# A closed ball compactification
of a maximal component via cores of trees

Giuseppe Martone, Charles Ouyang, and Andrea Tamburelli

###### Abstract.

We show that, in the character variety of surface group representations into
the Lie group $\mathrm{PSL}(2,\mathbb{R})\times\mathrm{PSL}(2,\mathbb{R})$,
the compactification of the maximal component introduced by the second author
is a closed ball upon which the mapping class group acts. We study the
dynamics of this action. Finally, we describe the boundary points
geometrically as $(\overline{A_{1}\times A_{1}},2)$-valued mixed structures.

###### Contents

  1. 1 Introduction
  2. 2 Background
  3. 3 Core of product of trees
  4. 4 Thurston’s compactification
  5. 5 Fixed point for the mapping class group action
  6. 6 $\overline{\mathfrak{a}^{+}}$-valued measured laminations and mixed structures

##  1\. Introduction

A recurring theme in higher Teichmüller theory is to relate surface group
representations into higher rank Lie groups with geometric objects. Taking its
cue from classical Teichmüller theory, one is often interested in studying the
degeneration of these associated geometric objects when the representation
leaves all compact sets in the character variety. The celebrated Thurston
compactification of Teichmüller space regards Fuchsian representations as
marked hyperbolic metrics, where degenerating families of hyperbolic metrics
subconverge to projectivized measured laminations. One key aspect of this
compactification is that it is a closed ball upon which the mapping class
group acts. In years following, there have been numerous different
perspectives of the Thurston compactification, using a variety of methods,
topological, geometric, analytic and algebraic (see [Bon88, Bes88, Pau88,
Wol89, MS84, Bru88]).

When the Lie group $\mathrm{PSL}(2,\mathbb{R})$ is replaced with a higher rank
one, the relevant geometric object is not always immediately clear. In rank 2
however, combined work of Schoen [Sch93], Labourie [Lab17], Loftin [Lof01],
Collier [Col16], Alessandrini-Collier [AC19], and Collier-Tholozan-Toulisse
[CTT19], provides a geometric interpretation to representations in the various
distinguished components of the relevant character variety. These components
are usually maximal components or Hitchin components, which maximize a
topological quantity, the Toledo invariant, or contain a deformation of the
classical Teichmüller space. Parreau [Par12] compactifies them by attaching at
infinity surface group actions on a Euclidean building.

This paper will primarily be concerned with the rank 2 semi-simple split Lie
group $G=\mathrm{PSL}(2,\mathbb{R})\times\mathrm{PSL}(2,\mathbb{R})$. The
product structure of $G$ makes our study more amenable towards techniques from
classical Teichmüller theory. For $S$ a closed, orientable, smooth surface of
genus $g>1$, work of Goldman [Gol88] shows the connected components of the
character variety $\chi(\pi_{1}(S),\mathrm{PSL}(2,\mathbb{R}))$ are determined
by the Euler number. In particular, the distinguished component with maximal
Euler number of $2g-2$ is the Teichmüller space Teich$(S)$. If we denote the
character variety for
$G=\mathrm{PSL}(2,\mathbb{R})\times\mathrm{PSL}(2,\mathbb{R})$ by
$\chi(\pi_{1}(S),G)$, then the connected components are merely products of the
connected components of $\chi(\pi_{1}(S),\mathrm{PSL}(2,\mathbb{R}))$. The
maximal component Max$(S,G)$ of $\chi(\pi_{1}(S),G)$ is the collection of
conjugacy classes of pairs of representations, each of which is a Fuchsian
representation. Hence
$\mathrm{Max}(S):=\mathrm{Max}(S,\mathrm{PSL}(2,\mathbb{R})\times\mathrm{PSL}(2,\mathbb{R}))$
is the product of two copies of Teichmüller space.

Elements in the component Max$(S)$ have a number of related geometric
interpretations. Schoen [Sch93] has shown these representations correspond to
equivariant minimal Lagrangians in $\mathbb{H}^{2}\times\mathbb{H}^{2}$. At
the same time, the group
$G=\mathrm{PSL}(2,\mathbb{R})\times\mathrm{PSL}(2,\mathbb{R})$ is the isometry
group of AdS3, and Mess [Mes07] has shown the holonomy representations of
GHMC-AdS3 manifolds are precisely the ones in Max$(S)$. Krasnov-Schlenker
[KS97] have shown to each GHMC-AdS3 manifold, there is a unique equivariant
space-like maximal surface, whose image under the Gauss map is the
aforementioned minimal Lagrangian.

In seeking a compactification of Max$(S)$ via degeneration of geometric
objects, the second author in his thesis [Ouy19] showed the natural limits to
the minimal Lagrangians were given by cores of $\mathbb{R}$-trees dual to
measured laminations. These are topologically and group-theoretically defined
distinguished subcomplexes of the product of two trees, where some parts are
two-dimensional and the remaining parts are one-dimensional. Denote by
Core$(\mathcal{T},\mathcal{T})$, the space of cores in the product of trees
dual to measured laminations. Observe that there is a natural
$\mathbb{R}^{+}$-action on Core$(\mathcal{T},\mathcal{T})$ and denote by
$\mathbb{P}\mathrm{Core}(\mathcal{T},\mathcal{T})$ the resulting
projectivization. We equip Max$(S)$ and
$\mathbb{P}\mathrm{Core}(\mathcal{T},\mathcal{T})$ with the equivariant
Gromov-Hausdorff topology. One natural question one might ask is what exactly
is the topology of the resulting compactification. Our first main result is
the following.

######  Theorem A.

The disjoint union

$\mathfrak{B}=\mathrm{Max}(S)\sqcup\mathbb{P}\mathrm{Core}(\mathcal{T},\mathcal{T})$

is homeomorphic to a closed ball of dimension $12g-12$.

