)_{r_1}$ is $\delta_1$–close to either a transition point of either
$\alpha_0$ or a transition point  in$\overline{\ell}_0$.

If $(\beta_0)_{r_1}$ is $\delta_1$–close to any transition point  of $\overline{\ell}_0$, then by Lemma <ref>, all transition points of $\overline{\ell}_0$ is $\delta_0$ close to points of $\ell$, we have that $(\beta_0)_{r_1}$ is $(\delta_1+\delta_0)$-close to points of $\ell$ in the metric of $\Cay(G)$. This contradicts the choice of $r_2$. Thus $(\beta_0)_{r_1}$ is only $\delta_1$–close to a transition point in $\alpha_0$. This implies that
$\alpha_0$ can be $(3,0)$–redirected to $\beta_0$ at radius $r_1/2 \geq r'$. Observe that $(3,0) < f_\bfb((1,0)) +(0,1)$. Thus Lem:Inclusion-Criterion
implies that $\bfa \in \calU(\bfb, r)$.

§.§ Case II
Suppose on the other hand $\bfb$ is not transient.
Let $r_0$ be such that $\beta|_{\geq r_0}$ is in a bounded neighbourhood of $A$ and $\xi(\bfb) = *p_A$.  Let
\[f_G(\qq) : = \max\{ (9q, Q), f_A(\qq) \forall  A \in \calA\}.\]
Such a maximum function exists because all $A$'s are translations of a finite set of subgroups. For
a give $r>0$ let $r'$ be as in Lem:Inclusion-Criterion and let

\[
r'' : = R_\bfb( f_\bfb(1,0)+(0,1), f_\bfb(\qq_0), r').
\]

\[
r_1 =\max \{ 2r'', 2r_0\}.
\]

\[
R \gg  r_1+\delta_0+D(f_G(1,0))
\]

and let $W = B(\go, R)$ be the ball of radius $R$ in $\Cay(G)$ centered at $\go$.
Again, $W$ contains finite number of vertices. We now think of $W$ as a subset of $K$.

Let $\xi(\bfa) \in m(\xi(\bfb), W)$ with $\alpha_0$ as a central element. If $\xi(\bfa) = \xi(\bfb)$, the case is trivial.
Thus we assume that $\alpha_0$ eventually leaves the mono-directional set $A$ at a point $x$ with a $(1,0)$–transition
point for $\alpha_0$.

Since $\xi(\bfa) \in m(\xi(\bfb), W)$, there exists a geodesic ray or segment $\ell$ in $K$ that connects
$\xi(\bfa)$ to $\xi(\bfb)$ and is disjoint from $W$. Again since $\bfb \neq \bfa$, $|\ell| \geq 3$. Thus by Lemma <ref> there exists geodesic line $\overline{\ell}_0$  whose projection to $K$ is boundedly close to $\ell$. In particular, we can choose for $\alpha_0$ to start at a vertex $x' \neq x$ in $A$. Since $\alpha_0$ and $\overline{\ell}_0$ are both geodesic lines connecting $\xi(\bfb)$ to $\xi(\bfa)$, by Lemma <ref>, there exists a $(1,0)$-transition point  $p \in \overline{\ell}_0$ such that

\[d (p, x) \leq D(f_\bfa(1,0)).\]

Furthermore, by by Lemma <ref>,

\[ d(p, \ell) \leq \delta_0.\]

\[d(x, \ell) \leq D(f_\bfa(1,0)) + \delta_0.\]

Since $\ell$ avoid $W$,  all points of $x$ has $\Cay(G)$-norm greater than $R$ and thus

\[ \Norm{x} \geq R-D(f_\bfa(1,0)) + \delta_0 \geq R-D(f_G(1,0)) + \delta_0 \geq r_1\geq r''.\]

Since $x \in A$ there exists a $\qq_0$–ray in $A$ which can be $f_\bfb(\qq_0)$ redirected to $\beta_0$ after $r''$.Thus $\alpha_0$ can be  $f_\bfb(\qq_0)$-redirected to $\beta_0$ at

\[ r'' = R_\bfb( f_\bfb(1,0)+(0,1), f_\bfb(\qq_0), r'),\]

thus $\alpha_0$ can be $(f_\bfb(1,0)+(0,1))$–redirected to $\beta_0$ at radius $r'$. Thus Lem:Inclusion-Criterion implies that $\bfa \in \calU(\bfb, r)$. And we are done.

The combination of all of the preceding results proves the homeomorphism:

Let  $G$ be a relatively hyperbolic group with respect to subgroups $P_1, P_2,...P_k$. Assume that the Cayley graphs of the subgroups $P_i$'s are mono-directional sets, then  the quasi-redirecting boundary $\partial G$ is homeomorphic to $\partial_B G$.

Since the map $\xi \from \partial X \to \partial_B X$ is 1-1, onto, and both $\xi$ and $\xi^{-1}$ is continuous, we conclude that $\xi \from \partial G \to \partial_B G$ is a homeomorphism.

§ THE HAIRY PARKING LOT

In this section we analyze the quasi-redirecting boundary of a metric space $X$ which demonstrates how
$\partial X$ could be non-Hausdorff when the space is not an ATM space.  The space $X$ we construct
here is a proper CAT(0) metric space and it is a modification of an example first analyzed by
Cashen [6]. We show that the partial relation in $P(X)$ is not symmetric which means
$\partial X$ is not Hausdorff. However, $\partial X$ is still compact. We also use to this example to justify
our definition of the topology for $\partial X$ (see Subsection <ref>) .

Let $\EE^2$ be the Euclidean plane and $B(1)$ be the open ball of radius 1 around the origin.
Let $Y = \EE^2 - B(1)$. In the polar coordinate in $\EE^2$, we can write

\[
Y = \{ (\rho, \theta) \st \rho \geq 1\}.
\]

Let $Z$ be the union of $Y$ and an infinite number of rays, attached at
a net of points in $Y$ (say, the points in $Y$ with integer Cartesian coordinates). Then $Z$ resembles
the plane with a hole in the center and infinitely many hairs attached. Let $X = \widetilde Z$ be the universal cover of $Z$.
We fix a lift of the point $(1,0)$ in $Z$ and denote it by $\go$. This defines a polar coordinate for $\widetilde Y \subset X$
where $\theta \in (-\infty, \infty)$, $\rho \geq 1$ and $\go = (1,0)$.

Note that $X$ is CAT(0). We fix 3 geodesic rays in $X$ (note that the first coordinate is the radius and
the second coordinate is the angle):

\begin{align*}
\alpha_+ \from \RR_+ \to X, && \alpha_+(t) &= (1,t)\\
\alpha_- \from \RR_+ \to X, && \alpha_-(t) &= (1,-t)\\
\zeta \from \RR_+ \to X, && \zeta(t) &= (t+1, 0)
\end{align*}

Also, for every hair attached at the point $(\rho, \theta)$ (which we refer to as the $(\rho, \theta)$–hair) we pick a quasi-geodesic
$\gamma_{\rho,\theta}$ exiting this hair, namely,

\[
\gamma_{\rho, \theta} \from \RR_+ \to X, \qquad \gamma_{\rho, \theta}(t) =
\begin{cases}
(1, t) & t \in [0, \theta] \\
(t-\theta + 1, \theta) & t \in [\theta, \theta + \rho-1]\\
\text{exits along the hair} & t \geq \theta+\rho -1
\end{cases}.
\]

This is a uniform quasi-geodesic ray and the geodesic ray exiting the $(\rho, \theta)$–hair stays
uniformly close to $\gamma_{\rho, \theta}$, however, $\gamma_{\rho, \theta}$ is easier to describe.
Let $\bfa_\pm = [\alpha_\pm]$, $\bfz = [\zeta]$ and $\bfc_{\rho,\theta} = [\gamma_{\rho, \theta}]$.

