moreover that if $R_{1}\cap R_{2}=\emptyset$, then $P_{1}\cap P_{2}=\emptyset$
as well. Hence assuming the the two pieces $P_{1}$ and $P_{2}$ have non-empty
intersection amounts to say that $R_{1}\cap R_{2}=\\{r\\}$, therefore we need
to show that in the quotient the line $r$ is a single point, which amounts to
show that for all $x,y\in r$ we have $d_{\mu}(x,y)=0$. Assume by contradiction
$d_{\mu}(x,y)>0$, then there exists a line $l\in\mathrm{supp}(\mu)$ that
intersects transversely $r$, contradicting the fact that $r$ is special (i.e.
doesn’t intersect any other line in $\mathrm{supp}(\mu)$), as wanted.

  2. (2)

_axiom transversals_. Recall $\mathfrak{R}$ denotes the family of
complementary regions determined by the lifts of the special multi-curve $m$.
For every $x\in X_{\mu}$, let $T_{x}=\\{x\\}$ if $x$ is in the interior of
$R_{i}$, for $R_{i}\in\mathfrak{R}$, and, otherwise, let $T_{x}$ be the subset
of $X_{\mu}$ given by
$T_{\mu}\coloneqq(\widetilde{X}-\cup_{i}\accentset{\circ}{R_{i}})/\\{d_{\mu}=0\\}$
equipped with the restriction of the metric $d_{\mu}$. Since $T_{\mu}$ is
isometric to $X_{m}$, $T_{\mu}$ is $0$-hyperbolic. Moreover, the transversal
condition is trivially satisfied.

  3. (3)

_axiom straight chain triangle_. A straight chain in this setting corresponds
to a sequence $(y_{1},\cdots,y_{n})$, where $y_{i}$ are the points obtained by
collapsing lifts of simple closed curves in $m$. If a straight chain triangle
is contained in more than one piece, we show that the interior point of one of
the chains must intersect one of the points in another side chain. Indeed, let
the straight chain triangle be composed of side chains $C_{1},C_{2},C_{3}$,
and suppose that the endpoints of $C_{1}$, $x$ and $y$, are in distinct pieces
$P_{1},P_{2}$. Then, there is an interior point of $C_{1}$, say $y_{i}$, that
is the equivalence class of a lift of a special geodesic $\widetilde{m_{i}}$,
separating regions $R_{1},R_{2}$ corresponding to the pieces $P_{1},P_{2}$.
Since $\widetilde{m_{i}}$ separates $\widetilde{X}$ into two components, the
straight chain $C_{2}\cup C_{3}$, which also connects $x$ and $y$, must also
contain $y_{i}$.

∎

Let $P=R/{\\{d_{\mu}=0\\}}$ be a piece corresponding to a region $R$ which is
a lift of the subsurface $X_{i}\subseteq X$. The subsurface $X_{i}$ is a
hyperbolic surface with geodesic boundary, and its universal cover is
isometric to $R$. Moreover the pseudo-distance $d_{\mu_{i}}$ on
$R=\widetilde{X_{i}}$ coincides with the restriction of $d_{\mu}$ on $R$. It
follows that $P=(X_{i})_{\mu_{i}}$, i.e. the piece $P$ is the dual space of
the current $\mu_{i}\in\operatorname{Curr}(X)$, understood as a geodesic
current in $\operatorname{Curr}(X_{i})$ restricted to a region $R$. Precisely,
there is an inclusion $\iota\colon X_{i}\xhookrightarrow{}X$ inducing a
continuous injective pushforward of geodesic currents
$\iota_{*}\colon\operatorname{Curr}(X_{i})\to\operatorname{Curr}(X)$ (see
[EM18, Section 4.2]), which sends $\mu_{i}$ as a current on $X_{i}$ to
$\mu_{i}$ as a current on $X$. This allows us to restate the previous Theorem
7.11 as follows.

######  Theorem 7.12.

The dual space $X_{\mu}$ is a metric tree-graded space where the underlying
tree is the dual tree of the special multi-curve $m$ and the pieces are the
dual spaces of the subcurrents $\mu_{i}$ of $\mu$ on the subsurfaces $X_{i}$,
restricted to the regions determined by the lifts of the special multicurve.

We conclude this subsection by showing that the pseudo-distance $d_{\mu}$ on
$X_{\mu}$ can be computed from the pseudo-distances $d_{\mu_{i}}$ on the
pieces $P_{i}$.

In this setting, we define a _chain_ between two points $x,y\in\widetilde{X}$
is a sequence of points
$C=(\overline{x_{0}},\overline{x_{1}},\cdots,\overline{x_{n+1}})$ with
$\overline{x_{0}}=\overline{x}$ and $\overline{x_{n+1}}=\overline{y}$ such
that any two consecutive points $\overline{x_{i}}$ and $\overline{x_{i+1}}$
are in the same piece $P_{i}$, with $P_{i}\neq P_{i+1}$.

It follows from the definition that given $P_{j}$ and $P_{j+1}$, the point
$\overline{x_{j+1}}$ is on the common geodesic boundary $c_{j}$. We call a
chain _straight_ if it does not go ‘back and forth’, i.e., if $c_{j}\neq
c_{j+1}$ for $j=1,\dots n-1$. If a chain $C$ is straight, then the
corresponding ordered sequence of boundary geodesics $(c_{1},\dots,c_{n})$ are
precisely _the_ geodesics separating $\overline{x}$ from $\overline{y}$. Each
piece $P_{i}$ is naturally endowed with the induced pseudo-distance
$d_{\mu_{i}}$. We define the _length of a chain_ to be

$l(C)=\inf_{C}\sum_{i=1}^{n+1}d_{\mu_{i}}(\overline{x_{i}},\overline{x_{i+1}}),$

where the $\inf$ is taken over all chains $C$ joining $\overline{x}$ to
$\overline{y}$. It is enough to consider the $\inf$ over straight chains $c$.
Moreover, since the curves $c_{i}$ are special geodesics, there is no line in
the support of $\mu$ intersecting them, and hence the distance between any two
points on the same special geodesic $c_{i}$ is zero. It follows that
$d(\overline{x_{j}},\overline{x_{j+1}})$ does not depend on the choice of
$\overline{x_{j}}$ on $c_{j}$. This means that all straight chains from
$\overline{x}$ to $\overline{y}$ have the same length, and hence we may as
well define $d(\overline{x},\overline{y})$ as the length of _any_ straight
chain.

######  Lemma 7.13.

Let $C=(\overline{x_{0}},\cdots,\overline{x_{n+1}})$ be a straight chain from
$x$ to $y$ in $\widetilde{X}$. The pseudo-distance $d_{\mu}$ on
$\widetilde{X}$ can be expressed as

$d_{\mu}(\overline{x},\overline{y})=\sum_{i=1}^{n+1}d_{\mu_{i}}(\overline{x_{i}},\overline{x_{i+1}}).$

###### Proof.

Let $\gamma$ be the hyperbolic geodesic joining $\overline{x}$ to
$\overline{y}$.

By the above arguments we can consider, without loss of generality, $C$ to be
the chain such that $\overline{x_{j}}=\gamma\cap c_{j}$. Since the pseudo-
distance $d_{\mu}$ is straight, it follows that

$d_{\mu}(\overline{x},\overline{y})=\inf_{C}\sum_{i=0}^{n}d_{\mu_{i}}(\overline{x_{i}},\overline{x_{i+1}})=\sum_{i=0}^{n}d_{\mu_{i}}(\overline{x_{i}},\overline{x_{i+1}}),$

where the last equality follows from the fact that $\overline{x_{i}}$ and
$\overline{x_{i+1}}$ belong to the piece $P_{i}$. ∎

###  7.1. Properties of the decomposition

We recall the definition of graph of groups, as in [Kap09, 10.2]. Let $Y$ be a
finite graph where each edge is oriented. We assume that to each vertex $v$ of
$Y$ is assigned a _vertex_ group $G_{v}$ and to each edge $e$ is assigned an
_edge_ group $G_{e}$. Each inclusion $v\xhookrightarrow{}e$ of a vertex into
an edge (as the initial of terminal vertex) corresponds to a monomorphism
$h_{ev}\colon G_{e}\to G_{v}$. The collection

$(Y,\\{G_{e},G_{v},h_{ev}\colon\mbox{ where }e,v\mbox{ are edges and vertices
of }Y\\})$

is called a _graph of groups_ $(\mathcal{G},Y)$, where $\mathcal{G}$ is the
data of all vertex groups, edge groups and monomorphisms, and $Y$ is the
underlying graph. We denote with $\mathcal{G}^{0}$ and $\mathcal{G}^{1}$ the
set of vertices and edges of $Y$, respectively. When we don’t need to specify
the underrlying graph $Y$, we will refer to the graph of groups simply as
$\mathcal{G}$.

The fundamental group $\pi_{1}(\mathcal{G})$ of a graph of groups
$\mathcal{G}$ is defined as

$\pi_{1}(\mathcal{G})=\langle
G_{v},t_{e}:v\in\mathcal{G}^{0},e\in\mathcal{G}^{1}|t_{e}t_{\overline{e}}=1,{t_{e}}^{-1}h_{e}(g)t_{e}=h_{\overline{e}}(g)\text{
for all }g\in G_{e},e\in\mathcal{G}^{1}\rangle$

We have seen that any geodesic current $\mu$ decomposes as in 7.2, where each
component $\mu_{i}$ is supported on a subsurface $X_{i}$. The decomposition of
$X$ in the subsurfaces $X_{i}$ is given by the family of so-called _special
geodesics_ $\\{c_{1},\dots,c_{n}\\}$. Note that a special geodesic does not
need to be separating.

Given $\mu\in\mathrm{Curr}(X)$ we define a graph of groups $(\mathcal{G},Y)$
as follows. For each subsurface $X_{i}$ we define a vertex $v_{i}$, and for
each special curve $c$ we define an edge $e_{i,j}$ between the vertices
$v_{i}$ and $v_{j}$ if the curve $c$ is the boundary of $X_{i}$ and $X_{j}$.
Note that we may have $i=j$ when $c$ is not separating, and $e_{i,j}$ is in
this case a loop based at $v_{i}=v_{j}$.

For each edge $e\in\mathcal{G}^{1}$ we put $G_{e}\coloneqq\mathbb{Z}$, and for
each $v\in\mathcal{G}^{0}$ we put $G_{v_{i}}\coloneqq\pi_{1}(X_{i})$. Let
$e\in\mathcal{G}^{1}$ be an edge joining $v_{i}$ to $v_{j}$, and let $c$ be
the special geodesic bounding $X_{i}$ and $X_{j}$. The monomorphisms
$h_{v_{i},e}:G_{e}\to G_{v_{i}}$ is given by
$\iota_{*}:\mathbb{Z}\to\pi_{1}(X_{i})$, i.e. the induced map from the
inclusion $c\hookrightarrow X_{i}$ at the level of fundamental groups.