It is not too difficult to see from the construction of this compactification,
that the action of the mapping class group extends continuously to the
boundary. Following Thurston, we study the action of the mapping class group
$\mathrm{MCG}(S)$ on our compactification $\mathfrak{B}$.

######  Proposition 1.1.

Suppose $\phi\in\mathrm{MCG}(S)$ and $\phi(x)=x$ for some
$x=(x_{1},x_{2})\in\mathfrak{B}$.

  1. (1)

If $\phi$ is periodic, then $x_{1}$ and $x_{2}$ are any two points fixed by
$\phi$ in the Thurston compactification of Teichmüller space such that
$(x_{1},x_{2})\in\mathfrak{B}$.

  2. (2)

If $\phi$ is pseudo-Anosov, then $(x_{1},x_{2})\in\partial\mathfrak{B}$ and
$x_{1}=0$, or $x_{2}=0$ or $x_{1}=x_{2}$.

The action of the mapping class group appears to be more interesting if we
consider its action on a natural quotient of $\mathfrak{B}$. In fact, given a
maximal representation $\rho$, there is a unique equivariant minimal
Lagrangian $\widetilde{\Sigma}_{\rho}$ in
$\mathbb{H}^{2}\times\mathbb{H}^{2}$. The induced metric on
$\widetilde{\Sigma}_{\rho}$ descends to a negatively curved Riemannian metric
on $S$. We denote by $\mathrm{Ind}(S)$ the space of such metrics. It turns out
that $\mathrm{Ind}(S)=\mathrm{Max}(S)/S^{1}$. Similarly, the distance on the
core of the product of two trees dual to a pair of measured laminations can be
recovered from a mixed structure, that is a hybrid geometric object on $S$
that is in part a measured laminations and in part a finite area flat metric
induced by a meromorphic quadratic differential on subsurfaces glued along
annuli. The space of projectivized mixed structure can then be identified with
the boundary of $\mathrm{Ind}(S)$ in the length spectrum topology ([Ouy19]).
The mapping class group acts on $\overline{\mathrm{Ind}(S)}$ and we prove the
following:

######  Theorem B.

Assume $\phi\in\mathrm{MCG}(S)$ fixes
$\mu\in\partial\overline{\mathrm{Ind}(S)}$.

  1. (1)

If $\mu$ is purely flat, then $\phi$ is periodic.

  2. (2)

If $\mu$ is properly mixed, then $\phi$ is not pseudo-Anosov.

Note that the remaining case of $\mu$ a purely laminar mixed structure is
handled by Nielsen-Thurston’s classification theorem. Theorem 5.10 will give a
more detailed description of item (2) in Theorem B when $\phi$ is reducible.
In particular, we will show that the subdivision of $S$ induced by $\mu$ is a
refinement of the one induced by $\phi$ if $\mu$ has no trivial parts.

The absence of a product structure for the other simple split Lie groups of
rank 2 makes the study of the topology of any compactification considerably
more difficult. Furthermore, for
$\mathrm{PSL}(2,\mathbb{R})\times\mathrm{PSL}(2,\mathbb{R})$, quadratic
differentials are intimately related to pairs of measured laminations, and for
higher order differentials, which appear for the other rank 2 cases, there are
no obvious analogous topological objects. However, it is possible to describe
our compactification without explicit references to $\mathbb{R}$-trees, and we
conjecture this perspective can be extended to the other rank 2 Lie groups. In
particular, given any Lie algebra $\mathfrak{g}$ with Cartan subalgebra
$\mathfrak{a}$ and positive Weyl chamber $\mathfrak{a}^{+}$, we define
$\overline{\mathfrak{a}^{+}}$-valued measured laminations and
$(\overline{\mathfrak{a}^{+}},k)$-mixed structures obtained by gluing these
vector-valued laminations together with $1/k$-translation surfaces of finite
area along annuli. We can rephrase our main result as follows:

######  Theorem C.

The boundary of $\mathrm{Max}(S)$ can be identified with the space of
$(\overline{A_{1}^{+}\times A_{1}^{+}},2)$-mixed structures on $S$, which is
thus topologically a sphere of dimension $12g-13$.

Moreover, we prove in Lemma 6.7 that $(\overline{A_{1}^{+}\times
A_{1}^{+}},2)$-mixed structures are dual to the subcomplexes of an Euclidean
building introduced and studied in [Par15]. Theorem C has the advantage to be
easily adaptable to other higher Teichmüller components (see Conjecture 6.6
for the precise statements in rank $2$).

### Historical remarks

In analogy with the classical case, compactifications of higher Teichmüller
spaces are fruitfully studied using different techniques and perspectives. As
mentioned above, Parreau [Par12] compactifies the character variety of surface
group representations into noncompact semisimple connected real Lie groups
with finite center using Euclidean buildings. For Hitchin and maximal
connected components, one can obtain additional information on the boundary
points by using the ($\Theta$-)positivity properties of the representations as
in [Ale08, BP17, FG06, Le16, Mar19a, Mar19b, Par15]. For rank two Lie groups,
the second and third authors used analytic methods to study degenerations of
geometric objects associated to these representations in [Ouy19, OT21, OT20].
In a series of papers Burger, Iozzi, Parreau and Pozzetti [BIPP17, BIPP19,
BIPP21a] use geodesic currents and real algebraic geometric methods to study
compactifications of general character varieties. We refer to [BIPP21b] for an
account of their work and point of view. Furthermore, in independent work
[BIPP21c] they describe a compactification of the product of $n$ copies of
Teichmüller space via the projectivization of the product of $n$ copies of the
space of measured laminations and vector valued mixed structures. This will
lead to a (a priori different) compactification using projectivized cores when
$n=2$.

### Acknowledgements

Part of this work was carried out when the first and second authors were
visiting Rice University during Summer 2021. We thank the department for their
hospitality. We thank Francis Bonahon for helpful comments on an earlier
version of this manuscript. We thank Beatrice Pozzetti for helpful comments on
this manuscript and for pointing out a mistake in a previous version of the
statement of Theorem B and Lemma 6.7. The third author acknowledges support
from the National Science Foundation under grant NSF-DMS:2005501.