We claim that

\[
P(X) = \big\{ \bfa_+, \bfa_-, \bfz \big\} \ \cup \ \big\{ \bfc_{r, \theta} \big\} _{\text{$(r,\theta)$--hairs}} .
\]

Also, $\bfa_\pm \preceq \bfz$ and otherwise, no other classes are comparable.
To see this, we note that all quasi-geodesic rays that exit the $(\rho, \theta)$–hair are in the same
class (since the tails of these rays coincide) and are not comparable to other quasi-geodesics (since
the hairs have only one point of contact with the rest of the space). We only need to show that every
quasi-geodesic ray that stays in $\widetilde Z$ can be quasi-redirected to $\zeta$. This can be
done using a logarithmic spiral. We do this explicitly for the $\alpha_+$. For $T \geq 1$, define

\begin{equation} \label{Eq:log-spiral}
\alpha_T\from \RR_+ \to X, \quad \alpha_T(t)=
\begin{cases}
\alpha_+(t) = (1, t) & t \in [0, T] \\
\big((t-T+1), T - \ln(t-T+1) \big) & t \in \left[T,  (e^T+ T -1)\right]\\
\big((t-T+1), 0\big) & t \in \big[(e^T + T -1), \infty\big)
\end{cases}.
\end{equation}

To see that $\alpha_T$ is a quasi-geodesic, note that, for $t \in \big[T, (e^T + T -1)\big]$, we have

\begin{equation} \label{Eq:Norm}
\left\Vert \frac{d}{dt}\alpha_T(t)  \right\Vert \leq
\left| \frac{d\rho}{dt} \right| + \rho  \left| \frac{d\theta}{dt} \right|
= 1 + (t-T+1) \cdot \frac{1}{(t-T+1)}  \leq 2.
\end{equation}

For other values of $t$, $\Norm{\dot{\alpha_T}(t)} =1$.
To see the lower bound, we use the fact that

\[
d\big( (r, \theta), (r', \theta') \big) \geq \max\big(|r-r'|, |\theta- \theta'|\big) \geq \frac 12 \big( |r-r'|+  |\theta- \theta'| \big).
\]

For example, we can estimate the distance between $\go$ and $\alpha_T(t)$, for $t \in [T, (e^T + T -1)]$, by

\begin{equation} \label{Eq:Lower}
d (\go, \alpha_T(t)) \geq \frac 12  \left( (t-T)  + (T - \ln(t-T+1)) \right)
= \frac 12( t- \ln(t-T+1)).
\end{equation}

Recall that, for $A > 0$, we have (the right hand side is the equation of a tangent line to the graph of $y=\ln(x)$)

\[
\ln(x) \leq \frac{x-A}{A} + \ln(A).
\]

Therefore, setting $x = (t-T+1)$ and $A=2$, we have

\begin{equation} \label{Eq:ln}
t- \ln(t-T+1)) \geq t - \left(\frac{t-T-1}{2}\right)  - \ln(2)
\geq \frac t2 - \ln(2).
\end{equation}

Combining (<ref>) and (<ref>) (and rounding up the additive error) we get

\[
d (\go, \alpha_T(t)) \geq \frac t4 -1.
\]

A similar calculation for other pairs of points shows that $\alpha_T$ is a $(4,1)$–ray.

Since the family $\alpha_T$ gives uniform quasi-redirections from $\alpha_+$ to $\zeta$
at every radius, we can conclude $\bfa_+ \preceq \bfz$. The proof of $\bfa_- \preceq \bfz$ is similar.
However, as it was noted in [6], the geodesic $\alpha_+$ is Morse. Hence,
any quasi-geodesic with end points on $\alpha_+$ stays in a bounded neighborhood of $\alpha_+$.
Thus, $\zeta$ cannot be quasi-redirected to $\alpha_+$. The same also holds for $\alpha_-$.
Hence, $\bfz$, $\bfa_+$ and $\bfa_-$ are distinct classes.

The same argument as above shows that every $\gamma_{\rho, \theta}$ can be $(4,1)$–redirected to $\zeta$
at radius comparable to $(\rho+\theta)$. That is, for a sequence $\bfc_{\rho_n, \theta_n}$, we have

\[
\bfc_{r_n, \theta_n} \to \bfz  \qquad \Longleftrightarrow \qquad  (\rho_n + \theta_n) \to \infty.
\]

In fact, if $(\rho+\theta) \geq r$ then $\bfc_{\rho, \theta} \in \calU(\zeta, r)$ and hence
$P(X) -  \calU(\zeta, r)$ is finite. In particular, $\partial X$ is compact.

§.§ Is there an Out-Topology?

Our goal throughout this paper has been to define a simple natural analogue of Gromov boundary.
The idea that $P(X)$ should be considered as the set of points at infinity seems quite natural to us
since the directions that can be quasi-redirected to each other do not represent truly different
directions. We then defined a notion of a cone topology on these classes essentially by saying that classes that
are close to $\bfa \in P(X)$ are those that can still be redirected to $\bfa$ at a large radius.

However, a priori, one could define the cone topology in the opposite way, namely, classes that are close to $\bfa \in P(X)$
are those that $\bfa$ can be redirected to at a large radius. It seems like we have made an arbitrary
choice here that needs justification. Why should a neighborhood be define by quasi-redirection towards
$\bfa$ (the In-topology) as opposed to away from $\bfa$ (the Out-topology)? Is there a well-defined notion of Out-topology?

To be more precise, let $X$ be a metric space where the assumptions 0, 1 and 2 hold.
We can attempt to define neighborhoods for the Out-topology as before:
For $\bfa \in P(X)$, $r>0$ and $F \from [1,\infty) \times [0,\infty) \to [1,\infty) \times [0,\infty)$ , define

\begin{align*}
\Uout(\bfa, r, F) := \Big  \{  \Gb \in P(X) \cup X \, \ST
&\text{ every $\qq$--ray $\alpha \in \bfa$ can be $F_\bfa(\qq)$--redirected } \\
& \, \text{to central element $\beta_0 \in \Gb$ at radius $r$.}  \Big \}
\end{align*}

In Sec:boundary, the choice of $F_\bfa$ was not important as long as $F_\bfa$
was large enough due to Lem:F which is why $F$ is not included in the notation
$\calU(\bfa, r)$. However,  the analogue of Lem:F
does not hold for sets $\Uout$.

To see this in the hairy parking lot example, we consider such neighborhoods around $\bfa_+$
for a given function $F$. It turns out, if we enlarge $F$ slightly, say set $F'= F+ (1,0)$,
then $\Uout(\bfa_+, r, F)$ does not contain $\Uout(\bfa_+, R, F')$ no matter how large $R$ is.