The following result is obtained by simply invoking classical Bass-Serre
theory (see [Ser03]).

######  Proposition 7.14.

The fundamental group $\pi_{1}(X)$ has a graph of groups decomposition. In
particular, $\pi_{1}(X)$ is isomorphic to $\pi_{1}(\mathcal{G},Y)$.
Furthermore, the fundamental group $\pi_{1}(\mathcal{G})$ acts on the
simplicial tree $T=\mathcal{T}(m)$ dual tree of the special multi-curve
$m=\sum_{j}a_{j}s_{j}$. The factor graph $T/\pi_{1}(\mathcal{G})$ is
isomorphic to $Y$, and for such action we have

  1. (1)

$\mathrm{stab}_{\pi_{1}(\mathcal{G})}(v)\cong G_{v}$ for all $v\in T^{0}$;

  2. (2)

$\mathrm{stab}_{\pi_{1}(\mathcal{G})}(e)\cong G_{e}$ for all $e\in T^{1}$.

##  8\. Actions

The dual spaces $X_{\mu}$ are naturally equipped with a $\pi_{1}(X)$-action.
We study the properties of this action.

Recall that $d_{\mu}$ denotes both the pseudo-distance in $\widetilde{X}$ as
well as the induced distance on $X_{\mu}$. We will distinguish them by writing
points in $\widetilde{X}$ with the overline notation
$\overline{x}\in\widetilde{X}$, and points in $X_{\mu}$ without it, $x\in
X_{\mu}$.

######  Definition 8.1 (Action).

Let $\pi_{\mu}\colon\widetilde{X}\to X_{\mu}$ be the natural quotient
projection. Given $x\in X_{\mu}$, let $\overline{x}\in\pi_{\mu}^{-1}(x)$. Let
$g\in\pi_{1}(X)$. Define $g\cdot x=\pi_{\mu}(g(\overline{x}))$.

######  Lemma 8.2.

The fundamental group $\pi_{1}(X)$ acts by isometries on $X_{\mu}$.

###### Proof.

First, we show that the action in Definition 8.1 is well-defined. Indeed,
suppose that $\overline{x},\overline{y}\in\pi_{\mu}^{-1}(x)$, i.e.,
$d_{\mu}(\overline{x},\overline{y})=0$, i.e.
$\frac{1}{2}\mu(G[\overline{x},\overline{y}))+\frac{1}{2}\mu(G(\overline{x},\overline{y}])=0$.
Since $\mu$ is $\pi_{1}(X)$-invariant, we have

$\displaystyle 0$
$\displaystyle=\frac{1}{2}\mu(gG[\overline{x},\overline{y}))+\frac{1}{2}\mu(gG(\overline{x},\overline{y}])=$
$\displaystyle\frac{1}{2}\mu(G[g\cdot\overline{x},g\cdot\overline{y}))+\frac{1}{2}\mu(G(g\cdot\overline{x},g\cdot\overline{y}])=$
$\displaystyle d_{\mu}(g\cdot\overline{x},g\cdot\overline{y}),$

which shows that $g\cdot\overline{y}\in\pi_{\mu}^{-1}(g\cdot y)$, and thus
shows that the action is well-defined. The same computation for two arbitrary
points $x,y\in X_{\mu}$ with lifts $\overline{x},\overline{y}$ shows that the
action is by isometries. ∎

######  Definition 8.3 (Translation length).

For $g\in\pi_{1}(X)$, we define

$\ell_{\mu}(g)\coloneqq\inf_{x\in X_{\mu}}d_{\mu}(x,g\cdot x).$

######  Lemma 8.4.

For $g\in\pi_{1}(X)$,

$\ell_{X_{\mu}}(g)=i(\mu,g)$

and thus

$\ell_{X_{\mu}}(g^{n})=n\ell_{X_{\mu}}(g)$

###### Proof.

The first can be found in [BIPP21a, Lemma 4.7]. The second result follows from
general properties about Bonahon’s intersection number. ∎

###  8.1. Coboundedness

######  Definition 8.5 (Cobounded/cocompact).

Let $X$ be a metric space. An action $(G,X)$ is said to be _cobounded_ if
there exists a bounded set $B\subset X$ so that $X=GB$, i.e.,
$X=\cup_{g}g(B)$. An action $(G,X)$ is said to be _cocompact_ if there exists
a compact set $K\subset X$ so that $X=GK$, i.e., $X=\cup_{g}g(K)$.

Let $X,Y$ be metric spaces and $f\colon X\to Y$ be map (not necessarily
continuous). We say that $f$ is _bornologous_ if for every $R>0$, there is
$S>0$ so that if $d_{X}(x,y)<R$ then $d_{Y}(f(x),f(y))<S$. We say that $f$ is
_large scale Lipschitz_ if there exist constants $c>0$ and $A$ so that

$d_{Y}(f(x),f(y))\leq c\cdot d_{X}(x,y)+A.$

$f$ is a _quasi-isometric embedding_ if there exist constants $c>1$ and $A$ so
that

$1/c\cdot d_{X}(x,y)-A\leq d_{Y}(f(x),f(y))\leq c\cdot d_{X}(x,y)+A.$

$f$ is _coarsely surjective_ if there exists a constant $C$ so that for every
$y\in Y$ there is $z\in g(X)$ so that $d_{Y}(z,y)<C$. We say that $f$ is a
_quasi-isometry_ if it is a coarsely surjective quasi-isometric embedding.

######  Lemma 8.6 ([Roe03, Lemma 1.10]).

Let $X$ be a length space and $Y$ a metric space. $f$ is bornologous if and
only if it is large scale Lipschitz.

######  Proposition 8.7.

Let $\mu\in\operatorname{Curr}(X)$. Then $\pi_{\mu}\colon\widetilde{X}\to
X_{\mu}$ is large scale Lipschitz.

###### Proof.

Given $R>0$, we will find $S_{R}>0$ so that if $d_{\widetilde{X}}(x,y)<R$,
then $d_{\mu}(\pi_{\mu}(x),\pi_{\mu}(y))<S_{R}$. Let $K\subset\widetilde{X}$
be a compact fundamental domain of the action of $\pi_{1}(X)$ on
$\widetilde{X}$, and let $D$ be the hyperbolic diameter of $K$. If
$d_{\widetilde{X}}(x,y)\leq D$, then there exist $g_{1},g_{2}\in\pi_{1}(X)$
(depending on $x,y$), so that $x\in g_{1}K$ and $y\in g_{2}g_{1}(K)$ (where
$g_{1},g_{2}$ could be the identity), and $g_{2}g_{1}(K)$ is adjacent to
$g_{1}K$. Indeed, otherwise $d_{\widetilde{X}}(x,y)>D$, contradicting the
choice of $D$. Thus, $x,y\in g_{1}(K)\cup g_{2}g_{1}(K)$. Let $G(K)$ be the
set of geodesics intersecting $g_{1}(K)\cup g_{2}g_{1}(K)$. We have, by
subadditivity of $\mu$ and $\pi_{1}(X)$ invariance, that

$\displaystyle d_{\mu}(\pi_{\mu}(x),\pi_{\mu}(y))$
$\displaystyle=d_{\mu}(x,y)=\frac{1}{2}\\{\mu(G(x,y])+\mu(G[x,y))\\}\leq\mu(G[x,y])$
$\displaystyle\leq\mu(G(g_{1}K)\cup
G(g_{2}g_{1}K))\leq\mu(G(g_{1}K))+\mu(G(g_{2}g_{1}K))$ $\displaystyle=2\mu(K)$

is finite (and independent of $x,y$). Furthermore, $\mu(G(K))$ is finite,
since $K$ is compact. Thus, we have proven that if $d_{\widetilde{X}}(x,y)\leq
D$, then $d_{X_{\mu}}(\pi_{\mu}(x),\pi_{\mu}(y))\leq 2\mu(G(K))$. If
$D<d_{\widetilde{X}}(x,y)\leq 2D$, the same proof now using three
consecutively adjacent fundamental regions will yield that
$d_{X_{\mu}}(x,y)\leq 3\mu(G(K))$. An induction argument then proves the
general case. ∎

######  Lemma 8.8.

Let $\mu$ be any geodesic current. The action of $\pi_{1}(X)$ on $X_{\mu}$ is
cobounded.

###### Proof.

By Proposition 8.7, there exist constants $c>0$ and $A\geq 0$ so that for all
$\overline{x},\overline{y}\in\widetilde{X}$,

$d_{\mu}(\pi_{\mu}(\overline{x}),\pi_{\mu}(\overline{y}))\leq c\cdot
d_{\widetilde{X}}(\overline{x},\overline{y})+A.$

Thus, for every $x\in X_{\mu}$, and $r>0$, we have

$B_{\widetilde{X}}(\overline{x},r)\subseteq B_{\mu}(x,c\cdot r+A)$

where $\overline{x}\in\pi_{\mu}^{-1}(x)$. Since $\pi_{1}(X)$ acts cocompactly
on $\widetilde{X}$, there exists a compact subset $K\subseteq\widetilde{X}$ so
that $\widetilde{X}=\cup_{g}gK$. Take a hyperbolic ball
$B_{\widetilde{X}}(\overline{x},r)$ so that

$K\subseteq B_{\widetilde{X}}(\overline{x},r).$

Then, by the above,

$K\subseteq B_{\mu}(x,c\cdot r+A).$

Thus, by equivariance of $\pi_{\mu}$, we have

$X_{\mu}=\bigcup_{g\in\pi_{1}(X)}gB_{\mu}(x,c\cdot r+A),$

so the action is cobounded, as we wanted to see. ∎

If $\mu$ has no atoms, we can moreover show the action is cocompact.

######  Lemma 8.9.

If $\mu$ is a current with no atoms, then the action of $\pi_{1}(X)$ on
$X_{\mu}$ is cocompact.

###### Proof.

Since $\pi_{1}(X)$ acts cocompactly on $\widetilde{X}$, there exists a compact
subset $K\subseteq\widetilde{X}$ so that $\widetilde{X}=\cup_{g}gK$. Let
$K^{\prime}=\pi_{\mu}(K)$, which is compact by continuity of $\pi_{\mu}$.
Thus, by equivariance of $\pi_{\mu}$, we have

$X_{\mu}=\bigcup_{g\in\pi_{1}(X)}gK^{\prime}$

as we wanted to see. ∎

###  8.2. Boundedness and compactness of balls

We show that a geodesic current $\mu$ is filling if and only if all balls in
$X_{\mu}$ (equivalently, balls in the $d_{\mu}$-pseudo metric) are bounded.
Moreover, we show that if $\mu$ has no atoms and it is filling, then $X_{\mu}$
is a proper metric space.