##  2\. Background

###  2.1. Foliations, laminations and $\mathbb{R}$-trees

We recall some classical facts about measured foliations and laminations. This
material can be found in [FLP21]. Let $S$ be a closed, orientable, smooth
surface of genus $g>1$. A measured foliation is a singular foliation (with
$k$-pronged singularities) equipped with a measure on transverse arcs,
invariant under transverse homotopy.

If $S$ is given a hyperbolic metric $\sigma$, then a measured lamination is a
closed set of disjoint simple geodesics on $(S,\sigma)$ together with a
transverse measure. There is a natural homeomorphism between the space
$\mathcal{MF}(S)$ of measured foliations on $S$ and the space
$\mathcal{ML}(S)$ of measured laminations on $(S,\sigma)$, so that the role of
$\sigma$ is an auxiliary one. Thurston showed $\mathcal{MF}(S)$ is
topologically trivial, being a ball of dimension $6g-6$. The space
$\mathbb{P}\mathcal{MF}(S)$ is the boundary of Teichmüller space under the
Thurston compactification.

If $S$ is given a complex structure $J$, then to any holomorphic quadratic
differential $q=q(z)dz^{2}$, one may consider the foliation obtained by
integrating the line field $q(v,v)>0.$ When further given the transverse
measure defined by $\int_{\alpha}|\operatorname{Im}(\sqrt{q})|$, the resulting
measured foliation is called the horizontal foliation of $q$. Likewise
integrating the line field $q(v,v)<0$ and taking the measure
$\int_{\alpha}|\operatorname{Re}(\sqrt{q})|$ gives the vertical foliation of
$q$. The theorem of Hubbard-Masur [HM79] states that for a fixed Riemann
surface $(S,J)$ and any measured foliation $\mathcal{F}$ on $S$, there is a
unique holomorphic quadratic differential $q$, whose horizontal foliation is
Whitehead equivalent (i.e. it differs at most by isotopies or expanding or
collapsing pronged singularities along straight arcs) to $\mathcal{F}$.

Any measured foliation $\mathcal{F}$ on $S$ lifts to a measured foliation
$\widetilde{\mathcal{F}}$ on the universal cover $\widetilde{S}$. Taking the
leaf space of $\widetilde{\mathcal{F}}$ together with a distance induced by
the pushforward of the transverse measure gives an $\mathbb{R}$-tree. When an
$\mathbb{R}$-tree is constructed from a measured foliation in this way, the
$\mathbb{R}$-tree comes equipped with a $\pi_{1}(S)$-action from
$\widetilde{\mathcal{F}}$. This action is small, that is, the stabilizer of an
arc never contains a free group of rank 2, and minimal, that is, the action
does not fix any proper subtree. A result of Skora [Sko96] says that any
$\mathbb{R}$-tree with a $\pi_{1}(S)$-action which is both small and minimal
is constructed from a measured foliation on $S$. Such $\mathbb{R}$-trees are
said to be dual to a measured foliation, and for our purposes, all
$\mathbb{R}$-trees we consider will be dual to a measured foliation.

###  2.2. Half-translation surfaces, flat metrics and mixed structures

A Riemann surface equipped with a holomorphic quadratic differential $q$ is
called half-translation surface. This terminology comes from the fact these
can be realized by gluing polygons in $\mathbb{C}$ via translations or
rotations of angle $\pi$.

A half-translation surface is naturally endowed with a singular flat metric
$|q|$, where the singularities are at the zeros of $q$. Duchin-Leininger-Rafi
[DLR10] have studied the degeneration of unit-area quadratic differential
metrics, and have shown the limits are precisely projectivized (quadratic)
mixed structures. A mixed structure is a collection of integrable meromorphic
quadratic differential metrics on subsurfaces and measured laminations on
other subsurfaces, glued along flat annuli to recover the surface $S$. Trivial
examples of mixed structures include singular flat metrics on $S$ and measured
laminations on $S$. We say that a mixed structure is _properly mixed_ if it
has a flat piece but it is not a singular flat metric. Mixed structures, when
the meromorphic differential is cubic or quartic, appear in the
compactification of Hitchin components for $\mathrm{SL}(3,\mathbb{R})$ and
$\mathrm{Sp}(4,\mathbb{R})$ (see [OT21], [OT20]).

A measured lamination $\lambda$ on $S$ is said to fill if the complement
$S\setminus\lambda$ is a disjoint union of topological disks. A pair
$\mathcal{F}_{1},\mathcal{F}_{2}$ of measured foliations on $S$ is said to
fill or is transverse if for any third foliation $\mathcal{G}$, one has
$i(\mathcal{F}_{1},\mathcal{G})+i(\mathcal{F}_{2},\mathcal{G})>0$. Here
$i(\cdot,\cdot)$ denotes the Bonahon intersection pairing, which generalizes
the topological intersection number between curves. We remark that the
intersection number for the corresponding measured laminations is the same,
therefore we can define filling for a pair of measured laminations
analogously. Notice that given a holomorphic quadratic differential $q$, the
vertical and horizontal foliations of $q$ fill. Conversely, the result by
Gardiner-Masur [GM91] says that given any pair of filling measured foliations,
there exists a unique Riemann surface structure and a unique holomorphic
quadratic differential which realizes the original pair as its vertical and
horizontal foliation (up to Whitehead equivalence). In particular, a pair of
filling measured foliations will determine a unique half-translation surface
structure and consequently a unique singular flat quadratic differential
metric.