In fact, let us analyze what subset of $X$ can be reached via a $(q,Q)$–ray that matches $\alpha_+$
up to a radius $r$. Of course, all point $(\rho, \theta)$
where $\theta \geq r$ can be reached. But to make $\theta$ smaller, we have to move
away from the set $\{\rho=1\}$ at a linear rate. We argue that the
most efficient way is along a log spiral as in (<ref>). Consider, for $A>0$
and $f \from \RR_+ \to \RR_+$, the ray

\begin{equation} \label{Eq:log-spiral-2}
\gamma \from \RR_+ \to X, \qquad \gamma(t)=
\begin{cases}
\alpha_+(t) = (1, t) & t \in [0, r] \\
\big( A \cdot (t-r+1) , r- f(t-r) \big) & t \geq r
\end{cases}.
\end{equation}

This path follows $\alpha_+$ up to a radius $r$ and then moves away from the set $\{\rho=1\}$
at a constant speed of $A$ while reducing $\theta$ by a function $f$. Let us analyze how fast $\gamma$
can unwind around $\alpha_+$ (that is, how large $f$ can be) and stay a quasi-geodesic.
As in (<ref>), for $t \geq r$, we have

\[
\Norm{\dot{\gamma}(t)} \leq
\left| \frac{d\rho}{dt} \right| + \rho  \left| \frac{d\theta}{dt} \right|
= A + (t-r +1) f'(t-r).
\]

If $\gamma$ is a $\qq$–ray, $\Norm{\dot{\gamma}(t)}$ has to be bounded with a bound depending on $\qq$. That is, for a constants $B_\qq$
depending on $\qq$, we have

\[
f'(t-r) \leq \frac {B_\qq}{t-r+1} \quad \Longrightarrow \qquad
f(t-r) \leq B_\qq \ln(t-r+1). \tag{$f(0) =0$}
\]

Note that $B_\qq$ is increasing as a function of $\qq$.

If $(\theta, \rho)$ is a point on $\gamma$, for $\theta \in (-\infty, r]$, we have

\begin{align*}
\theta =  r- f(t-r) \geq r -  B_\qq \ln(t-r+1)
&& \Longrightarrow && \ln(t-r+1) &\geq \frac{r- \theta }{B_\qq}  \\
&& \Longrightarrow  && (t -r+1) &\geq e^{\frac{r- \theta}{B_\qq}}.
\end{align*}

\begin{equation} \label{gamma}
\rho = A \cdot (t-r+1) \geq A \cdot e^{\frac{r- \theta}{B_\qq}} \geq  C_{r,\qq} \cdot e^{\frac{- \theta}{B_\qq}}.
\end{equation}

For some function $C_{r, \qq}$ depending on $r$ and $\qq$.
Similarly, if $\gamma'$ is a $\qq'$–ray that matches $\alpha_+$ up to a radius $R$
for any $(\rho, \theta)$ on such a geodesic we have

\begin{equation} \label{gamma'}
\rho \geq  C_{R,\qq'} \cdot e^{\frac{- \theta}{B_{\qq'}}}.
\end{equation}

If $\qq=(q,Q)$, $\qq'= (q', Q')$ and $q' >q$, we have $B_{\qq'}  > B_{\qq}$. Therefore, no matter how large
$R$ is compared to $r$, as $\theta \to -\infty$,

\[
C_{R,\qq'} \cdot  e^{\frac{- \theta}{B_{\qq'}}} \leq  C_{r,\qq} e^{\frac{- \theta}{B_\qq}}.
\]

That is, the set of point that could be on $\gamma'$ is not contained in the set of point
that could be on $\gamma$ and, for every $R>r$, we can find a point $(\rho, \theta)$ where
(<ref>) holds but (<ref>) does not.  If $(\rho, \theta)$ is any such point, then

\[
\bfc_{\rho, \theta} \not \in \Uout(\bfa, r, F_\bfa)
\qquad \text{but} \qquad
\bfc_{\rho, \theta} \in \Uout(\bfa, R, F').
\]

This causes some complication with the definition of Out-topology.
For the sets $\Uout$ to define a topology, in analogy with Lemma:Ubeta, we need to know that
the following:  There is a family of functions $F_\param$ and, for every $r > 0$ and every $\bfb \in \Uout(\bfa, r\,)$,
there is $R>0$ such that

\[
\Uout(\bfb, R , F_\bfb) \subset \Uout(\bfa, r, F_\bfa).
\]

The argument above can be modified to show that this does not hold in general for
arbitrarily large functions $F_\param$. For example, we can set $\bfa=\bfa_+$, and choose $\gamma_{1,R}$ to be a hair
attached to $\alpha_+$ at a distance $R$ from $\go$. Then
$\gamma_{1,R} \in \Uout(\bfa_+, R)$, but if $F_{\bfc_{1,R}}$ is even slightly larger than
$F_\bfa$, we have

\[
\Uout(\bfc_{1,R}, R , F_{\bfc_{1,R}}) \not \subset \Uout(\bfa, r, F_\bfa).
\]

That is, if we want to have a Out-Topology, we have to choose the functions  $F_\param$ in a very delicate way
and it is not clear if this is possible. And the Out-topology would change if you changed the functions

Another issue is that, even if it was possible to define an Out-topology, the restriction of the Out-topology
to natural subspaces such as the sublinearly Morse boundary or even the contracting boundary (see [7])
does not match the previously defined topologies on these spaces. For the sake of brevity, we leave
the details to reader except to mention that, similar calculations as above shows that, for every $F$, it is possible
to find a sequence $\rho_n, \theta_n, R_n$ such that

\[
R_n \to \infty, \quad \theta_n \to -\infty \quad\text{and}\quad  \bfc_{\rho_n, \theta_n} \in \Uout(\alpha_+, R_n , F).
\]

However, if $\theta_n \to -\infty$, the sequence $\bfc_{\rho_n, \theta_n}$ does not converge to
$\alpha_+$ in the usual topology of the Morse-boundary (following either [7] or [43]).

§ THE CROKE-KLEINER GROUP

In this section, we show that the hairy parking lot example (Section <ref>) is not an artificial example and indeed the same phenomena  happens in a finitely presented group, in particular, in right angled Artin groups. As a special case,
we closely examine the Croke-Kleiner Group acting on the universal cover of the  Salvetti Complex (denoted, as usual, by $X$).
Our main goal is to prove Thm:CK. The proof showcases the rich structure of space
of quasi-geodesic rays in the Croke-Kleiner Group. We also give a classification of the set $P(X)$
by examining the growth rate of the excursion of geodesic rays (see Thm:Classification).
In the hairy parking lot example, every direction is either sublinearly Morse (in fact Morse) or non-Hausdorff point. However,
it turns out that, in the Croke-Kleiner Group, there exists unexpected equivalence classes of directions that  are rank-one and QI-invariant, but are not sublinearly Morse. In addition, these points are not the non-Hausdorff points of the space. This adds to the knowledge that rank-one is not a good predictor of QI-invariant, as a class of rank-one directions is exhibited to not be QI-invariant in  [41]. Lastly, this example shows that neither
the Bass-Serre trees nor the hierarchically hyperbolic structures of such spaces is in general accurate in identifying QI-invariant directions at boundary at infinity.

In the interest of brevity, the exact calculation of the constants of quasi-geodesic rays constructed in this section
are omitted since these calculations are standard and are similar to the arguments presented in the paper so far.