The following proof is adapted from [Glo17, Proposition 2.8]. We claim no
originality, but supply details and point out a gap in that proof.

######  Proposition 8.10.

Let $\overline{x}\in\widetilde{X}$ and $\mu\in\operatorname{Curr}(X)$. The
following are equivalent:

  1. (1)

The current $\mu$ is filling;

  2. (2)

for every $\overline{x}\in\widetilde{X}$, $B_{\mu}(\overline{x},0)$ is
bounded;

  3. (3)

for every $\overline{x}\in\widetilde{X}$ and every $r\geq 0$,
$B_{\mu}(\overline{x},r)$ is bounded.

###### Proof.

3\. $\Rightarrow$ 2\. is obvious.

2\. $\Rightarrow$ 1\. Assume $\mu$ is not filling, then there exists a
geodesic line $g\in\mathcal{G}(\widetilde{X})$ which does not intersect any
line in $\mathrm{supp}(\mu)$. It follows that
$d_{\mu}(\overline{x},\overline{y})=0$ for any $\overline{x},\overline{y}$ on
the geodesic $\gamma$. In particular $\gamma\subset B_{\mu}(\overline{x},0)$,
and hence $B_{\mu}(\overline{x},0)$ is not bounded, contradiction.

1\. $\Rightarrow$ 3\. Now we assume that $\mu$ is filling. By Lemma 8.14,
there exists $R>0$ so that $B_{\mu}(\overline{x},r)\subset
B_{\widetilde{X}}(\overline{x},R)$. Since $B_{\widetilde{X}}(\overline{x},R)$
is bounded, it follows that $B_{\mu}(x,\varepsilon)$ is bounded.

∎

######  Lemma 8.11.

Let $\mu$ be a filling geodesic current. There exist a constant $C(\mu)>0$ and
a constant $\varepsilon(C,\mu)>0$, so that if
$d_{\widetilde{X}}(\overline{x},\overline{x})>C+1$ then
$d_{\mu}(x,y)>\varepsilon$.

###### Proof.

The surface $X$ has been endowed with a fixed hyperbolic structure. Let
$\phi:T^{1}Z\times\mathbb{R}\to T^{1}X$ denote the geodesic flow on $T^{1}X$
and $\pi\colon T^{1}X\to X$ the canonical projection, and $p:\widetilde{X}\to
X$ the universal covering. Define $r\colon T^{1}X\to\mathbb{R}$ to be the
_first return time_ of $\phi_{t}$ to the support of $\mu$

$r(v)\coloneqq\inf\\{t\in\mathbb{R}:\pi(\phi(v,t))\cap\kern-6.99997pt|\kern
6.99997ptp(\mathrm{supp}\mu)\neq\emptyset\\}$

In other words, $r(v)$ is the first time when the geodesic emanating from
$v\in T^{1}X$ intersects the support of $\mu$ transversely. Notice that $r(v)$
is finite since $\mu$ is filling.

Since the function $r$ is upper semi-continuous (see, for example, [MGT21,
Lemma 7.3]), it follows that it admits an upper bound $C>0$ on the compact set
$T^{1}X$. By lifting $r$ to $\widetilde{X}$ we have the upper bound $r(v)\leq
C$ for all $v\in T^{1}\widetilde{X}$.

Now let us fix $\overline{x},\overline{y}\in\widetilde{X}$ such that
$d_{\widetilde{X}}(\overline{x},\overline{y})\geq C+1$. Then the segment
$[\overline{x},\overline{y}]$ must intersect transversely some geodesic line
in the support of $\mu$, and therefore $\mu(G[\overline{x},\overline{y}])>0$.
If for every $n$, we could find
$\overline{x_{n}},\overline{y_{n}}\in\widetilde{X}$ so that
$d_{\widetilde{X}}(\overline{x_{n}},\overline{y_{n}})\geq C+1$ and
$\mu(G[\overline{x_{n}},\overline{y_{n}}])<\frac{1}{n}$, then compactness of
$X$ would yield points
$\overline{x_{\infty}},\overline{y_{\infty}}\in\widetilde{X}$ with
$d_{\widetilde{X}}(\overline{x_{\infty}},\overline{y_{\infty}})\geq C+1$ but
$\mu(G[\overline{x_{\infty}},\overline{y_{\infty}}])=0$, which would
contradict the choice of $C$. Thus, there must exist a uniform lower bound
$\varepsilon>0$ so that for every $\overline{x},\overline{y}\in\widetilde{X}$
such that $d_{\widetilde{X}}(\overline{x},\overline{y})>C+1$, we have
$\mu(G[\overline{x},\overline{y}])>\varepsilon>0$. This shows that

$B_{\mu}(\overline{x},\varepsilon)\subset
B_{\widetilde{X}}(\overline{x},C+1),$

and thus $B_{\mu}(\overline{x},\varepsilon)$ is bounded. ∎

######  Remark 8.12.

Note that in [Glo17, Proposition 2.8] it is claimed that
$B_{\mu}(\overline{x},\varepsilon)$ is compact. We contest this: in fact, it
is not true in general. Observe that since, by Lemma 8.11,
$B_{\mu}(\overline{x},\varepsilon)$ is bounded, its compactness is equivalent
to $B_{\mu}(\overline{x},\varepsilon)$ being closed in $\widetilde{X}$. At the
same time,

$B_{\mu}(\overline{x},\varepsilon)=\\{\overline{y}\in\widetilde{X}\colon
d_{\mu}(\overline{y},\overline{x})\leq\varepsilon\\}$

which is closed if and only if $d_{\mu}(\overline{y},\cdot)$ is lower
semicontinuous (in the topology of $\widetilde{X}$ induced by the hyperbolic
metric). However, we know from Propositions 4.7 and 4.12 this happens if and
only if $\mu$ has no atoms. We collect this in the next proposition.

A metric space $(X,d)$ is _proper_ if closed balls are compact.

######  Proposition 8.13.

Given a geodesic current $\mu$ on $X$. If $\mu$ is filling and has no atoms,
then $X_{\mu}$ is proper.

###### Proof.

Assume $\mu$ is filling and has no atoms. Let $x\in X_{\mu}$ and
$B=B_{\mu}(x,r)$ be a closed $d_{\mu}$-ball. The preimage $\pi_{\mu}^{-1}(B)$
is the closed $d_{\mu}$-ball in $\widetilde{X}$ (in the pseudo-metric
$d_{\mu}$), which by Proposition 8.11 is bounded. If $\mu$ has no atoms, then
$\pi_{\mu}$ is continuous, thus $\pi_{\mu}^{-1}(B)$ is closed. Altogether,
this shows $\pi_{\mu}^{-1}(B)$ is compact, and continuity of $\pi_{\mu}$ again
implies $B$ is compact, so $X_{\mu}$ is proper. ∎

######  Lemma 8.14.

Consider a filling geodesic current $\mu$ on a closed hyperbolic surface $X$.
There exists a constant $\varepsilon(\mu,X)>0$ so that for all
$r\geq\varepsilon$, there exists a constant $R(r)>0$, so that for all
$\overline{x}\in\widetilde{X}$,

$B_{\mu}(\overline{x},r)\subset B_{\widetilde{X}}(\overline{x},R).$

###### Proof.

Let $C$ and $\varepsilon$ be the constants given by Lemma 8.11. Let
$\overline{x},\overline{y}\in\widetilde{X}$ with
$d_{\widetilde{X}}(\overline{x},\overline{y})\geq C+1$ and and
$d_{\mu}(\overline{x},\overline{y})\geq\varepsilon$. In other words, we have

$B_{\mu}(\overline{x},\varepsilon)=\\{\overline{y}\in\widetilde{X}\colon
d_{\mu}(\overline{x},\overline{y})\leq\varepsilon\\}\subseteq
B_{\widetilde{X}}(\overline{x},C+1)$

Now we proceed by induction on the distance
$d_{\widetilde{X}}(\overline{x},\overline{y})=N\cdot(C+1)$, for any
$N\in\mathbb{N}$. The induction basis has been proven already. Assume that the
following implication holds

$d_{\widetilde{X}}(\overline{x},\overline{y})\geq(N-1)\cdot(C+1)\Rightarrow
d_{\mu}(\overline{x},\overline{y})\geq(N-1)\cdot\varepsilon$

assume that $d_{\widetilde{X}}(\overline{x},\overline{y})\geq N\cdot(C+1)$. We
want to show that $d_{\mu}(\overline{x},\overline{y})\geq N\cdot\varepsilon$.

Let $\gamma\in\mathcal{G}({{\widetilde{X}}})$ be the geodesic line joining
$\overline{x}$ to $\overline{y}$ in ${\widetilde{X}}$. Since
$d_{\widetilde{X}}(\overline{x},\overline{y})\geq N(C+1)$, then there must
exist $\overline{z}\in\gamma$ such that
$d_{\widetilde{X}}(\overline{x},\overline{z})\geq(N-1)(C+1)$ and
$d_{\widetilde{X}}(\overline{z},\overline{y})\geq C+1$, and hence, by
induction hypothesis $d_{\mu}(\overline{x},\overline{z})\geq(N-1)(C+1)$ and
$d_{\mu}(\overline{z},\overline{y})\geq\varepsilon$. Since the pseudo distance
$d_{\mu}$ is straight, we have

$d_{\mu}(\overline{x},\overline{y})=d_{\mu}(\overline{x},\overline{z})+d_{\mu}(\overline{z},\overline{y})\geq(N-1)(C+1)+(C+1)=N(C+1)$

as wanted. ∎

######  Proposition 8.15.

Let $\mu\in\operatorname{Curr}(X)$. If $\operatorname{sys}(\mu)>0$, then
$\pi_{\mu}\colon\widetilde{X}\to X_{\mu}$ is a quasi-isometry.