###  2.3. Minimal Lagrangians in $\mathbb{H}^{2}\times\mathbb{H}^{2}$

A minimal Lagrangian $\widetilde{\Sigma}$ in
$\mathbb{H}^{2}\times\mathbb{H}^{2}$ is a minimal surface which is Lagrangian
with respect to the symplectic form $\omega\oplus-\omega$, where $\omega$ is
the standard Kähler form on $\mathbb{H}^{2}$. Any $\rho\in\mathrm{Max}(S)$
acts on $\mathbb{H}^{2}\times\mathbb{H}^{2}$, and Schoen [Sch93] has shown to
each such $\rho$, there is a unique $\rho$-equivariant minimal Lagrangian
$\widetilde{\Sigma}_{\rho}$ in $\mathbb{H}^{2}\times\mathbb{H}^{2}$, thereby
providing a geometric interpretation to representations in $\mathrm{Max}(S)$.
The second author [Ouy19] has studied the degeneration of these minimal
Lagrangians, and has shown that one may interpret the space
$\mathbb{P}\mathrm{Core}(\mathcal{T},\mathcal{T})$ as the boundary of the
maximal component $\mathrm{Max}(S)$.

###  2.4. Induced metrics and projectivized mixed structures

The induced metric on the unique $\rho$-equivariant minimal Lagrangian
descends to a metric on $S$. It is not too difficult (see [Ouy19, Proposition
4.2]) to see this metric is in fact negatively curved. Hence, by the result of
[Ota90], its marked length spectrum determines the metric. The marked length
spectrum is the data of both the curve class and the length of its geodesic
representative in the given homotopy class. Let Ind$(S)$ denote the space of
induced metrics coming from the $\rho$-equivariant minimal Lagrangians. Then
in fact one may embed Ind$(S)$ into the space of projectivized marked length
spectra. Its closure is then determined to be precisely the space Ind$(S)$
together with the projectivized mixed structures ([Ouy19, Theorem 5.5]).

##  3\. Core of product of trees

In this section we recall the notion of core of a product of trees and
describe its geometry in the case of trees dual to measured laminations. The
core of a product of two $\mathbb{R}$-trees can actually be defined for any
pair of $\mathbb{R}$-trees each admitting a $\pi_{1}(S)$-action. It is not
necessary that the $\mathbb{R}$-trees be dual to measured foliations. However,
we will specifically mention when particular properties of cores are germane
only to $\mathbb{R}$-trees dual to measured foliations. The main reference for
the material covered here is [Gui05].

Given an $\mathbb{R}$-tree $T$, a direction $\delta$ based at a point $p\in T$
is a connected component of $T\setminus\\{p\\}$. For a product $T_{1}\times
T_{2}$ of $\mathbb{R}$-trees, a quadrant $Q$ based at $(p_{1},p_{2})\in
T_{1}\times T_{2}$ is a product $\delta_{1}\times\delta_{2}$ of directions. If
the $\mathbb{R}$-trees $T_{1},T_{2}$ are equipped with a $\pi_{1}(S)$-action
by isometries, then we say a quadrant $Q$ is heavy if there exists a sequence
$\\{\gamma_{n}\\}\subset\pi_{1}(S)$ for which:

  1. i)

$\gamma_{n}\cdot p_{i}\in\delta_{i}$, and

  2. ii)

$d_{i}(\gamma_{n}\cdot p_{i},p_{i})\to\infty$ as $n\to\infty$ .

Otherwise the quadrant is said to be light. Following Guirardel [Gui05], the
core $\mathcal{C}(T_{1},T_{2})$ of $T_{1}\times T_{2}$ is

$T_{1}\times T_{2}\setminus\bigsqcup_{Q\,\,\text{light}}Q.$

When $T_{1}$ and $T_{2}$ are dual to measured laminations, the core
$\mathcal{C}(T_{1},T_{2})$ is always non-empty since the $\pi_{1}(S)$-actions
are irreducible ([Gui05, Proposition 3.1]).

However, even when $T_{1}$ and $T_{2}$ are dual to measured foliations, one
pathology may still occur: $\mathcal{C}(T_{1},T_{2})$ may be disconnected.
This happens, for instance, when $T_{1}=T_{2}$ and $T_{1}$ is dual to a
multicurve. However, in such cases, Guirardel introduced a canonical way of
extending the core to a connected subset of $T_{1}\times T_{2}$ with convex
fibers. With abuse of terminology, we will still refer to this canonical
extension as the core of $T_{1}\times T_{2}$. The following result completely
characterizes when this extension needs to be considered.

######  Definition 3.1.

Given two real trees $T$ and $T^{\prime}$ endowed with an action of
$\pi_{1}(S)$, we say that $T$ is a refinement of $T^{\prime}$ if there is an
equivariant map $f:T\rightarrow T^{\prime}$ such that for all $x,y,z\in T$ if
$z$ lies in the geodesic $[x,y]$ connecting $x$ and $y$, then $f(z)$ belongs
to $[f(x),f(y)]$.

######  Proposition 3.2 ([Gui05, Proposition 4.14]).

Let $T_{1}$ and $T_{2}$ be trees dual to measured laminations. Then the core
$\mathcal{C}(T_{1},T_{2})$ is disconnected if and only if $T_{1}$ and $T_{2}$
are refinements of a common non-trivial simplicial tree $T$.

For example the assumptions of Proposition 3.2 are satisfied if $T_{1}$ and
$T_{2}$ are dual to measured laminations $\lambda_{1}$ and $\lambda_{2}$ with
common isolated leaves.

When $T_{1}$ and $T_{2}$ are both dual to measured laminations $\lambda_{1}$
and $\lambda_{2}$, we can actually realize the core $\mathcal{C}(T_{1},T_{2})$
more concretely. Before describing this construction, we need the following
result, which can be seen as a special case of the decomposition theorem for
general geodesic currents in ([BIPP17], see also [BIPP19]) about how two
measured laminations interact on subsurfaces.