§.§ The Croke-Kleiner Group

For background on the Croke-Kleiner group, see [10, 11] and [41]. Here we follow the notation from
[10]. The Croke-Kleiner group is a group $G$
with the following presentation:

\[
G = \big\langle \,  a, b, c, d \st [a, b], [b, c], [c, d]  \, \big\rangle.
\]

Consider a tori complex as follows.

(-1,0) to [bend left] (1,0);
(-1.2,0.1) to [bend right] (1.2,0.1);
(0,0) ellipse (4 and 2);
[thick,blue] (0,0) ellipse (2.5 and 1.3);

(8,0) to [bend left] (10,0);
(7.8,0.1) to [bend right] (10.2,0.1);
(9,0) ellipse (4 and 2);
[thick,blue] (9,0) ellipse (2.5 and 1.3);
[thick,red,bend right=30] (9,-0.2) to (9,-2);
[thick,red,dashed,bend left=20] (9,-0.2) to (9,-2);

(17,0) to [bend left] (19,0);
(16.8,0.1) to [bend right] (19.2,0.1);
(18,0) ellipse (4 and 2);
[thick,red,bend right=30] (18,-0.2) to (18,-2);
[thick,red,dashed,bend left=20] (18,-0.2) to (18,-2);

at (0,-3) $S_1$;
at (9,-3) $S_2$;
at (18,-3) $S_3$;

the tori are glued along the curves isometrically .

Let $\go$ be a 0-cell. Let $a, b, c, d$ be four oriented 1-cells,
each an isometric copy of $[0,1]$, whose boundary is $\go$. Let $S_1$ be a torus that is an isometric copy of
$[0, 1] \times [0, 1]$, where the opposite sides are glued together to the loops $a$ and  $b$. Likewise, let $S_2$ be
a Euclidean square torus with 1-skeleton $b, c$ and $S_3$ be a Euclidean square torus with 1-skeleton $c, d$.
This tori complex is the Salvetti complex associated with this group. The universal cover of the Salvetti complex is the main space we study
in this section and we denote  it by $X$.
In this section, we show

The space $X$ satisfies Assumptions 0-2 and hence $\partial X$ is defined.
The relation $\preceq$ is not symmetric and $P(X)$ has a unique maximal element.
Therefore, $\partial X$ is not Hausdorff.

Note that Assumption 0 holds by Lem:cocompact. To check  Assumptions 1 and 2, we need to identify elements
of $P(X)$, choose a geodesic representative $\alpha_0$ in each class $\bfa \in P(X)$ and show that every $\qq$–ray
$\alpha\in \bfa$ can be $f_\bfa(\qq)$–redirected to $\alpha_0$ for some function $f_\bfa$ depending only on the class $\bfa$.

§.§ Blocks and the Bass-Serre tree

A block in $X$ is a convex infinite subset of $X$ that is a lift of either  $S_1 \cup S_2$,
or $S_2 \cup S_3$. Thus a block is isometric to the universal cover of the Salvetti complex of either of the following groups

\[
G_1 = \big\langle a, b, c \st [a, b], [b, c] \, \big\rangle \qquad \text{or} \qquad
G_2 =\big \langle b, c, d \st [b, c], [c, d] \, \big\rangle.
\]

In other words, each blocks comes with a co-compact action of a conjugate copy of either $G_1$ of $G_2$.
A flat in $X$ is a lift of $S_1, S_2$ or $S_3$. Given a pair of blocks, their intersection in $X$ is
either empty or a flat (in fact, always a lift of $S_2$). That is, a flat comes with a compact action of
a conjugate of the group $ \big\langle b, c \st [b, c] \, \big\rangle = \ZZ^2$.  We refer to these as $bc$–flats.

One can construct a graph where vertices are blocks and two vertices are
connected by an edge if and only if two blocks intersect in a flat. The resulting graph, which we denote by $T$,
is the Bass-Serre tree associated with amalgamated product decomposition of $G$:

\[
G = G_1 *_{ \langle \, b, c \, \st \, [b, c] \, \rangle} G_2.
\]

For $x \in X$, the $a$–axis passing through $x$ is the bi-infinite geodesic ray passing through points
$x \cdot a^n$, for $n \in \ZZ$. The $b$–axis, $c$–axis and $d$–axis are similarly defined.

Note that blocks have a product structure.  A $G_1$–block containing a point
$x \in X$ is a Euclidean product of the tree generated by $\langle \, a, c \, \rangle$ and the $b$–axis
centered at the point $x$. Similarly, a $G_2$–block containing $x \in X$ is a Euclidean product of the
tree generated by $\langle \, b, d \, \rangle$ and the $c$–axis centered at the point $x$.
We can use this product structure to quasi-redirect quasi-geodesics to each other.
For example, the arguments in product can be used to prove the following lemma.

For every $\qq \in [1, \infty]\times [0, \infty)$ and $\rho>0$, there is a
$\qq' \in [1, \infty]\times [0, \infty)$ such that the following holds.
Let $B$ be a block, $R \geq (1+\rho) \cdot r >0$ be a pair of radii and $\alpha$ and $\beta$ be two
$\qq$–rays. Assume $\alpha_r \in B$ and that $\beta|_{\geq R}$ starts at a point in $B$. Then, $\alpha$
can be $\qq'$–redirected to $\beta$ at radius $r$.

That is, we can transition from $\alpha|_r$ to $\beta|_{\geq R}$ as long as there is buffer between
them that has a product structure and a thickness that is a linear function of $r$.

§.§ The unique maximal element

We now start analyzing the set $P(X)$. Let $\go$ also denote the point in $X$ associated with
the identity element in $G$. Let $\zeta \from \RR_+ \to X$ be a geodesic ray in $X$ such that,
for a positive integer $n$,

\[
\zeta(n) = \go \cdot b^n.
\]

That is, $\zeta$ follows the positive $b$–axis passing through $\go$.
We start by showing that $\bfz=[\zeta]$ is the unique maximal element in $P(X)$.
The choice of $b$ here is arbitrary. The geodesic $\zeta_c$ following the $c$–axis is in the same block
at $\zeta$ and hence $\zeta_c \in \bfz$. Similarly, if $\zeta_a$ and $\zeta_d$ are geodesics following
the $a$–axis and the $d$–axis respectively, we have $\zeta_a \in [\zeta]$ since $\zeta$ and
$\zeta_a$ are in the same block and $\zeta_d \in [\zeta_c] = [\zeta]$ since $\zeta_c$ and $\zeta_d$
are in the same block.
Recall that $\qq$–rays of $X$ are assumed to be a continuous and emanating from $\go$.

Let $\alpha$ be a $\qq$–ray in $X$. Then $\alpha$ can be $\qq'$–redirected to $\zeta$ where
$\qq'$ depends only on $\qq$. In particular,  $\alpha \preceq \zeta$.

After perturbing $\alpha$ by a bounded amount, we can assume that $\alpha$ lies in the $1$–skeleton of
the universal cover of the Salvetti complex. In fact, we assume, there is a sequence $s_n \in \{ a, b, c, d\}$, for $n \in \NN$,
such that $\alpha(n) = s_1 \dots s_n$. Let $r>0$ be given. We construct a quasi-geodesic ray redirecting
$\alpha$ to $\zeta$ at radius $r$.

Using Lem:block-redirect, we can find a quasi-geodesic ray $\alpha'$, with constants depending only on
$\qq$, where $\alpha_r = \alpha'_r$
and such that $\alpha'$ eventually follows some $b$–axis. That is, there is $T>0$, such that for $t \geq T$,
we have $\alpha'(t+1) = \alpha'(t) \cdot b$.