###### Proof.

First, note that for every $x\in X_{\mu}$, $\pi_{\mu}^{-1}(x)$ is a bounded
subset of $\widetilde{X}$. Indeed, its hyperbolic diameter is upper bounded by
the maximum of the return time, as in the proof of Lemma 8.11. Making an
arbitrary choice of $z_{x}\in\pi_{\mu}^{-1}(x)$ for each $x\in X_{\mu}$, we
define a map $g\colon X_{\mu}\to\widetilde{X}$, given by $x\mapsto z_{x}$.
This map is bornologous by Lemma 8.14. It follows then by Lemma 8.6, that
there exist constants $c>0$ and $A$ so that

$d_{\widetilde{X}}(\overline{x},\overline{y})\leq c\cdot
d_{\mu}(\pi_{\mu}(\overline{x}),\pi_{\mu}(\overline{y}))+A.$

for every $\overline{x},\overline{y}\in g(X_{\mu})$. This, together with Lemma
8.7 shows that $\pi_{\mu}$ is a quasi-isometric embedding. We note that $g$ is
coarsely surjective: the upper bound $C$ of the return time satisfies the
property that, for every $\overline{x}\in\widetilde{X}$, there is
$\overline{y}\in g(X_{\mu})$ so that
$d_{\widetilde{X}}(\overline{x},\overline{y})<C$. From this it follows
$\pi_{\mu}$ is a quasi-isometry. ∎

The following definition can be found in [BH11, Definition I.8.2].

######  Definition 8.16 (Proper action).

Let $G$ a group acting by isometries on a metric space $X$. The action is said
to be proper if for each $x\in X$, there exists $r>0$, so that the set
$\\{g\in G\colon gB(x,r)\cap B(x,r)\neq\emptyset\\}$ is finite, where $B(x,r)$
denotes an open ball of radius $r$ centered at $x$.

The action $\pi_{1}(X)$ on $X_{\mu}$ is not proper in general. In this section
we characterize when it is.

Let $d^{*}$ denote the distance on $T^{1}\widetilde{X}$ and $d$ the distance
on $X$. We recall the following classical result in the dynamical properties
of the geodesic flow of a hyperbolic surface, known as the Closing lemma.

######  Lemma 8.17 ([Ebe79, 4.5.15]).

Given a compact set $C\subset T^{1}\widetilde{X}$ and $\xi>0$, there exists
$T\geq 0$, and $\delta>0$ such that if there is $t\geq T$, $v\in C$, and
$g\in\pi_{1}(X)$ and $d^{*}(g(v),g_{t}(v))<\delta$, then there is
$t^{\prime}\in\mathbb{R}$ with $|t^{\prime}-t|<\xi$ and $v^{\prime}\in
T^{1}\widetilde{X}$ with $d^{*}(v^{\prime},v)<\xi$ and
$g(v^{\prime})=g_{t^{\prime}}(v^{\prime})$.

Using this result, we can prove the following.

######  Proposition 8.18.

If $\mu$ is a geodesic current with a subcurrent of type 2 in its
decomposition, then the action of $\pi_{1}(X)$ on $X_{\mu}$ is not proper.

###### Proof.

This follows from elaborating on the proof of [BIPP21a, Proposition 5.1]. Let
$\mu_{i}$ be a type 2 current in the decomposition of $\mu$, and $Y$ be the
subsurface $X_{i}$ on which $\mu_{i}$ is supported in the decomposition of
$\mu$. Recall that a geodesic $c$ is recurrent in $\mathring{Y}$ if there
exists a sequence $(t_{n})$ in $\mathbb{R}$ with $\lim_{n}|t_{n}|=\infty$ and
the sequence $(c(t_{n}))$ staying in a fixed compact subset $K$ of
$\mathring{Y}$. Up to reparameterization we may assume that the sequence
$(t_{n})$ is monotone increasing with $\lim t_{n}=+\infty$. Let $v\in
T^{1}\mathring{Y}_{p}$ be an accumulation point of the sequence
$(c^{\prime}(t_{n}))$. Up to enlarging $K$ we may assume $p\in\mathring{K}$.
Let $s\colon\mathbb{R}\to X$ be the unit speed geodesic with
$s^{\prime}(0)=v$, and $\gamma\subset\widetilde{X}$ be a lift of $s$. Let
$\overline{B}(\widetilde{p},\eta)$ be the closed ball centered at
$\widetilde{p}$ of radius $\eta$ for the hyperbolic metric in $\widetilde{X}$.
In [BIPP21a, Proposition 5.1] it is shown that for every $\varepsilon>0$, and
for all $x\in\gamma$ except at most countably many, there exists $\eta>0$ so
that
$\mu(G(\overline{B}(\widetilde{p},\eta)\setminus\\{\gamma\\})<\varepsilon$.
Decreasing $\eta$ if necessary, we can in addition assume $\eta$ is smaller
than $d_{X}(p,\partial X)$ and the injectivity radius at $p$, so that the
projection from $\widetilde{X}$ to $X$ sends
$\overline{B}(\widetilde{p},\eta)$ to
$\overline{B}(p,\eta)\subset\mathring{X}$ isometrically. Since $c$ is not
closed, $c^{\prime}(t_{n})$ is never tangent to $s$. We can assume $c(t_{n})$
is not in $s$ and, by passing to a subsequence, the points $c(t_{n})$ are on
the same side of $s$ in $\overline{B}(p,\eta)$. Now let $0<\xi<\eta$ and $C$
be the compact set consisting of unit tangent vectors based at a point of
$\bar{B}(p,\eta)$. Let $T$ and $\delta$ be the corresponding constants given
by Lemma 8.17. We may assume $\delta<\eta$, and choose $n_{0}\in\mathbb{N}$ so
that $d^{*}(c_{n}^{\prime}(t),v)<\delta$ for all $n\geq n_{0}$. Then, pick
$n>m>n_{0}$ so that $t_{n}-t_{m}>T_{k}$ and thus, by the hypotheses,
$d^{*}(c_{n}^{\prime}(t),c_{m}^{\prime}(t))<\delta$. Now, we construct a
sequence of curves by taking a sequence of shrinking $\varepsilon$’s in the
above construction, yielding corresponding $\eta_{k}$, as follows. Letting
$\varepsilon=1/k$, we get a number $\eta_{k}>0$ satisfying the properties
above, and we can construct a loop $\alpha_{k}$ by concatenating a geodesic
segment $c([0,t_{n}-t_{m}])$ and the geodesic segment connecting $c(t_{n})$
and $c(t_{m})$ within $\overline{B}(p,\eta_{k})$. As in the proof of [BIPP21a,
Proposition 5.1], the closed lemma gives us a closed geodesic $c_{k}$ in the
homotopy class of $\alpha_{k}$ represented by an element $g_{k}\in\pi_{1}(X)$
with the key property that $i(\mu,[g_{k}])<\frac{1}{k}$. We now analyze the
hyperbolic geometry of any of its lifts. A lift of $\alpha_{k}$ in
$\widetilde{X}$ is a piecewise geodesic path consisting of long geodesic
segments of hyperbolic length larger than $T_{k}$, and short geodesic segments
of length $1$. Each endpoint $\widetilde{\alpha_{k}}^{\pm}$ of such lift is
contained in intervals $I_{k}\subset\partial\widetilde{X}$ of diameter
shrinking to $0$ as $k$ goes to infinity. Thus, $\alpha_{k}^{+}$ (resp.
$\alpha_{k}^{-}$) converges to some $\alpha^{+}$ (resp. $\alpha_{-}$) in
$\partial\widetilde{X}$. Observe that the endpoints $\alpha_{k}^{\pm}$ are the
same as the endpoints of the lift of the closed geodesic $c_{k}$ stabilized by
$g_{k}$, since the periodic orbits $c_{k}$ is in the homotopy class of
$\alpha_{k}$. Thus, if we also denote $\widetilde{c_{k}}$ the geodesic lifts
of the closed geodesics $c_{k}$, we have that $\widetilde{c_{k}}$ converges to
some geodesic $\widetilde{c}$ in the Gromov-Hausdorff topology of
$\mathcal{G}(\widetilde{X})$. Observe, too, that $\ell_{X}(g_{k})\asymp T_{k}$
(where $\asymp$ means asymptotically equal up to multiplicative and additive
error independent of $k$), since the hyperbolic length of $\alpha_{k}$ is
$T_{k}+1$ and $\widetilde{\alpha_{k}}$ and $\widetilde{c_{k}}$ are quasi-
isometric. Therefore (up to taking a subsequence), the $g_{k}$ are hyperbolic
isometries of different translation lengths, and thus distinct elements of
$\pi_{1}(X)$. We also note that, by construction, $i(\mu,g_{k})<\frac{1}{k}$.
By picking $\overline{x}\in\widetilde{c}$, letting
$x=\pi_{\mu}(\overline{x})$, and taking $B_{\mu}(x,r)$, for an arbitrary
$r>0$, by the proof of [MZ19, Proposition 3.14], we have

$d_{\mu}(\bar{x},g_{k}\bar{x})<i(\mu,g_{k})+2d_{\mu}(\bar{x},A_{c_{k}})<\frac{1}{k}+2r<3r$

for all $k$ larger than $1/r$. Thus, we have that, for every $r$,

$B_{\mu}(\bar{x},3r)\cap g_{k}B_{\mu}(\bar{x},3r)\neq\emptyset$

for infinitely many distinct $g_{k}\in\pi_{1}(X)$. This shows the action of
$\pi_{1}(X)$ is not proper on $X_{\mu}$. ∎

######  Proposition 8.19.

If $\operatorname{sys}(\mu)>0$, then $\pi_{1}(X)$ acts properly on $X_{\mu}$.

###### Proof.

Suppose not, i.e., suppose there exists $x$ so that for all $r>0$, the set
$\\{g\in\pi_{1}(X)\colon gB_{\mu}(x,r)\cap B_{\mu}(x,r)\neq\emptyset\\}$ is
infinite. Consider a sequence $(g_{n})$ of elements in
$\pi_{1}(X)\cap\\{g\in\pi_{1}(X)\colon gB_{\mu}(x,r)\cap
B_{\mu}(x,r)\neq\emptyset\\}$. Choose $r>\varepsilon$, where $\varepsilon$ is
the constant in Lemma 8.14, so that $B_{\mu}(\overline{x},r)\cap
g_{i}(B_{\mu}(\overline{x},r))\neq\emptyset$. By Lemma 8.14, we have,

$B_{\mu}(\overline{x},r)\cap g_{i}(B_{\mu}(\overline{x},r))\subset
B_{\widetilde{X}}(\overline{x},R)\cap
g_{i}(B_{\widetilde{X}}(\overline{x},R)).$

This contradicts that the action of $\pi_{1}(X)$ on $\widetilde{X}$ is proper.
∎

The following theorem follows immediately from the previous propositions.

######  Theorem 8.20.

The action of $\pi_{1}(X)$ on $X_{\mu}$ is proper if and only if $\mu$ has no
type 2 subcurrents in its decomposition.