######  Lemma 3.3.

Let $\lambda_{1}$ and $\lambda_{2}$ be measured laminations on $S$. Then there
is a system of non-trivial, pairwise non-homotopic, disjoint, simple closed
curves $\gamma_{1},\dots,\gamma_{n}$ such that on each connected component
$S^{\prime}$ of $S\setminus\cup_{j}\gamma_{j}$ either

  1. i)

$\lambda_{1}+\lambda_{2}$ is a (possibly zero) measured lamination on
$S^{\prime}$; or

  2. ii)

$\lambda_{1}$ and $\lambda_{2}$ are transverse and fill $S^{\prime}$, i.e. for
all measured lamination $\nu$ on $S^{\prime}$ we have
$i(\lambda_{1},\nu)+i(\lambda_{2},\nu)\neq 0$.

###### Proof.

Consider a maximal collection of non-trivial, pairwise non-homotopic,
disjoint, simple closed curves $\gamma_{j}$ such that

$i(\lambda_{1},\gamma_{j})+i(\lambda_{2},\gamma_{j})=0.$

We claim that this collection of curves satisfies the requirement of the
lemma. Indeed, let $S^{\prime}$ be a connected component of
$S\setminus\cup_{j}\gamma_{j}$. We need to show that if the pair
$(\lambda_{1},\lambda_{2})$ does not fill the subsurface $S^{\prime}$, then
$\lambda_{1}+\lambda_{2}$ is a lamination on $S^{\prime}$, or, equivalently,
$\lambda_{1}$ and $\lambda_{2}$ are nowhere transverse on $S^{\prime}$. The
claim is clearly true if the support of either $\lambda_{1}$ or $\lambda_{2}$
does not intersect $S^{\prime}$, so we can assume that both have support on
$S^{\prime}$. Because the pair $(\lambda_{1},\lambda_{2})$ does not fill
$S^{\prime}$ by assumption, there is a measured lamination $\nu$ on
$S^{\prime}$ such that $i(\lambda_{1},\nu)+i(\lambda_{2},\nu)=0$. On the other
hand, by hypothesis, $i(\lambda_{1},\gamma)+i(\lambda_{2},\gamma)\neq 0$ for
all non-peripheral simple closed curves $\gamma$ on $S^{\prime}$. Therefore,
the measured lamination $\nu$ does not contain isolated closed leaves. Let us
first consider the case in which $\nu$ fills the subsurface $S^{\prime}$, in
the sense that the complement of $\nu$ only consists of disks and annuli. We
note that then necessarily the support of $\nu$ must contain the support of
$\lambda_{1}$ and $\lambda_{2}$ because otherwise $\lambda_{1}$ and
$\lambda_{2}$ would intersect $\nu$ transversely somewhere. But this implies
that $\lambda_{1}$ and $\lambda_{2}$ are nowhere transverse, being both
contained in the support of a measured lamination. We now reduce the general
case to this, by showing that $\nu$ must fill $S^{\prime}$. Assume the
opposite, and let $S^{\prime\prime}\subset S^{\prime}$ be a subsurface filled
by $\nu$. Note that at least one between $\lambda_{1}$ and $\lambda_{2}$
intersects the boundaries of $S^{\prime\prime}$ transversely. Without loss of
generality we assume it is $\lambda_{1}$. Since $\nu$ fills
$S^{\prime\prime}$, the support of $\lambda_{1}$ intersects $\nu$
transversely, but this contradicts the fact that $i(\nu,\lambda_{1})=0$. ∎

The last ingredient we need is an explicit realization of a tree $T_{\lambda}$
dual to a measured lamination $\lambda$. The construction goes as follows (see
[MO93] for more details). Fix an auxiliary hyperbolic metric on $S$ and
identify $\widetilde{S}$ with $\mathbb{H}^{2}$. Let $\widetilde{\lambda}$ be
the lift of $\lambda$ under the covering map $\pi:\mathbb{H}^{2}\rightarrow
S$. We define the metric space $\mathrm{pre}(T_{\lambda})$ where points of
$\mathrm{pre}(T_{\lambda})$ are the connected components of
$\mathbb{H}^{2}\setminus\widetilde{\lambda}$ and the distance is computed as
follows: if $x,y\in\mathrm{pre}(T_{\lambda})$ correspond to connected
components $C_{x},C_{y}$ of $\mathbb{H}^{2}\setminus\widetilde{\lambda}$ then

$d_{\lambda}(x,y)=\inf\left\\{\int_{\gamma}d\widetilde{\lambda}\ \Big{|}\
\gamma:[0,1]\rightarrow\mathbb{H}^{2},\ \gamma(0)\in C_{x},\ \gamma(1)\in
C_{y}\right\\}$

The tree $T_{\lambda}$ is then the unique $\mathbb{R}$-tree that contains
$\mathrm{pre}(T_{\lambda})$ such that any point of $T_{\lambda}$ lies in a
segment with vertices in $\mathrm{pre}(T_{\lambda})$. Note that we have a
natural projection map
$p_{\lambda}:\mathbb{H}^{2}\setminus\widetilde{\lambda}\rightarrow
T_{\lambda}$. If $\lambda$ has no isolated leaves, this map extends
continuously to a map, still denoted by $p_{\lambda}$, defined on the entire
$\mathbb{H}^{2}$. Otherwise, the continuous extension is obtained by first
replacing each isolated leaf $\ell$ in $\widetilde{\lambda}$ with a strip
$\ell\times[-\epsilon,\epsilon]$ endowed with a uniform measure with total
mass equal to $\widetilde{\lambda}_{|_{\ell}}$.