Again, we can assume that $\alpha'$ lies in the $1$–skeleton of the universal cover of the Salvetti complex.
We now construct a spiral quasi-geodesic redirecting $\alpha'$ to $\zeta$. This is, in spirit, the same
idea as Eq:log-spiral-2. Meaning, we try to backtrack along the segment $\alpha'[0,T]$,
however, to stay a quasi-geodesic, we move exponentially far away from $\alpha'$ in each block.
To be more precise, let

\[
\alpha'(T) =v_1 w_1 \dots v_k w_k,\quad
\text{where} \quad v_i \not \in G_2  \quad\text{and}\quad  w_i \not \in G_1.
\]

It is possible that we have to start with a word in $G_2$ or end with a word in $G_1$, but in
these cases the construction is similar. For $t \in [0,2T]$, we define
$\gamma(t) = \alpha'(t)$. That is,

\[
\gamma(2T) = v_1 w_1 \dots v_k w_k \cdot b^T.
\]

We start by moving along the $c$–axis to

\[
\gamma(4T) = v_1 w_1 \dots v_k w_k \cdot b^T \cdot c^{2T} =
v_1 w_1 \dots v_k\cdot c^{2T} \cdot w_k b^T.
\]

This is still a quasi-geodesic. At the worst case moving along the $c$–axis cancels with
$w_k$. But the presence of $b^T$ in the end ensures that, any point along this segment is at least
a distance $T$ from $\alpha'[0,T]$. Now we can undo $w_k b^T$ to get:

\[
\gamma(5T+|w_k|) = v_1 w_1 \dots v_k \cdot c^{2T}.
\]

This is also a quasi-geodesic since the distance between this segment and $\gamma[0, 2T]$
is at least $T$. We can now proceed in this way: first we add a large power of $b$ to get

\[
\gamma(9T+|w_k|) = v_1 w_1 \dots w_{k-1} v_k \cdot c^{2T} \cdot b^{4T}=
v_1 w_1 \dots w_{k-1}  \cdot b^{4T} \cdot v_k c^{2T},
\]

and then we undo $v_k c^{2T}$ to get

\[
\gamma(11T+|w_k| +|v_k|) = v_1 w_1 \dots w_{k-1}  \cdot b^{4T}.
\]

As before, the large power of $b$ ensures that this path is still a quasi-geodesic. We can let
$T_1 = 11T+|w_k| +|v_k|$ and proceed inductively. To establish the pattern, we repeat the above
procedure one more time. We will add $c^{T_1}$ and undo $w_{k-1} b^{4T}$ to reach

\[
\gamma(2T_1 + 4 T + |w_{k-1}|) =  v_1 w_1 \dots w_{k-2} v_{k-1}  \cdot c^{T_1}.
\]

We then add $b^{2T_1}$ and undo $v_{k-1}  \cdot c^{T_1}$ to reach

\[
\gamma(5T_1 + 4 T + |w_{k-1}| + |v_{k-1}|) =  v_1 w_1 \dots w_{k-2}  \cdot b^{2T_1}.
\]

This is the same at multiplying $\gamma(T_1)$ on the right by

\[
c^{T_1} \cdot b^{-4T} \cdot w_{k-1}^{-1} \cdot b^{2T_1} \cdot c^{-T_1} \cdot v_{k-1}^{-1}
\]

one letter at a time. Note that the powers of $b$ do not cancel since $w_k \not \in G_1$.
And then we set
\[T_2 = 5T_1 + 4 T + |w_{k-1}| + |v_{k-1}|\]
and proceed as before. When all $v_i$
and $w_i$ disappear, we have $\gamma(T_k)$ is on $\zeta$, and we can continue along
$\zeta$ for $t \geq T_k$.

§.§ Recurrent quasi-geodesic rays

We define a map $\Psi_\alpha \from \RR_+ \to T$ (recall that $T$ is the Bass-Serre tree) as follows:
Let $A_0 \in T$ be the $G_1$–blocks containing $\go$. Let the $u_1>0$ be the supremum of times
$t$ such that $\alpha(t) \in A_0$ and let $A_1 \not = A_0$ be the block $\alpha$ enters immediately after
it exists $A_0$. We now define $u_i$ and $A_i$ inductively. Let $u_i > u_{i-1}$ be the supremum of times
$t$ such that $\alpha(t)$ is contained in the block $A_{i-1}$ and let $A_i \not = A_{i-1}$ be the block
$\alpha$ enters immediately afterwards. Now define

\[
\Psi_\alpha(t) = A_i \qquad\text{for}\qquad t \in [u_i, u_{i+1}).
\]

This is a quasi-Lipschitz map where contestants depend on $\qq$.

It is possible that $u_i = \infty$ for some $i$. This means that $\alpha$ visits the block $A_i$
infinitely many times. In this case, we say $\alpha$ is recurrent.
Otherwise, we say $\alpha$ is transient.

If $\alpha$ is recurrent, then $\alpha \in \bfz$.

Let $A_i$ be the terminal block for $\alpha$. Let $\beta$ be a quasi-geodesic that
eventually stays in $B$ following the $b$–axis. Lem:block-redirect implies that
$\beta \preceq \alpha$. Also, we can use the argument of Prop:zeta-is-top  to build
a spiral from $\zeta$ to $\beta$.  That is, $\zeta \preceq \beta$. Hence, $\zeta \preceq \alpha$.
But we already know $\alpha \preceq \zeta$ by Prop:zeta-is-top . This finishes the proof.

§.§ Geodesic representatives in each class

Now assume $\alpha$ is not recurrent. Note that the sequence $A_i$ is an embedded path in
$T$, that is, a sequences of vertices where $A_i$ is adjacent to $A_{i+1}$ without
repeating. Since $T$ is a tree, this path is a geodesic in $T$ limiting to some end $\xi$ of $T$.
By <cit.>, there is at least one geodesic ray $\alpha_0 \in X$ whose itinerary
is the sequence $A_i$. We show that $\alpha_0$ can always be redirected to $\alpha$. First,
we recall the following lemma that follows from <cit.> and <cit.>.

Let $x,y$ be two points in $A_{i+2}$, let $[\go, x']$ and $[\go, y']$ be subsegments of geodesic
segments $[\go, x]$ and $[\go, y]$ such that $x', y' \in A_i$. Let $[\go, z]$ be the intersection
$[\go, x'] \cap [\go, y']$ (possibly $z = \go$). Then, there is an embedded Euclidean triangle
in $X$ where the edges are the geodesic segments $[z, x']$, $[z, y']$ and $[x',y']$.

For $\alpha$ and $\alpha_0$ as above, we have $\alpha_0 \preceq \alpha$. In particular, all
geodesics with itinerary $A_i$ are in the same class.

vertex =[circle,draw,fill=black,thick, inner sep=0pt,minimum size=2pt]

[black!15] (1,0) – (4,1.5) – (4,0) – cycle;
[vertex] (a) at (0,0)  [label=180:$\go$]   ;

at (11.6 , 2) $\alpha|_{\geq R}$;
[thick] (0,0)–(11, 0) node [right] $\alpha_0$;
[vertex] (x) at (5, 2) [label=-90:$x$]  ;
[vertex] (z) at (1,0) [label=-90:$z$]  ;
[vertex] (x') at (4, 1.5) [label=90:$x'$]  ;
[vertex] (y') at (4, 0) [label=-90:$y'$]  ;
[vertex] (ar) at (2, 0) [label=-90:$(\alpha_0)_r$]  ;

[dashed, thick](1,0)–(5,2);
[dashed, thick] (x')–(y');
[red, thick] (2,0)–(3,0) – (3,1) – (4, 1.5);

5 cm2cm
7 cm2.2cm
8 cm3.4cm
9 cm2.2cm
10 cm3.2cm

$\alpha_0$ can be redirected to $\alpha$ at radius $r$.