###### Proof.

If $\operatorname{sys}(\mu)>0$, then $\mu$ has no type 2 subcurrents in its
decomposition. Proposition 8.19 implies the action of $\pi_{1}(X)$ on
$X_{\mu}$ is proper. If $\operatorname{sys}(\mu)=0$ then either $\mu$ has a
subcurrent of type 2 in its decomposition, in which case Proposition 8.18
implies the action is not proper, or $\mu$ has no components of type 2 in its
decomposition. In this case, by the graph of groups decomposition in
Proposition 7.14, all the pieces in the tree graded space structure of
$X_{\mu}$ correspond to duals of subcurrents of type 1, and thus of positive
systole $\mu_{i}$ in their respective subsurfaces $X_{i}$. Then, by
Proposition 8.19 it follows the action of $\iota_{*}(\pi_{1}(X_{i}))\subset X$
on $X_{\mu_{i}}$ is proper. Hence, by the graph of groups decomposition of
$\pi_{1}(X)$, it follows that the action of $\pi_{1}(X)$ on $X_{\mu}$ is
proper. ∎

###  8.3. Freeness

The action of $\pi_{1}(X)$ on $X_{\mu}$ is not always free. In this section we
characterize when it is.

The following lemma relates stabilizers of points of $X_{\mu}$ and the
topology of the support of a measured lamination $\mu$, and it’s just an
observation.

######  Lemma 8.21.

Let $\mu$ be a non-trivial geodesic current. Let $\mathcal{C}$ be the set of
connected components of $\widetilde{X}\setminus\operatorname{supp}\mu$.

  1. (1)

If $x\in C\in\mathcal{C}$, then the stabilizer of $\pi_{\mu}(x)$ is equal to
the (set-wise) stabilizer of $C$.

  2. (2)

If $x\in\gamma\in\operatorname{supp}\mu$, so that $\gamma$ doesn’t intersect
any geodesic in $\operatorname{supp}\mu$, then the stabilizer of
$\pi_{\mu}(x)$ in $X_{\mu}$ is the stabilizer of $\gamma$ in $\widetilde{X}$.

  3. (3)

If $\overline{x}\in\gamma\in\operatorname{supp}\mu$, so that $\gamma$
intersects some geodesic in $\operatorname{supp}\mu$, then the stabilizer of
$\pi_{\mu}(\overline{x})$ in $X_{\mu}$ is trivial.

###### Proof.

  1. (1)

Clear.

  2. (2)

Clear.

  3. (3)

Let $\gamma^{\prime}\in\operatorname{supp}(\mu)$ so that
$\gamma\cap\gamma^{\prime}\neq\emptyset$. Then $\pi_{\mu}^{-1}(x)$ is a proper
geodesic subsegment $\eta$ of $\gamma$, and the (set-wise) stabilizer of
$\eta$ is trivial.

∎

######  Lemma 8.22.

Let $\mu$ be a non-trivial geodesic current. Then $\operatorname{sys}(\mu)>0$
if and only if $\mu$ consists of one and only one type 1 component.

###### Proof.

Since $\mu$ is non-trivial, then it has at least one component. If $\mu$ has
more than one type 1 component, then, by the decomposition Theorem 7.2, if
there is at least one special curve in $c\in\mathcal{E}_{\mu}$. But then
$i(\mu,c)=0$, so $\operatorname{sys}(\mu)=0$. If $\mu$ has no type 1
component, since $\mu$ is non-trivial, it must have some non-trivial type 2 or
type 3 component. A type 3 component yields a non-trivial stabilizer as
before. A unique type 2 component means, by Theorem [BIPP21a, Theorem 1.7],
that $\operatorname{sys}(\mu)=0$. ∎

######  Lemma 8.23.

Let $\mu$ be a non-trivial geodesic current. Then the action of $\pi_{1}(X)$
on $X_{\mu}$ is free if and only if $\mu$ has only one component in its
decomposition.

###### Proof.

Suppose that $\mu$ has more than one component in its decomposition. Then,
there is at least one curve $c\in\mathcal{E}_{\mu}$, which is a simple closed
curve not intersected by any other geodesic of the support of $\mu$. By Lemma
8.21, if $\gamma$ is a lift of $c$, the stabilizer of $\pi_{\mu}(\gamma)\in
X_{\mu}$ is infinite cyclic, so the action of $\pi_{1}(X)$ on $X_{\mu}$ is not
free. Now, suppose $\mu$ has only one component in its decomposition. We show
that the action is free. Assume first $\operatorname{sys}(\mu)>0$. If
$g\in\pi_{1}(X)$ had a fixed point, i.e., $g(x)=x$ for some $x\in X_{\mu}$.
Then $\ell_{X}(g)=0$, i.e., $i(\mu,[g])=0$, contradicting that the systole is
positive. Assume, to finish, that $\operatorname{sys}(\mu)=0$. If
$g\in\pi_{1}(X)$ had a fixed point, i.e., $g(x)=x$ for some $x\in X_{\mu}$.
Then $\ell_{X}(g)=0$, i.e., $i(\mu,[g])=0$. If $c=[g]$ is a closed curve, then
$i(\mu,c)=0$, contradicting that $\mu$ is a type 2 subcurrent (and thus
filling as a measured lamination within its support subsurface). ∎

######  Remark 8.24.

We give here a brief comment on isometry types and axes, which will not be
used in the sequel. Recall that for Gromov hyperbolic spaces, there is an
analogous classification of isometries in three types: hyperbolic, parabolic
and elliptic. The classification can be described also in terms of the number
of fixed points at infinity (see [CDP90, Chapter 10]). Observe that from the
previous results, it follows that $g\in\pi_{1}(X)$ acts on $X_{\mu}$ as an
elliptic or hyperbolic isometry. Let $g\in\pi_{1}(X)$. If $i(\mu,[g])>0$ then
$\ell_{X_{\mu}}(g)>0$, and thus $g$ is hyperbolic (by [Fuj15, 2.2]). Suppose
$i(\mu,[g])=0$. If $[g]$ is a special geodesic in the decomposition of $\mu$,
then it follows it has a fixed point, as in the proof of Lemma 8.23.
Otherwise, it’s in one of the subsurfaces $X_{i}$ of the decomposition of $X$.
By the same argument as in Lemma 8.23, it must be in a subsurface that does
not contain any projections of leaves in the support of $\mu$, and so $g$ has
a fixed point. Similarly, one can define a notion of axis of a hyperbolic
element $g$ as

$T_{g}=\\{x\in X_{\mu}\colon d_{\mu}(x,g(x))=\ell_{X_{\mu}}(g)\\}.$

By [MZ19, Proposition 4.4], it follows that the hyperbolic axis $A_{g}$ of $g$
is contained in $\pi_{\mu}^{-1}(T_{g})$. In general, $T_{g}$ is geodesic
(whenever $X_{\mu}$ is equipped with a geodesic structure) and corresponds to
the sequence of complementary regions $R_{i}$ in $\widetilde{X}$ traversed by
$A_{g}$, as in the setup of the decomposition theorem of Section 7. Finally,
if two hyperbolic elements $g,h$ in $X_{\mu}$ have intersecting hyperbolic
axes $A_{g},A_{h}$ in $X$, then their axes $T_{g}$ and $T_{h}$ in the dual
also intersect.

######  Theorem 8.25.

If $\operatorname{sys}(\mu)>0$, then $\pi_{1}(X)$ acts properly, coboundedly
and freely on $X_{\mu}$.

###### Proof.

It follows by Lemma 8.8, Lemma 8.19 and Lemma 8.23. ∎

##  9\. Completeness

Recall that a metric space $(X,d)$ is _complete_ if every Cauchy sequence
converges. In this section we characterize when a dual $X_{\mu}$ is complete
in terms of the structural decomposition of $\mu$ (see Theorem 7.2). The main
theorem of this section, Theorem 9.5, shows that, in the absence of atoms for
$\mu$, $X_{\mu}$ is complete if and only it has no type 2 components in its
structural decomposition, i.e., components which are non-discrete measured
laminations. In the presence of atoms, $X_{\mu}$ is complete if and only it is
a multicurve. When $\mu$ is not a multi-curve, we show one can always find
non-convergent Cauchy sequences, due to, essentially, the discontinuities of
the projection map $\pi_{\mu}$. We split the proof into two subsections. In
the first one, we analyze the case of multi-curves and currents with type 2
components, and in the second, we prove the general result.

###  9.1. Multi-curves and currents with type 2

In Definition 3.1 we introduced the dual of geodesic currents. The first
observation is that, by the definition of $X_{\mu}$, if $\mu$ is a multi-
curve, any two points in $X_{\mu}$ are a uniform $d_{\mu}$ distance apart.
Therefore, it follows vacuously that

######  Lemma 9.1.

If $\mu$ is a multi-curve, then the dual $X_{\mu}$ is complete.

We will now show that if $\mu$ is a non-discrete measured lamination, i.e. a
type 2 current, then $X_{\mu}$ is not complete.

We will make use of the notion of _exotic ray_ of a lamination as defined by
T. Torkaman and Y. Zhang in [TZ21]. A _geodesic ray_ $r$ on a complete
hyperbolic surface of finite area $X=\widetilde{X}/\Gamma$ is a geodesic
isometric immersion $r:[0,\infty)\to X$.

Given a lamination $\Lambda$ on $X$ which is not a multi-curve, the
intersection $i(r,\Lambda)$ between the ray and the lamination is generically
infinite. There are two obvious cases when a ray has finite intersection with
$\Lambda$:

  1. (1)

The ray $r$ is asymptotic to a leaf of $\Lambda$;

  2. (2)

The ray $r$ is eventually disjoint from $\Lambda$.

Nevertheless, these are not the only two possibilities. Namely, there exist
geodesic rays such that $i(r,\Lambda)<\infty$ but are neither asympotic to a
leaf of $\Lambda$, nor eventually disjoint. Such rays are called _exotic
rays_.

######  Theorem 9.2 ([TZ21, Theorem 1.1]).

Let $\Lambda$ be a non-multi-curve lamination. Then there exist exotic rays
for $\Lambda$.

We will make use of the above result to prove the non-completeness for
geodesic currents $\mu$ with type 2 components in their decomposition.

######  Lemma 9.3.

Let $\nu$ be a geodesic current with a type 2 component $\mu$ in its
decomposition. Then the current dual space $X_{\nu}$ is incomplete.