There is also another way of realizing the tree dual to a measured lamination
using the language of measured foliations. Let $\mathcal{F}$ denote the
measured foliation corresponding to the measured lamination $\lambda$ under
the homeomorphism between $\mathcal{MF}(S)$ and $\mathcal{ML}(S)$. Let
$\widetilde{\mathcal{F}}$ be its lift to $\mathbb{H}^{2}$. Then the tree
$T_{\lambda}$ can be defined as the quotient $\mathbb{H}^{2}/\sim$, where
$\sim$ denotes the equivalence relation

$x\sim y\Leftrightarrow d_{\mathcal{F}}(x,y)=0$

and

$d_{\mathcal{F}}(x,y)=\inf\\{i(\widetilde{\mathcal{F}},\gamma)\ |\
\gamma:[0,1]\rightarrow\mathbb{H}^{2},\ \gamma(0)=x,\ \gamma(1)=y\\}.$

More concretely, $T_{\lambda}$ identifies with the leaf space of
$\widetilde{\mathcal{F}}$ with distance given by integrating the measure of
$\widetilde{\mathcal{F}}$ along arcs transverse to the leaves. We denote by
$\pi_{\lambda}$ the natural projection
$\pi_{\lambda}:\mathbb{H}^{2}\rightarrow T_{\lambda}$.

We are now ready to describe the core of a product of two trees $T_{1}$ and
$T_{2}$ dual to measured laminations $\lambda_{1}$ and $\lambda_{2}$ on $S$.
Lemma 3.3 furnishes a decomposition of $S$ into subsurfaces that we lift to a
decomposition of $\mathbb{H}^{2}$. The regions of this decomposition come in
two flavors according to whether they project to subsurfaces where
$\lambda_{1}+\lambda_{2}$ is a lamination or to subsurfaces where the pair
$(\lambda_{1},\lambda_{2})$ fills. Following the statement of Lemma 3.3, we
call these regions of type $i)$ and type $ii)$, respectively. On the regions
$\Omega\subset\mathbb{H}^{2}$ of type $i)$, the union of the lifts
$\widetilde{\lambda}_{1}$ and $\widetilde{\lambda}_{2}$ can be regarded as the
lift of the measured lamination $\lambda_{0}=\lambda_{1}+\lambda_{2}$. We
denote by $T_{0}$ the tree dual to $\lambda_{0}$. Note that, for each region
$\Omega\subset\mathbb{H}^{2}$ of type $i)$, we have a map
$p_{0}:=p_{\lambda_{0}}$, as defined before, and two natural collapsing maps
$c_{j}:T_{0}\rightarrow T_{j}$ for $j=1,2$. On the regions of type $ii)$, we
replace the measured laminations $\widetilde{\lambda}_{i}$ with the
corresponding measured foliations $\widetilde{\mathcal{F}}_{i}$ and consider
the projections $\pi_{i}:=\pi_{\lambda_{i}}$ as described previously.
Following Guirardel ([Gui05, Example 4], [Gui05, Proposition 6.1]), the core
$\mathcal{C}(T_{1},T_{2})$ is the image of the map $F:\widetilde{S}\rightarrow
T_{1}\times T_{2}$ defined as follows:

$F(x)=\begin{cases}(\pi_{1}\times\pi_{2})(x)\hskip 56.9055pt\text{if $x$
belongs to a region of type $ii)$}\\\ (c_{1}\times c_{2})({p_{0}}(x))\hskip
41.25641pt\text{if $x$ belongs to a region of type $i)$}\ .\end{cases}$ (1)

Note that $F$ is well-defined and continuous on the boundary
$\widetilde{\gamma}$ between two different regions of $\widetilde{S}$ because
$\widetilde{\gamma}$ is the lift of a curve $\gamma_{j}$ given by Lemma 3.3
which, by definition, has vanishing intersection number with $\lambda_{0}$,
$\mathcal{F}_{1}$, and $\mathcal{F}_{2}$, hence
$(\pi_{1}\times\pi_{2})(\widetilde{\gamma})$ and $(c_{1}\times
c_{2})({p_{0}}(\widetilde{\gamma}))$ is a single point.

It follows from this explicit description of $\mathcal{C}(T_{1},T_{2})$ that
the core is, in general, a 2-dimensional subcomplex of $T_{1}\times T_{2}$
that is invariant under the diagonal action of $\pi_{1}(S)$. Moreover, the
$2$-dimensional pieces of $\mathcal{C}(T_{1},T_{2})$ are exactly the images of
regions of type $ii)$ and are foliated by two families of transverse
foliations. Their quotients under the group action are the union of the
subsurfaces of $S$ in which $\lambda_{1}$ and $\lambda_{2}$ fill, endowed with
the foliations $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ ([Gui05, Example 4]).
In particular, the 2-dimensional pieces of $\mathcal{C}(T_{1},T_{2})$ are the
universal covers of half-translation surfaces. On the other hand, the images
under $F$ of regions $\Omega$ of type $i)$ are $1$-dimensional subcomplexes of
$T_{1}\times T_{2}$. Each such $\Omega$ can be seen as the universal cover of
a subsurface $S^{\prime}$ of $S$ where the restriction of
$\lambda_{1}+\lambda_{2}$ is a measured lamination. Let $T_{1}^{\prime}\subset
T_{1}$ and $T_{2}^{\prime}\subset T_{2}$ be the corresponding subtrees. It
turns out ([Gui05, Section 6]) that $F(\Omega)$ is an $\mathbb{R}$-tree that
is a common refinement of $T_{1}^{\prime}$ and $T_{2}^{\prime}$ if endowed
with the distance

$d_{0}(x,y)=d_{1}(x_{1},y_{1})+d_{2}(x_{2},y_{2})\ \ \ \
\text{$x=(x_{1},x_{2}),y=(y_{1},y_{2})\in T_{1}^{\prime}\times
T_{2}^{\prime}$}$

where $d_{j}$ denotes the distance on $T_{j}$.

######  Lemma 3.4.

The $\mathbb{R}$-tree $(F(\Omega),d_{0})$ is isometric to the tree dual to the
measured lamination $\lambda_{0}=\lambda_{1}+\lambda_{2}$ restricted to
$S^{\prime}$.