Pick $r>0$. Let $(\alpha_0)_r$ be in $A_i$. We  choose $R>2r$ and $j> i+2$ such that $\alpha|_{\geq R}$
starts at a point $x \in A_j$ and there is a point $y'$ on $\alpha_0 \cap A_{j-2}$ with $\Norm{y'} \geq 2r$.
It follows from Part (I) of surgery that the concatenation $\gamma=[\go, x] \cup \alpha|{\geq R}$ is a
$(3q, Q)$–quasi-geodesic. Also, by Lem:Sector, for a point $x' \in [\go, x] \cap A_{j-2}$,
there is a flat triangle containing points $z, x'$ and $y'$. If $\Norm{z} \geq r$, then $\gamma$ already
gives a redirection from $\alpha_0$ to $\alpha$ at radius $r$. Otherwise, the Euclidean triangle
can be used, similar to product, to redirect $\alpha_0$ to $\gamma$ at radius $r$.
(See Fig:triangle). This finishes the proof.

For every $\qq$, there is $\qq'$ such that the following holds.
Assume $\alpha$ is a $\qq$-ray that is transient with itinerary $A_i$ and let $\alpha_0$ be a geodesic
with the same itinerary. Then either $\alpha \in \bfz$ or $\alpha$ can be $\qq'$–redirected to $\alpha_0$.

We use Part (II) of surgery. If there is a sequence of radii $r_i \to \infty$ such that
$d_X((\alpha_0)_{r_i}, \alpha) \leq r_i/2$, then $\gamma$ can be $(9q, Q)$–redirected to $\alpha_0$
and we are done. Otherwise, there is $r_0$ such that, for every $r \geq r_0$,

\begin{equation} \label{Eq:r2}
d_X((\alpha_0)_r, \alpha) \geq r/2.
\end{equation}

We show that, in this case, $\zeta$ can be redirected to $\alpha$. Since $\alpha$ can always be redirected to
$\zeta$, this implies that $\alpha \in \bfz$.

By <cit.>, the itinerary of $\alpha_0$ determines the combinatorial path.
Using the language of [11], let $x_k$ be the points on the $bc$-flat shared by $A_k$ and $A_{k+1}$
that determines the quarter-planes on the walls. Let $w_k \in G$ be the word such that $x_k \cdot w_k = x_{k+1}$.
Then the concatenation of words $w_k$ is a quasi-geodesic rays $\alpha_1$ with uniform constants $\qq_0$ and with
the same itinerary as $\alpha_0$. That is, we can write the $\qq_0$–ray  $\alpha_1$ as an infinite sequence

\[
w_1 w_2 w_3 \dots
\qquad\text{where}\qquad w_{2i+1} \not \in G_2  \qquad\text{and}\qquad w_{2i} \not \in G_1,
\]

or perhaps we start with $G_2$ instead of $G_1$.
Since we assume in this paper without loss of generality that quasi-geodesics are continuous, it follows from the template structure of $X$ that the entry point of $\alpha$ into $A_k$ is aligned
with $x_k$ either along the $b$–axis of the $c$–axis. That is, there are powers $p_k \in \ZZ$ such that

\[
\alpha(u_k) = x_{k} \cdot b^{p_k} \qquad\text{for even $k$ and}\qquad
\alpha(u_k) = x_{k} \cdot c^{p_k} \qquad\text{for odd $k$,}
\]

or, again, with opposite parity.

Let $\alpha^{A_i}_1$ be the point where  $\alpha_1$ enters $A_i$, note that if a $\qq$-ray satisfies  Equation (<ref>),  $p_k$ is comparable and differ by a uniform multiplicative constant to the
$d(\alpha^{A_k}_1, \alpha(u_k))$. Thus we observe that $p_k$ (powers of $b$ or $c$) comparable and differ by a uniform multiplicative constant to $u_k$. Recall also that
from $\alpha(u_k)$ to $\alpha(u_{k+1})$, the path $\alpha$ has to replace the powers of $b$ with the powers of $c$ (or vice versa)
and  also it has to travel along $w_k$.  Therefore, there is a constant $\rho_0>0$
depending on $\qq$ such that,
for $k>0$ large enough,

\begin{equation} \label{Eq:uk}
u_{k+1} - u_k \geq \rho_0 \cdot (u_k + |w_k|).
\end{equation}

Let $r>0$ be given. We attempt to build a spiral quasi-geodesic $\gamma$ chasing after $\alpha$
following the outline of the proof of Prop:zeta-is-top .
We build the quasi-geodesic $\gamma$ inductively. For $t \in [0,r]$, define
$\gamma(t) = \zeta(t) = b^t$. Then we proceed exactly as in the proof of Prop:zeta-is-top  trying
to get to the block $A_i$ as fast possible while staying a quasi-geodesic. To accomplish this, the length of
time $\gamma$
spends in each $A_i$ has to grow exponentially. However, we do not have to double the length
each time; the powers of $b$ and $c$ in the proof of Prop:zeta-is-top  can grow as a
slow exponential function. And how slow these powers grow will determine the quasi-geodesic constants
of $\gamma$.

That is, for every $\rho>0$, there is $\qq_\rho$, such that the amount of time $\gamma$
stays in each $A_i$ grows by a factor of $(1+\rho)$ only, and then we can proceed along
the word $w_i$ with a speed of $1/\rho$.  Here $\qq_\rho \to \infty$ as $\rho \to 0$. However, for a fixed $\rho$,
we can build a spiral that visits the block $A_k$ at time $t_k$ where

\begin{equation} \label{Eq:tk}
t_0 = r \qquad\text{and}\qquad t_{k+1} \leq (1+\rho) \cdot t_k + \rho \cdot |w_k|.
\end{equation}

If we choose $\rho < \rho_0$, Equations (<ref>) and (<ref>) imply the sequence $t_k$ grows more
slowly than the sequence $u_k$. That is, for $k$ large enough

\[
\Norm{\alpha(u_{k+1})} \geq 2 \Norm{\gamma(t_k)}.
\]

Now Lem:block-redirect implies that $\gamma$ can be redirected to $\alpha$. Since this is true for every $r$,
$\zeta$ can be redirected to $\alpha$ and, in view of Prop:zeta-is-top , $\alpha \in \bfz$.

For every quasi-geodesic ray $\alpha$, the class $[\alpha]$ contains a geodesic representative and
$\alpha$ can be uniformly redirected to this geodesic.

Every quasi-geodesic can be uniformly quasi-redirected to $\zeta$. Hence, the Corollary holds if $\alpha \in \bfz$.
Otherwise,  by Lem:geodesic-to-alpha, $\alpha_0 \preceq \alpha$ and by Prop:zeta-or-geodesic
$\alpha$ can be uniformly quasi-redirected to $\alpha_0$. That is, $[\alpha_0] = [\alpha]$ and the corollary holds.