###### Proof.

Let $Y$ be the subsurface induced by the decomposition of $\nu$ where $\mu$ is
supported on. Let $r\colon[0,\infty)\to X$ be an exotic ray for $\mu$ in $Y$,
and let $T=\\{t\colon r(t)\in\operatorname{supp}(\mu)\\}$, i.e., the times of
intersection of $r$ with the support of $\mu$. Since $r$ is exotic, $T$ is
infinite. We extract a countable collection of these times $(t_{n})$ and
consider the sequence $\bar{x_{n}}\coloneqq r(t_{n})\in\widetilde{X}$. Also,
let $r_{n}$ denote the subray of $r$ from $t_{n}$ onwards. Since $r$ is
contained in $Y$, considering intersection with $\nu$ is the same as
considering intersection with $\mu$. Note that $i(\mu,r_{n})\to 0$ as $n$ goes
to infinity, since $i(\mu,r)$ is finite. For every $\varepsilon>0$, take $N>0$
large enough so that $i(\mu,r_{N})<\varepsilon$. Then, for every $n>m\geq N$,
we have
$i(\mu,r_{m})-i(\mu,r_{n})=\mu(G[\bar{x_{n}},\bar{x_{m}}])<\varepsilon$, so
the sequence $(\bar{x_{n}})$ is Cauchy. We are left to show that
$(\overline{x_{n}})$ does not converge. Assume by contradiction $x_{n}\to x\in
X_{\mu}$. Then pick any
$\overline{x}\in\pi_{\mu}^{-1}(x)\subseteq\widetilde{X}$ and a small ball
$B_{\varepsilon}(\overline{x})$ around it. It follows that there exists $N>0$,
so that $\overline{x_{n}}\in B_{\varepsilon}(\overline{x})$ for all $n>N$,
which is absurd as $(\overline{x_{n}})$ exits all compact sets in
$\widetilde{X}$. ∎

###  9.2. General Case

The following result is elementary, see [BH11, Chapter I.3, Corollary 3.8].

######  Lemma 9.4.

If $(X,d)$ is a proper metric space, then it is complete and locally compact.

We can now prove the main result of this section:

######  Theorem 9.5.

Let $\mu$ be a geodesic current.

  1. (1)

If $\mu$ has no atoms, then $X_{\mu}$ is complete if and only if $\mu$ has no
components of type 2 in its structural decomposition

  2. (2)

If $\mu$ has atoms, then $X_{\mu}$ is complete if and only if $\mu$ is a
multi-curve.

###### Proof.

  1. (1)

Suppose first $\mu$ has no atoms. If $\operatorname{sys}(\mu)>0$, we have
proven that $X_{\mu}$ is proper in Proposition 8.13, and therefore its
complete by Lemma 9.4. If $\operatorname{sys}(\mu)=0$ and $\mu$ has a
component of type 2 in its decomposition, then, by Lemma 9.3, $X_{\mu}$ is not
complete. If it has no components of type 2, then given a Cauchy sequence, we
can extract a subsequence that is contained in a piece of the tree graded
space structure (by Theorem 7.12), since in this case the transverse trees of
the tree graded structure are singletons, and pieces intersect at point. On
the other hand, the subdual space corresponding to the piece is proper, and
thus the subsequence is convergent. Hence, the original sequence is
convergent.

  2. (2)

Suppose now $\mu$ has atoms. If it is a multi-curve, then it is obviously
complete. If it is not, then it has non-atomic components or it has an
infinite set of atoms (non-discrete) in $\mathcal{G}(\widetilde{X})$ (both
cases can happen simultaneously, too). In the first case, we have a compact
set of geodesics $B$ containing an infinite uncountable collection of lifts of
non-closed geodesics $\mathcal{A}$ and an atom $\\{\gamma\\}$. Consider
$G=\mathcal{A}\cup\\{\gamma\\}$ as a subset of $\widetilde{X}$, and consider
an infinite sequence of points $(\overline{x_{n}})$ in
$B-G\subset\widetilde{X}$, and $\overline{x}\in\gamma$, so that
$\mu(G(\overline{x_{n}},\overline{x}))\to 0$. This is possible since $\mu(B)$
is finite, and we can pick a sequence of points approaching $\gamma$ along a
geodesic ray that intersects $B$ and $\gamma$. Since we have
$d_{\mu}(x_{n},x)>\frac{1}{2}\mu(\\{\gamma\\})$, the sequence is Cauchy but
has no limit. If $\mu$ it is purely atomic but non-discrete, a similar
argument as above can be performed, where now $\mathcal{A}$ is now replaced by
a (countably) infinite set of atoms inside of a compact $B$.

∎

##  10\. Topology

In this section we prove that the map sending a geodesic current to its dual
space is a continuous injection when the space of geodesic currents is
equipped with the natural weak∗-topology and the space of duals is equipped
with the also natural equivariant Gromov-Hausdorff topology introduced by
[Pau88].

###  10.1. Topologies in the space of duals

We start by describing a topology on the space of duals. Let $\mathcal{Z}$
denote the space of Gromov hyperbolic spaces $Z$ with a cobounded action of
$\pi_{1}(X)$ by isometries. A topology we can equip it with is the
_equivariant Gromov-Hausdorff topology_ , defined by the following family of
neighbhorhoods.

######  Definition 10.1 ($\varepsilon$-relation).

Given $Z\in\mathcal{Z}$, $K\subset Z$ a compact subset and
$P\subset\pi_{1}(X)$ a finite subset, we say that $(Z,K)$ is
$\varepsilon$-related to $(Z^{\prime},K^{\prime})$ if

  1. (1)

There exists a compact $K^{\prime}\subset Z^{\prime}$, and a relation
$\mathcal{R}$ between $K$ and $K^{\prime}$ so that for all $x,y\in K$ and
$x^{\prime},y^{\prime}\in K^{\prime}$, if $x\mathcal{R}x^{\prime}$ and
$y\mathcal{R}y^{\prime}$, then
$|d(x,y)-d(x^{\prime},y^{\prime})|<\varepsilon$; and

  2. (2)

For every $x\in K$, $x^{\prime}\in K^{\prime}$, and $\gamma\in P$, if
$\gamma(x)\in K$, and $x\mathcal{R}x^{\prime}$, then $\gamma(x^{\prime})\in
K^{\prime}$ and $\gamma(x)\mathcal{R}\gamma(x^{\prime})$.

We will think of the relation $\mathcal{R}$ as a bijection $\varphi\colon K\to
K^{\prime}$, and write
$(Z^{\prime},K^{\prime})\sim_{\varphi,\varepsilon}(Z,K)$ to denote that
$(Z^{\prime},K^{\prime})$ is $\varepsilon$-related to $(Z,K)$ via the relation
$\varphi$. We will also write $(Z^{\prime},K^{\prime})\sim_{\varepsilon}(Z,K)$
when we don’t need to be explicit about the relation. Now, given
$Z\in\mathcal{Z}$, $K\subset Z$ a compact subset and $\varepsilon>0$, we
define the subset $V(Z,K,P,\varepsilon)$ to be

$\\{Z^{\prime}\in\mathcal{Z}:(Z^{\prime},K^{\prime})\sim_{\varepsilon}(Z,K)\\}$

Using coboundedness of the action of $\pi_{1}(X)$, the same proof as in
[Pau89, Proposition 4.1] shows that if one takes the sets $K$ to be finite in
the above family of neighborhoods $V(Z,K,P,\varepsilon)$, the induced topology
is equivalent.

We recall that the space of geodesic currents is equipped with the weak∗
topology (see Section 2.4).

We will use the following family of basis of neighborhoods which induces a
topology coarser than the weak∗-topology of geodesic currents.

######  Proposition 10.2.

Let

$A=\\{\overline{x}\in\widetilde{X}\colon\mbox{ for all
}\nu\in\operatorname{Curr}(X),\nu(G[\overline{x}])=0\\}.$

For $\mu\in\operatorname{Curr}(X)$, let $\mathcal{W}_{\widetilde{X}}^{\mu}$
denote the family of sets

$W(\mu,C,\varepsilon)=\\{\nu\in\operatorname{Curr}(X)\colon|\nu(G[\overline{x},\overline{y}])-\mu(G[\overline{x},\overline{y}])|<\varepsilon\colon\overline{x},\overline{y}\in
C\\}$

where $\varepsilon>0$, and $C\subset\widetilde{X}$ ranges over all finite
subsets of the subset $A\subset\widetilde{X}$. Then subset
$A\subset\widetilde{X}$ is dense and $\mathcal{W}_{\widetilde{X}}^{\mu}$ is a
subbasis of neighborhoods at $\mu$ generating a topology coarser than the
weak∗-topology in $\mathcal{G}(\widetilde{X})$.

###### Proof.

We relate this family of neighborhoods to the topology generated by flow boxes
for geodesic currents as in measures on the projective tangent bundle $PT(X)$
of $X$, as described, for example, in [AL17, Lemma 3.4.4] or [Bon86]. We note
that, in the flow box topology of $PT(X)$, an $H$-shape
$(\tau_{L},\gamma,\tau_{R})$ is determined by a pair of arcs $\tau_{L}$,
$\tau_{R}$ on $X$, that we will furthermore assume to be geodesic, as well as
another geodesic segment $\gamma$ on $X$, transverse to both $\tau_{L}$ and
$\tau_{R}$, with one endpoint on the first and the other in the second. An
$H$-shape then consists of all geodesic arcs on $X$ homotopic to $\gamma$ and
transverse to $\tau_{L}$ and $\tau_{R}$. An $H$-shape $H$ defines a subset of
$PT(X)$ by considering $B_{H}$, the set of lifts to $PT(X)$ of geodesic
segments on $H$. A lift of $B_{H}$ to $PT(\widetilde{X})$ is then given by the
set of lifts to $PT(\widetilde{X})$ of $B_{H}$, i.e., a set of geodesic
segments on $\widetilde{X}$ with endpoints on lifts $\widetilde{\tau_{L}}$ and
$\widetilde{\tau_{R}}$. These lifts can be uniquely extended to bi-infinite
geodesics, obtaining $G(\tau_{L},\tau_{R})$, the set of geodesics intersecting
both $\tau_{L}$ and $\tau_{R}$. This is the set of bi-infinite geodesics
$G_{H}=G(\tau_{L})\cap G(\tau_{R})\subset\mathcal{G}(\widetilde{X})$. By
[Bon86, Lemma 4.2], it follows that the family of sets