###### Proof.

The tree $F(\Omega)$ inherits from $T_{1}^{\prime}\times T_{2}^{\prime}$ an
isometric action of $\pi_{1}(S^{\prime})$. We can define a length function

$\displaystyle\ell:\pi_{1}(S^{\prime})$
$\displaystyle\rightarrow\mathbb{R}^{+}$ $\displaystyle\gamma$
$\displaystyle\mapsto\lim_{n\to+\infty}\frac{1}{n}d_{0}(x,\gamma^{n}\cdot x)$

where $x$ is any point in $F(\Omega)$ (the definition is independent of the
choice of $x$). Since the action of $\pi_{1}(S^{\prime})$ on $F(\Omega)$ is
minimal and irreducible, by [GL17, Theorem A.5], the isometry class of
$(F(\Omega),d_{0})$ is completely determined by its length function. However,
it is clear from the definition of $\ell$ and $d_{0}$ that
$\ell=\ell_{0}:=\ell_{1}+\ell_{2}$, where $\ell_{j}$ denotes the analogously
defined length functions on $T_{1}^{\prime}$ and $T_{2}^{\prime}$. On the
other hand, $\ell_{0}$ is exactly the length function of the tree dual to the
measured lamination $\lambda_{0}$, and the claim follows. ∎

The ambient space $T_{1}\times T_{2}$ has, however, another natural distance
defined by

$d(x,y)=\sqrt{d_{1}(x_{1},y_{1})^{2}+d_{2}(x_{2},y_{2})^{2}}\ \ \ \
\text{$x=(x_{1},x_{2}),y=(y_{1},y_{2})\in T_{1}\times T_{2}$}\ .$

This induces a path metric $d_{\mathcal{C}}$ on the core $\mathcal{C}$ of
$T_{1}\times T_{2}$, where the $d_{\mathcal{C}}$-distance between two points
in the core is the infimum of the length of all paths connecting the points
and entirely contained in the core, where the length is computed using the
distance $d$. Guirardel showed ([Gui05, Proposition 4.9]) that the core is a
CAT(0)-space if endowed with this path distance $d_{\mathcal{C}}$. In
particular, since $F(\Omega)$ does not contain topological circles by Lemma
3.4, we can conclude that $F(\Omega)$ endowed with the restriction of
$d_{\mathcal{C}}$ is still an $\mathbb{R}$-tree.

We will denote by $\mathrm{Core}(\mathcal{T},\mathcal{T})$ the space of cores
of the product of two trees dual to measured laminations on $S$ endowed with
this path distance.

######  Proposition 3.5.

$\mathrm{Core}(\mathcal{T},\mathcal{T})$ is homeomorphic to
$\mathcal{ML}(S)\times\mathcal{ML}(S)$.

###### Proof.

Since the core of a product of trees is uniquely determined by the two
factors, the result follows immediately from the homeomorphism between the
space of trees dual to measured laminations and $\mathcal{ML}(S)$. ∎

We note that there is a natural $\mathbb{R}^{+}$-action on
$\mathrm{Core}(\mathcal{T},\mathcal{T})$ given by rescaling the induced metric
on the core, which, under the homeomorphism above, corresponds to the diagonal
action of $\mathbb{R}^{+}$ by scalar multiplication on the measures. We denote
by $\mathbb{P}\mathrm{Core}(\mathcal{T},\mathcal{T})$ the quotient
$\mathrm{Core}(\mathcal{T},\mathcal{T})/\mathbb{R}^{+}$. It follows that
$\mathbb{P}\mathrm{Core}(\mathcal{T},\mathcal{T})$ is homeomorphic to
$\mathbb{P}(\mathcal{ML}(S)\times\mathcal{ML}(S))$. In particular, it is
topologically a sphere of dimension $12g-13$.

##  4\. Thurston’s compactification

Recall that we denote by $\text{Max}(S)$ the space of conjugacy classes of
representations $\rho=(\rho_{1},\rho_{2})$ of the fundamental group of a
closed connected oriented surface $S$ of negative Euler characteristic into
the Lie group $\mathrm{PSL}(2,\mathbb{R})\times\mathrm{PSL}(2,\mathbb{R})$
such that $e(\rho_{1})+e(\rho_{2})=4g-4$. Here, $e$ denotes the Euler number
of the representation. It follows from [Gol88] that $\rho_{1}$ and $\rho_{2}$
are both Fuchsian representations. Therefore, as $\text{Max}(S)$ may be
thought of as the product of two copies of Teichmüller space, it is
homeomorphic to an open cell of dimension $12g-12$.

The main goal of this section is to prove Theorem A from the introduction,
which we restate below for the convenience of the reader.

######  Theorem 4.1.

The disjoint union

$\mathfrak{B}=\mathrm{Max}(S)\sqcup\mathbb{P}\mathrm{Core}(\mathcal{T},\mathcal{T})$

is homeomorphic to a closed ball of dimension $12g-12$.

We begin by recalling the topology placed on $\mathfrak{B}$. The maximal
component Max$(S)$ is naturally homeomorphic to the product of two copies of
Teichmüller space. This in turn, by the result of Schoen is homeomorphic to
the space of equivariant minimal Lagrangians in
$\mathbb{H}^{2}\times\mathbb{H}^{2}$. Under the Gromov-Hausdorff topology,
diverging sequences of minimal Lagrangians subconverge to the (projective)
core of a product of two trees ([Ouy19, Theorem 8.1]). The two trees are dual
to a pair of measured laminations, and the topology on $\mathfrak{B}$ is
compatible with the Thurston compactification on Teich$(S)\times$ Teich$(S)$
in the following way: if
$(\rho_{1,n},\rho_{2,n})\to[\lambda_{1},\lambda_{2}]$, then the associated
minimal Lagrangians converge to the core of $T_{1}\times T_{2}$, where $T_{i}$
is dual to $\lambda_{i}$.