§.§ Excursion

For a transient quasi-geodesic ray $\alpha$, whether or not $\zeta$ can be quasi-redirected to $\alpha$
depends the excursion function of $\alpha$, namely, the amount of progress $\alpha$ makes in each block.
We now make this precise. We work with the combinatorial path introduces in the previous section.
That is, we assume $\alpha_1$ is an infinite sequence

\[
w_1 w_2 w_3 \dots
\qquad\text{where}\qquad w_{2i+1} \not \in G_2   \qquad\text{and}\qquad w_{2i} \not \in G_2.
\]

We say the excursion is sublinear with respect to the distance in $X$ if

\[
\lim_{i \to \infty}\frac{ |w_k|}{\sum_{i=1}^{k-1} |w_i|} =0.
\]

In <cit.>, these class of rays were studied and it was proven than
they are sublinearly Morse. As we saw in Sec:kappa, every element of a sublinearly Morse
class $[\alpha_1]$ has to fellow travel $\alpha_1$ sublinearly. Therefore, for such quasi-geodesics,
$[\alpha_1] \not = \bfz$.

Assumption 0 follows from Lem:cocompact and Assumption 1 and 2 follow from
Cor:Geod-Rep. By Prop:zeta-is-top, $\bfz$ is the unique maximal element in $P(X)$.
And since the sublinearly Morse direction are different from $\bfz$, $P(X)$ has more than one point.
That is $\preceq$ is not symmetric and, by sym-Haus, $\partial X$ is not Hausdorff.

However, it turns out that sublinear excursion with respect to the distance in $X$ does not
give a characterization of directions that are different from $\bfz$. Indeed, we say the excursion of
$\alpha_1$ is sub-exponential with respect to the distance in $T$ if

\[
\lim_{i \to \infty}\frac{\log |w_i|}{i} =0.
\]

For a transient quasi-geodesic ray $\alpha_1$, $[\alpha_1] \not = \bfz$ if and only if
the excursion of $\alpha_1$ is sub-exponential with respect to the distance in $T$.

First we show that if the excursion of $\alpha_1$ is sub-exponential with respect to the distance in $T$
then $[\alpha_1] \not = \bfz$. We need to show that $\zeta$ cannot be quasi-redirected to $\alpha_1$.
That is, for every $\qq$, there is an $r>0$ such that there is no $\qq$–ray $\gamma$
with $\gamma|_r = \zeta|_r$ that is eventually equal to $\alpha_1$.

Assume, for contradiction that such $\gamma$ exists for every $r$.
The proof is similar to Prop:zeta-or-geodesic with $\gamma$ in this proof playing the role of
$\alpha$ in the previous one. As before, let $x_k$ be the points on the $bc$–flat
shared by $A_k$ and $A_{k+1}$ that determines the quarter-planes on the walls. In particular
$x_{k+1} = x_k \cdot w_k$. Let $t_k$ be the first time $\gamma$ enters the $bc$–flat
containing $x_k$ and let $\ell_k = d(\gamma(t_k), x_k)$. Then
$\gamma(t_k) \cdot  x_k^{-1}$ is either a power $b$ or a power of $c$ depending on parity.
That is, if the difference is a power of $b$ for $x_k$ then $\gamma(t_{k+1}) \cdot  x_{k+1}^{-1}$
is a power of $c$ and vice versa. Hence, during the interval $[t_k, t_{k+1}]$, the path $\gamma$
has to add $\ell_{k+1}$ powers of $c$, undo $\ell_k$ powers of $b$ and travel along $w_k$.
Since $\gamma$ is a $\qq$–ray, there is $\rho_\qq>0$ depending on $\qq$ such that

\begin{equation} \label{Eq:Low}
t_0 = r \qquad\text{and}\qquad
t_{k+1}-t_k \geq \rho_\qq \cdot (\ell_k + \ell_{k+1} + |w_k|).
\end{equation}

Another way to travel to $\gamma(t_k)$ is to go along the segments $w_i$ to $x_k$ and then
a distance of $\ell_k$ in the $bc$–plane containing $x_k$. Again, since $\gamma$ is
a $\qq$–ray, we have

\begin{equation} \label{Eq:High}
\left(\ell_k + \sum_{i=1}^{k-1} |w_i| \right) \geq \rho_\qq \cdot t_k.
\end{equation}

\[
\rho_1 = \frac{\rho_\qq}2 \qquad\text{and}\qquad 0 < \rho_0 < \rho_1.
\]

We will show by induction that, for $r$ large enough,

\begin{equation}  \label{Eq:Estimates}
t_k \geq r \cdot (1 + \rho_1)^k \qquad\text{and}\qquad
\ell_k \geq r \cdot \frac{\rho_q}2 \cdot (1 + \rho_1)^k.
\end{equation}

The base case follows immediately from (<ref>) and (<ref>).
Assuming these inequalities for $k$, we have (from (<ref>)) that

\[
t_{k+1} \geq t_k + \rho_\qq \cdot \ell_k \geq
r \cdot (1 + \rho_1)^k  + r \cdot \frac{\rho_q^2}2 \cdot (1 + \rho_1)^k =  r \cdot (1 + \rho_1)^{k+1}.
\]

On the other hand, since the excursion of $\alpha_1$ is sub-exponential with respect to the distance in $T$,
there is $C$ depending on $\rho_0$ such that $|w_i| \leq C \cdot (1+\rho_0)^i$. Hence,

\[
\sum_{i=1}^{k-1} |w_i|  \leq C \sum_{i=1}^{k-1} \frac{(1+\rho_0)^k - 1)}{(1+\rho_0) -1}
\leq \frac{C}{\rho_0} \cdot (1+ \rho_0)^k.
\]

Choose $r$ large enough such that

\[
\frac{r \cdot \rho_\qq}2 \geq \frac{C}{\rho_0}.
\]

Then, by (<ref>), we have

\[
\ell_{k+1} \geq \rho_q \cdot t_{k+1} - \sum_{i=1}^{k} |w_i|   \geq
\rho_q \cdot   r \cdot (1 + \rho_1)^{k+1} -\frac{C}{\rho_0} \cdot (1+ \rho_0)^{k+1}
\geq r \cdot \frac{\rho_q}2 \cdot (1 + \rho_1)^{k+1}.
\]

This finishes the proof of (<ref>). Since $\rho_0 < \rho_1$, this implies that
$\gamma$ arrives in $A_i$ long after $\alpha_1$ has left $A_i$ and the distance between $\gamma$
and $\alpha_1$ goes to infinity (instead of zero). Thus, there does not exit a $\qq$–ray
redirecting $\zeta$ to $\alpha_1$. But this holds for every $\qq$. Hence, $\alpha_1 \not \in \bfz$.