$W(\mu,G_{H},\varepsilon)=\\{\nu\in\operatorname{Curr}(X):|\nu(G_{H})-\mu(G_{H})|<\varepsilon\colon
G_{H}\subseteq\mathcal{G}(\widetilde{X})\text{ and }\nu(\partial G_{H})=0\\}$

as $\varepsilon>0$ ranges over all positive values, and the sets $G_{H}$ range
over all the sets of geodesics determined by flow boxes $B_{H}$, is a subbasis
at $\mu$ inducing a topology which, a priori, is coarser than that of the
weak∗-topology. Indeed, the condition $\nu(\partial B_{H})=0$ in Bonahon’s
result is equivalent to the condition $\nu(\partial G_{H})=0$. We finally note
that $W(\mu,C,\varepsilon)$ is a finite intersection of sets of type
$W(\mu,G_{H},\varepsilon)$, and thus also a neighborhood. We now address the
density of $A\subset\widetilde{X}$. We note that, by Proposition 4.6,
$\nu(\partial G[\overline{x},\overline{y}])\neq 0$ if and only if there is a
lift of a closed geodesic in the support of $\nu$ passing through
$\overline{x}$ or $\overline{y}$. Let $B$ denote the union of the lifts of all
closed geodesics in $X$. We will show that the complement of $B$ is dense.
Firstly, we notice that $B$ is the projection to the surface of a countable
union of disjoint Liouville measure zero sets in the unit tangent bundle.
Indeed, these correspond to the periodic orbits of the geodesic flow,
$\tilde{B}$. Since $\tilde{B}$ is geodesic flow invariant, by ergodicity of
the geodesic flow its complement must have full Liouville measure, and thus
must be dense (since the Liouville measure has full support). This argument
also works with any geodesic current of full support in place of the Liouville
current. Finally, we observe that $A$ contains the complement of $B$, so $A$
is a dense subset of points of $\widetilde{X}$. ∎

Let $\mathcal{D}(X)\subset\mathcal{Z}$ be subset of dual spaces of geodesic
currents of $X$ (it is a subset of $\mathcal{Z}$ by Proposition 6.12),
equipped with the subspace topology inherited from the equivariant Gromov-
Hausdorff topology. Let $\operatorname{Curr}(X)$ denote the space of geodesic
currents, equipped with the subspace weak∗-topology.

######  Theorem 10.3.

The map $\Psi\colon\operatorname{Curr}(X)\to\mathcal{D}(X)$ given by
$\mu\mapsto X_{\mu}$ is a continuous injection.

###### Proof.

  1. (1)

Continuity of $\Psi$. Let $X=X_{\mu}$, and $K=\\{x_{1},\cdots,x_{n}\\}\subset
X$, $P=\\{g\\}$. Given a standard neighborhood of the equivariant Gromov-
Hausdorff topology $U(X,K,P,\varepsilon)$, we find a neighborhood $W$ of the
subbasis introduced in Proposition 10.2. We recall that, a priori, the
topology it generates is coarser than the weak∗-topology, but this will be
enough to prove continuity. We shall now construct such a $W$ so that
$\Psi(W)\subset U$. First of all, by density of the set $A$ in Proposition
10.2, we can, if necessary, replace the points $x_{i}$ in $K$ by points
$x_{i}^{*}$ in $K^{*}\subset A$ so that
$|\mu(G(x_{i},x_{j}))-\mu(G(x_{i}^{*},x_{j}^{*}))|<\varepsilon/2$. Let
$\alpha_{1},\cdots,\alpha_{n}$ denote all the geodesic arcs of the type
$[\overline{x},\overline{y}]\subset\widetilde{X}$, where $x,y\in K^{*}$,
$x\neq y$, and $\pi_{\mu}(\overline{x})=x$ and $\pi_{\mu}(\overline{y})=y$.
Note that, if needed, we can also add more points to $K$ (and thus in $K^{*}$)
so that we get at least two distinct $\alpha_{i}$’s (we need at least 3
distinct points in $K^{*}$). Let
$W=\cap_{i=1}^{n}W_{\mu}(\alpha_{i},\varepsilon/2)$. We can then write

$\cap_{i}G(\alpha_{i})=\cap_{i\neq j}G(\alpha_{i},\alpha_{j})$

where $G(\alpha_{i},\alpha_{j})$ denotes the set of geodesics intersecting
$\alpha_{i}$ and $\alpha_{j}$ transversely, as in Proposition 10.2. Thus, we
can write $W=\cap_{i\neq j}W_{\mu}(\alpha_{i},\alpha_{j},\varepsilon/2)$,
where

$W_{\mu}(\alpha_{i},\alpha_{j},\varepsilon)=\\{\mu^{\prime}\in\operatorname{Curr}(X):|\mu(G(\alpha_{i},\alpha_{j}))-\mu^{\prime}(G(\alpha_{i},\alpha_{j}))|<\varepsilon/2\\}.$

Let $\mu^{\prime}\in W$. For every $x\in K\subset X_{\mu}$, pick one
$\overline{x}\in\pi_{\mu}^{-1}(x)\subset\widetilde{X}$, and let $\overline{K}$
be the finite set consisting of one $\overline{x}\in\pi^{-1}(x)$ for each
$x\in K$. Finally, let $K^{\prime}=\pi_{\mu^{\prime}}(\overline{K})$, where we
denote $\pi_{\mu^{\prime}}(\overline{x})=x^{\prime}$. We claim this defines an
$\varepsilon$-relation between $x\in K$ and $x^{\prime}\in K^{\prime}$. We
start by proving the first condition of Definition 10.1.

Note that by

$|\mu^{\prime}(G(\overline{x}^{*},\overline{y}^{*}))-\mu(G(\overline{x}^{*},\overline{y}^{*}))|<\varepsilon/2$

and the choice of $x_{i}^{*}\in K^{*}$ in relation to $x_{i}\in K$, it follows
that

$|d_{\mu^{\prime}}(x^{\prime},y^{\prime})-d_{\mu}(x,y)|<\varepsilon.$

Thus, the first condition of the $\varepsilon$-relation is satisfied. As for
the second condition, $g(x)\in K$ means, by equivariance of $\pi_{\mu}$, that
$g(\pi_{\mu}(\overline{x}))=\pi_{\mu}(g\overline{x})\in K$. Thus,

$g\overline{x}\in\pi_{\mu}^{-1}(K),$

and hence, by equivariance of $\pi_{\mu^{\prime}}$, we have

$g(x^{\prime})=\pi_{\mu^{\prime}}(g\overline{x})\in K^{\prime}$

which shows that $g(x)\mathcal{R}g(x^{\prime})$ and $g(x^{\prime})\in
K^{\prime}$, as we wanted. Thus, $X_{\mu^{\prime}}\in U(X,K,P,\varepsilon)$.
This shows continuity of $\Psi$ with respect to the topology in
$\operatorname{Curr}(X)$ induced by the subbasis of Proposition 10.2. Since
that topology is coarser than the weak∗-topology, this proves continuity of
$\Psi$.

  2. (2)

Injectivity of $\Psi$. If $X_{\mu}=X_{\mu^{\prime}}$, then in particular
$\ell_{\mu^{\prime}}([g])=\ell_{\mu^{\prime}}([g])$ for all $g\in\pi_{1}(X)$,
i.e., by Lemma 8.4 $i(\mu,g)=i(\mu^{\prime},g)$ for all $g\in\pi_{1}(X)$, and
thus $\mu=\mu^{\prime}$.

∎

It seems plausible that the weak∗-topology is, in general, strictly finer than
the equivariant Gromov-Hausdorff topology. On the other hand, the subspace
topology in the of duals of measured laminations yields an embedding (see the
next subsection 10.2).

###  10.2. Relation to Paulin’s work on $\mathbb{R}$-trees

Theorem 10.3 can be seen as a generalization of Paulin’s result [Pau89, Main
Theorem].

Let $G$ be a finitely generated group. The action of $G$ on an
$\mathbb{R}$-tree $T$ is said to be _minimal_ if the only invariant subtrees
are $\emptyset$ and $T$. An _end_ of an $\mathbb{R}$-tree $T$ is an
equivalence class of rays in $T$, with two rays identified if their
intersection is a ray. The action of $G$ on $T$ is said to be _irreducible_ if
there is no end of $T$ fixed by every element of $G$. The action is said to be
_reducible_ when it is not irreducible. Let $\mathcal{T}(G)$ the set of
equivalence classes of $\mathbb{R}$-trees, containing more than one point,
endowed with a minimal irreducible action of $G$, where two $\mathbb{R}$-trees
are identified whenever there is an isometry from one onto the other commuting
with the actions.

Paulin defines the _axial topology_ (or _translation length topology_) on
$\mathcal{T}(G)$ by the family of neighborhoods

$V(P,T,\varepsilon)=\\{T^{\prime}\in\mathcal{T}\colon|\ell_{T}(g)-\ell_{T^{\prime}}(g)|\mbox{
for all }g\in P\\},$

as $P$ ranges over all finite subsets of $G$, and $\varepsilon>0$. One can
define a similar topology for any collection of metric spaces $\mathcal{D}$
acted on by a fixed group of isometries $G$. One can also endow
$\mathcal{T}(G)$ with the equivariant Gromov-Hausdorff topology. When
$G=\pi_{1}(X)$ is a surface group, and one restricts furthermore to small
actions of $G$, Skora showed [Sko90, Theorem 3.3] (see also [Hub22, 13.6])
that $\mathcal{T}(G)$ corresponds to $\mathcal{D}_{\mathcal{ML}(X)}$, the
subset of dual spaces $X_{\lambda}$ where $\lambda\in\mathcal{ML}(X)$.
Therefore, by Proposition 8.4 and [Pau89, Main Theorem], it follows that the
restriction of our map $\Psi$ to $\mathcal{ML}(X)$,

$\Psi\colon\mathcal{ML}(X)\to\mathcal{D}_{\mathcal{ML}}(X),$

is a homeomorphism onto its image, where $\mathcal{ML}(X)$ is equipped with
the subspace weak∗-topology, and $\mathcal{D}_{\mathcal{ML}}(X)$ is equipped
with the equivariant Gromov-Hausdorff topology. Our result is thus a partial
generalization of Paulin’s work. On the other hand, Paulin’s result is more
general than ours, since it applies to all irreducible and minimal $G$-actions
on $\mathbb{R}$-trees, for any finitely generated group $G$.