We now prove the other direction. Assume that the excursion of $\alpha_1$ is not sub-exponential with respect to
the distance in $T$. This implies that

\begin{equation} \label{Eq:quantifiers}
\exists \rho>0 \quad \forall r>0 \quad \exists \, k >0 \quad \text{such that} \quad
|w_k| \geq r  \cdot (1+\rho)^k.
\end{equation}

\[
\forall \rho>0 \quad \exists r>0 \quad \forall k >0 \quad \text{we have} \quad
|w_k| \leq r \cdot \rho \cdot (1+\rho)^k,
\]

which implies

\[
\forall \rho>0 \quad \lim_{k \to \infty}\frac{\log |w_k|}{k} \leq \rho
\qquad \Longrightarrow \qquad  \lim_{k \to \infty}\frac{\log |w_k|}{k}=0.
\]

Let $\rho$ be as in (<ref>) and $r>0$ be given. We construct a $\qq$–ray
$\gamma$ (where $\qq$ depends on $\rho$) that quasi-redirect $\zeta$ to $\alpha_1$
at radius $r$. Let $k$ be the first index where the inequality in (<ref>) holds. That is,

\[
|w_k| \geq r \cdot (1+\rho)^k
\qquad\text{and}\qquad |w_i| < r \cdot (1+\rho)^i,
\qquad\text{for}\qquad 1 \leq i <k.
\]

Set $t_0 = r$ and, for $i=1, \dots, (k-1)$ set

\[
t_i = r \cdot (1 + \rho) ^i
\qquad\text{and}\qquad
\ell_i = r \cdot \rho \cdot (1 + \rho)^i
\]

Then, assuming $\rho$ is small enough such that

\[
\rho + \rho(1+\rho) + 1 \leq 2,
\]

we have

\begin{align*}
t_i + \frac{\rho}2(\ell_i + \ell_{i+1} + |w_i| )
& \leq  r \cdot (1 + \rho) ^i  + r\cdot \frac{\rho}2 \Big( \rho \cdot (1 + \rho)^i + \rho \cdot (1 + \rho)^{i+1} +  (1+\rho)^i\Big)  \\
& \leq  r \cdot (1 +  \rho) ^i  + r\cdot \rho \cdot (1 + 4 \rho)^i  =  t_{i+1}.
\end{align*}

Therefore, for an appropriate $\qq$ depending on $\rho$, there is enough time for a $\qq$–ray $\gamma$ that starts a distance
$\ell_i$ from $x_i$ to reach a point that is distance $\ell_{i+1}$ from $x_{i_l}$ while maintaining a distance $\ell_i$
from the $\alpha_1$. Since $\ell_i \geq \rho\, t_i$, this path is indeed a quasi-geodesic. Note that
$\rho/2$ is playing the role of $\rho_\qq$ in (<ref>).

We have shown that there exists a quasi-geodesic ray $\gamma$ where $\gamma|_r = \zeta|_r$
such that $\gamma$ that reached $A_k$ at time $t_k$ and a distance from $x_k$ is

\[
\ell_k = r \cdot \rho \cdot (1+\rho)^k < r \cdot (1+\rho)^k = |w_k|.
\]

Hence, by Lem:block-redirect, we can connect $\gamma$ to $x_{k+1}$ and continue along $\alpha_1$
while staying a quasi-geodesic. This gives a redirection from $\zeta$ to $\alpha_1$ at radius $r$.

Since this holds for every $r$, we can conclude that if the excursion of $\alpha_1$ is not sub-exponential with respect
to the distance in $T$ then $\alpha_1 \in \bfz$.

§.§ Description of $P(X)$

Let us review what we know about $P(X)$ so far. To every transient quasi-geodesic ray $\alpha$, we can associated
an itinerary $A_i$, which is an infinite embedded path in $T$ exiting an end $\xi$. Given such $\xi$,
there is a preferred quasi-geodesic $\alpha_1$ exiting $\xi$ that passes through the points $x_k$.
We set $w_k = x_{k+1} \cdot x_k^{-1}$ and refer to $|w_k|$ as the excursion of $\alpha_1$ in the block $A_i$.
Then $[\alpha_1]$ is different from $\bfz$ if and only if the excursion of $\alpha_1$ is sub-exponential with
respect to the distance in $T$. That is, every class in $P(X)$ is either $\bfz$ or $[\alpha_1]$ for such $\alpha_1$.
To finish the description of $P(X)$ we need to show all these classes are different.

Let $\alpha$ and $\beta$ be transient quasi-geodesic rays with different itineraries. If $\beta \preceq \alpha$
then $\zeta \preceq \alpha$.

Let $A_i$ be the itinerary for $\alpha$, $B_i$ be the itinerary for $\beta$ and let $k$ be the largest index where
$A_k = B_k$. Let $r>0$ be given. Consider a quasi-geodesic ray $\gamma$ constructed as before
such that $\gamma|_r = \zeta|_r$ and $\gamma$ follows the itinerary $A_i$. Let $t_k$ be the first time
$\gamma(t_k) \in A_k$ and $\ell_k - d(\gamma(t_k), x_k)$. Now choose $R \gg \ell_k$ and consider
a quasi-geodesic $\beta'$ quasi-redirecting $\beta$ to $\alpha$ at radius $R$. Then $\beta'$ arrives at and leaves
$A_k$ much later than $\gamma$. Hence, by Lem:block-redirect, we can redirect $\gamma$ to
$\beta'$, that is, construct a quasi-geodesic ray $\gamma'$ where $\gamma[0, t_k\ = \gamma'[0, t_k]$
and $\gamma'$ is eventually equal to $\beta'$. Since $\beta'$ is eventually equal to $\alpha$, $\gamma'$
quasi-redirects $\zeta$ to $\alpha$ at radius $r$. This can be done for every $r$ with uniform constants. Hence
$\zeta \preceq \alpha$.

If $\alpha_1$ and $\beta_1$ are transient quasi-geodesic rays whose excursion is sub-exponential with
respect to distance in $T$, then $[\alpha_1] \not = [\beta_1]$.

If $\alpha_ \preceq \beta_1$, then by Lem:different-or-zeta, $\zeta \prec \alpha_1$ which implies
$[\alpha_1] = \bfz$. But this contradicts Thm:Classification.

Therefore, we can think of $P(X)$ as a quotient of $\partial T$ where all the direction
whose excursion is not sub-exponential with respect to distance in $T$ are collapsed into one point $\zeta$.

§.§ An enlargement of the sublinearly Morse boundary

Finally, we check that not all directions are sublinearly Morse. Choose a sequence $\rho_n >0$ such that
$\rho_n \to 0$ as $n \to \infty$.
Fix a constant $C\geq1$ and, for each $n$, choose a power $k_n$ large enough such that

\begin{equation} \label{Eq:rho_n}
(1+\rho_n)^{k_n} \geq C \, k_n + \sum_{m=1}^{n-1}(1+\rho_m)^{k_m}.
\end{equation}

Then construct a quasi-geodesic $\alpha_1$ as before where

\[
|w_i| = \begin{cases}
(1+\rho_n)^{k_n} & \text{if $i = k_n$ for some $n$}\\
C & \text{otherwise}
\end{cases}.
\]

Eq:rho_n implies that, for every $n$,

\[
\frac{|w_{k_n}|}{\sum_{i=1}^{k_n-1} |w_i|} \geq 1,
\]

meaning, the excursion is not sublinear with respect to distance in $X$. Hence $\alpha_1$ is not
sublinearly Morse. However, since $\rho_n \to 0$, we have

\[
\frac{\log |w_i|}{i} \leq \rho_n \qquad\text{for $i \geq k_n$}
\qquad\Longrightarrow\qquad
\lim_{i\to \infty} \frac{\log |w_i|}{i} =0.
\]

That is, the excursion is sub-exponential with respect to distance in $T$.  These seem to be interesting new classes
of quasi-geodesic rays that warrant more study.

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