###  10.3. Relation to Cantrell–Oregón-Reyes and Sapir’s work

Our map $\Psi$ descends to a continuous injection

$\mathbb{P}\Psi\colon\mathbb{P}\operatorname{Curr}(X)\to\mathbb{P}\mathcal{D}(X),$

where $\mathcal{D}(X)$ is equipped with the equivariant Gromov Hausdorff
topology. Here $\mathbb{P}\mathcal{D}(X)$ denotes the projectivization of
$\mathcal{D}(X)$, where $X_{\mu}$ and $X_{\nu}$ are equivalent if and only if
there exists a constant $C>0$, so that

$d_{\mu}(\pi_{\mu}(x),\pi_{\mu}(y))=C\cdot
d_{\mu^{\prime}}(\pi_{\mu^{\prime}}(x),\pi_{\mu^{\prime}}(y))$

for all $x,y\in\widetilde{X}$. We note that even though
$\mathbb{P}\operatorname{Curr}(X)$ is a compact metrizable space,
$\mathbb{P}\mathcal{D}(X)$ with the equivariant Gromov Hausdorff is not
Hausdorff in general (see [Pau88, Chapter 4]), so one cannot conclude the
induced continuous injection is an embedding.

The subspace
$\mathbb{P}\operatorname{Curr}_{\mathit{fill}}(X)\subseteq\mathbb{P}\operatorname{Curr}(X)$
consisting of filling geodesic currents can be equipped with a distance, by
recent work of Sapir [Sap22], defined as follows

$d_{\mathit{fill}}([\mu],[\nu])\coloneqq\sup_{c\in\mathit{Curves}(X)}\log\frac{i(\mu,c)}{i(\nu,c)}+\sup_{c\in\mathit{Curves}(X)}\log\frac{i(\nu,c)}{i(\mu,c)}.$

This distance coincides with the symmetrization of the Thurston distance on
Teichmuller space, when restricted to
$\operatorname{Curr}_{\operatorname{Teich}(X)}(X)$, i.e., the embedded image
of Teichmüller space in $\mathbb{P}\operatorname{Curr}(X)$ by [Thu98, Theorem
8.5]. On the other hand, there is also a natural distance $d_{\mathcal{D}}$ on
$\mathbb{P}\mathcal{D}_{\mathit{fill}}(X)$, defined in recent work of Oregón-
Reyes [OR22, Definition 1.2], which can also be expressed, by [OR22, Lemma
3.5], as

$d_{\mathcal{D}}([\mu],[\nu])\coloneqq\sup_{g\in\pi_{1}(X)}\log\frac{\ell_{X_{\mu}}(g)}{\ell_{X_{\mu^{\prime}}}(g)}+\sup_{g\in\pi_{1}(X)}\log\frac{\ell_{X_{\mu^{\prime}}}(g)}{\ell_{X_{\mu}}(g)}.$

Indeed, by Proposition 8.15, $\mathbb{P}\mathcal{D}_{\mathit{fill}}(X)$
corresponds to a subspace of the space $\mathcal{D}(\pi_{1}(X))$ (this also
follows from [CR23, Theorem 1.11]), and one can endow it with the restriction
of that metric. By [OR22, Lemma 3.5], and Proposition 8.4, it follows that the
restriction of $\Psi$ to $\mathbb{P}\operatorname{Curr}_{\mathit{fill}}$,

$\mathbb{P}\Psi\colon\mathbb{P}\operatorname{Curr}_{\mathit{fill}}\to\mathbb{P}\mathcal{D}_{\mathit{fill}}(X)$

is an isometry with respect to the metrics $d_{\mathit{fill}}$ and
$d_{\mathcal{D}}$. Furthermore, the topology induced by $d_{\mathit{fill}}$
coincides with the subspace weak∗-topology, by Theorem 2.14. The topology
induced by $d_{\mathcal{D}}$ coincides with the topology of translation
lengths or axial topology on $\mathbb{P}\mathcal{D}_{\mathit{fill}}(X)$ by
[OR22, Lemma 3.5]. Since by [EM18, Lemma 3.5] and [EM18, Corollary 3.8]
$\mathbb{P}\operatorname{Curr}_{\mathit{fill}}(X)$ is dense and open in
$\mathbb{P}\operatorname{Curr}(X)$, then the extension of
$\mathbb{P}\Psi|_{\mathbb{P}\operatorname{Curr}_{\mathit{fill}}(X)}$ to the
closure is the same as $\Psi$.

##  Appendix A Equivalence of notions of tree graded spaces

In this section we show that, under the assumption that $X$ is geodesic, the
Drutu-Sapir definition of tree graded space as in Definition 7.6 and our
Definition 7.9 are equivalent.

######  Proposition A.1.

If $(X,d)$ is a geodesic metric space, then Definition 7.9 is equivalent to
Definition 7.6.

###### Proof.

Assume throughout that $(X,d)$ is geodesic metric space and also that it has a
collection of pieces that intersect in at most one point, i.e., _axiom
pieces_. Let $\Delta=xyz$ be a geodesic triangle with geodesic sides
$\gamma^{1}=\gamma_{xy}$, $\gamma^{2}=\gamma_{yz}$ and
$\gamma^{3}=\gamma_{xz}$. Suppose $\Delta$ is contained in more than one
piece. We want to show $\Delta$ is non-simple. We can decompose each
$\gamma^{i}$, according to Proposition 7.8, into a piece-wise geodesic
consisting of geodesic segments $\alpha^{i}_{j}$ contained in pieces (we will
call them _$\alpha$ -segments_) and geodesic segments $\beta^{i}_{k}$ each
contained in a transversal $T_{x}$ (we will call them _$\beta$ -segments_),
for $i=1,2,3$. The endpoints of the $\beta$ for each $i$, give corresponding
straight contact chains $(y^{i}_{1},\cdots,y^{i}_{\ell_{i}})$. Since $\Delta$
is contained in more than one piece, at least one of the the $\beta$ segments
are non-degenerate. By _axiom straight chain triangle_ , one of the straight
chains shares an interior point with another side chain say
$y^{1}_{i}=y^{3}_{j}$. This means that their corresponding subsegments
$\beta^{1}_{i}$ and $\beta^{3}_{j}$ share an endpoint, and thus $\Delta$ is
non-simple.

$\alpha$
---
$P_{1}$
---
$P_{2}$
---
$P_{3}$
---
$\beta$
---
$y_{1}^{1}$
---
$y_{1}^{\ell_{1}}$
---
$y_{3}^{1}$
---
$y_{2}^{1}$
---
$y_{2}^{\ell_{2}}$
---
$y_{1}^{\ell_{1}}$
---

$P_{1}$
---
$P_{2}$
---
$P_{3}$
---
$y_{1}^{1}$
---
$y_{1}^{\ell_{1}}$
---
$y_{3}^{1}$
---
$y_{2}^{1}$
---
$y_{2}^{\ell_{2}}$
---
$y_{1}^{\ell_{1}}$
---

Figure A.1. From triangles to straight chain triangles: Proof of implication
from Definition 7.6 to Definition 7.9. From a straight chain we generate a
triangle with simple subsegments. The axiom triangles forces self-intersection
of the triangles, giving two cases depending on which types of subsegments
intersect. The left figure exhibits the cases of two $\beta$-segments
intersecting, and the right one the case of two $\alpha$-segments intersecting

Now we show that Definition 7.6 implies Definition 7.9. Given _axiom
triangles_ , the (axiom transversals) is [DS05, Lemma 2.12] and [DS05, Lemma
2.13].

We will see now that the axiom contact chain triangle also follows from _axiom
triangles_.

Let $\Delta=xyz$ be a contact chain triangle, where $x,y,z$ are three points
in $X$ contained in at least two distinct pieces. Suppose, say, that $x\in
P_{1}$, and let $y\in P_{2}$ and $z\in P_{3}$, where $P_{2}$ and $P_{3}$ could
potentially be the same, but both distinct from $P_{1}$. Denote by
$C_{i}\coloneqq(y^{i}_{1},\cdots,y^{i}_{\ell_{i}})$ the straight chains
connecting the points $x,y,z$ pairwise, for $i=1,2,3$. We can connect the
contact points in these chains by simple geodesic segments that we will call,
as before, $\alpha^{i}_{j}$ and $\beta^{i}_{k}$, depending on whether their
interiors are in pieces or in $T_{x}$, respectively. Let $\hat{C_{i}}$ denote
the corresponding piece-wise geodesic realizing the straight chain $C_{i}$.
They together form a corresponding geodesic triangle $\hat{\Delta}$ which is
contained in more than one piece, so it must be non-simple. Thus, there must
be a point of intersection $z$ between, say, the interior of $\hat{C_{1}}$ and
a point in $\hat{C_{3}}$. The geodesics $[z,y^{1}_{1}]$ and $[z,y^{3}_{1}]$
both intersect the piece $P_{1}$ only at their endpoints $y^{1}_{1}$ and
$y^{3}_{1}$, and thus, by [DS05, Lemma 2.4], we have $y^{1}_{1}=y^{3}_{1}$.
Similarly, the geodesics $[z,y^{1}_{\ell_{1}}]$ and $[z,y^{2}_{\ell_{3}}]$
both intersect, say $P_{2}$, at $y^{1}_{\ell_{1}}$ and $y^{2}_{\ell_{3}}$, so
$y^{1}_{\ell_{1}}=y^{2}_{\ell_{3}}$. Then, a similar argument shows that
$y^{3}_{\ell_{1}}=y^{2}_{1}$. Thus, since $\alpha^{i}_{j}$ are all simple
geodesics, it follows that $\alpha^{1}_{1}$, $\alpha^{2}_{1}$ and
$\alpha^{1}_{\ell_{1}}$ must be singletons and equal to the points of contact
$y_{1},y_{2},y_{3}$ at $P_{1},P_{2}$ and $P_{3}$, respectively. Then, the
$\beta$ segments adjacent to each $y_{i}$, since they share $y_{i}$, must be
contained in the same $T_{x}$. If $z$ is in a $\beta$ segment, a similar
argument using geodesics emanating from $z$ and Lemma [DS05, Lemma 2.4], as
before, shows that all other $\beta$ segments are contained in the same
$T_{x}$, and thus $\Delta=\Delta_{1}\cup A$, where $\Delta_{1}$ is a triangle
made of $\beta$ segments and contained in $T_{x}$ and $A$ is a disjoint union
of $\alpha$-segments. If $z$ is in an $\alpha$ segment, a similar argument
shows that $\Delta=\Delta_{1}\cup A\cup B$, where $\Delta_{1}$ is a triangle
made of $\beta$ segments contained in a single $T_{x}$, $A$ is a union of
$\alpha$ segments, and $B$ is a union of $\beta$ segments. In any of the two
cases, since $T_{x}$ is $0$-hyperbolic and $\Delta_{1}$ is contained in
$T_{x}$, the triangle $\Delta_{1}$ must be non-simple, and thus there must be
two $\beta$ subsegments corresponding to different sides of $\Delta$ sharing
an endpoint, which shows that one of the corresponding straight contact chains
shares an interior point with another, thus contradicting the axiom contact
chain triangle. ∎

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