Fix a complex structure $J$ on $S$ and denote by $X$ the Riemann surface
$(S,J)$. Then for any hyperbolic metric $h\in\text{Teich}(S)$ there is [ES64]
a unique [Har67] harmonic map $w_{h}\colon X\to(S,h)$ in the homotopy class of
the identity. Harmonicity of $w_{h}$ ensures that the Hopf differential
$q_{h}=\left(w_{h}^{*}h\right)^{(2,0)}$ is a holomorphic quadratic
differential on $X$. The vector space $\text{QD}(X)$ of holomorphic quadratic
differentials on $X$ has a natural norm given by the $L^{2}$-norm with respect
to the uniformizing hyperbolic metric $\sigma$ of $X$. With an abuse of
notation, we will still denote by $X$ the hyperbolic surface $(S,\sigma)$. The
map which assigns to a point in Teichmüller space its corresponding Hopf
differential is a homeomorphism [Wol89].

###### Proof of Theorem 4.1.

By Theorem 6.13 of [Ouy19], the space
$\text{Max}(S)\sqcup\mathbb{P}\mathrm{Core}(\mathcal{T},\mathcal{T})$ is
naturally homeomorphic to
$\text{Teich}(S)\times\text{Teich}(S)\sqcup\mathbb{P}(\mathcal{ML}(S)\times\mathcal{ML}(S))$,
so it suffices to prove the latter is homeomorphic to a closed ball of
dimension $12g-12$.

As $\mathbb{P}(\mathcal{ML}(S)\times\mathcal{ML}(S))$ is homeomorphic to a
sphere of dimension $12g-13$, the remainder of the proof consists of
describing how to attach this topological space to the open cell
$\text{Teich}(S)\times\text{Teich}(S)$ to obtain a closed ball.

We start by fixing a complex structure $J$ on $S$. Let $X=(S,J)$ be the
resulting Riemann surface. By the Wolf parameterization [Wol89]

$\text{Teich}(S)\times\text{Teich}(S)\cong\mathrm{QD}(X)\oplus\mathrm{QD}(X)$

via the map $\Phi(\rho_{1},\rho_{2})=(q_{\rho_{1}},q_{\rho_{2}})$. We equip
$\mathrm{QD}(X)\oplus\mathrm{QD}(X)$ with the norm

$\|q\|=\max\left(\|q_{1}\|,\|q_{2}\|\right),$

and consider

$\mathrm{BPQD}(X)=\\{q=(q_{1},q_{2})\colon\|q\|<1\\}.$

which is, topologically, a ball of dimension $12g-12$. We will need the
following lemma.

######  Lemma 4.2.

The map

$\displaystyle\beta\colon\mathrm{QD}(X)\oplus\mathrm{QD}(X)$
$\displaystyle\to\mathrm{BPQD}(X)$ $\displaystyle q=(q_{1},q_{2})$
$\displaystyle\mapsto\frac{4q}{1+4\|q\|}$

is continuous, injective, and proper. Hence $\beta$ is a homeomorphism.

###### Proof.

Suppose $\beta(q_{1},q_{2})=\beta(\phi_{1},\phi_{2})$. It follows then that
$q_{1}=k\phi_{1}$ and $q_{2}=k\phi_{2}$ for some $k\in\mathbb{R}$. Writing out
$\beta(q_{1},q_{2})=\beta(q_{1}/k,q_{2}/k)$, basic algebra shows $k=1$.
Continuity and properness follow by inspection. ∎

We will now describe the attaching map. Consider the map

$\psi:\text{Teich}(S)\times\text{Teich}(S)\sqcup\mathbb{P}(\mathcal{MF}(S)\times\mathcal{MF}(S))\to\overline{\text{BPQD}(X)}$

defined by

$\psi(x)=\begin{cases}\ \ \beta(\Phi(x))\ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{if
}x\in\text{Teich}(S)\times\text{Teich}(S)\\\
\lim\limits_{n\to+\infty}\beta(\Phi(x_{n}))\ \ \ \ \ \ \text{if
}x\in\mathbb{P}(\mathcal{MF}(S)\times\mathcal{MF}(S))\text{ and }x_{n}\to
x\end{cases}$

We show first that the map $\psi$ is well-defined. Suppose
$x_{n}=(X_{1,n},X_{2,n})\to x$ and
$x^{\prime}_{n}=(X^{\prime}_{1,n},X^{\prime}_{2,n})\to x$, where
$x=[\lambda_{1},\lambda_{2}]\in\mathbb{P}(\mathcal{MF}(S)\times\mathcal{MF}(S))$.
That is to say, there exists sequences of real numbers $c_{n},d_{n}$ for which
the rescaled hyperbolic surfaces $\widetilde{X}_{i,n}/c_{n}$ and
$\widetilde{X^{\prime}}_{i,n}/d_{n}$ converge to $\mathbb{R}$-trees
$T_{i},T^{\prime}_{i}$ dual to laminations $\lambda_{i}$ and
$\lambda_{i}^{\prime}$ such that
$[(\lambda_{1},\lambda_{2})]=[(\lambda_{1}^{\prime},\lambda_{2}^{\prime})]$.
By [Wol89], the sequences $c_{n}$ and $d_{n}$ can be taken to be
$\|\Phi(x_{n})\|$ and $\|\Phi(x_{n}^{\prime})\|$. Note that, a priori,
$\lambda_{i}^{\prime}=k\lambda_{i}$ for some $k>0$. With such rescaling, the
harmonic maps $h_{i,n}:\widetilde{X}\to\widetilde{X}_{i,n}/c_{n}$ converge to
the harmonic map $h_{i}:\widetilde{X}\to T_{i}$ given by projection onto the
leaf space of the measured foliation $\widetilde{\mathcal{F}}_{i}$
corresponding to $\widetilde{\lambda}_{i}$ [Wol95, Corollary 5.2]. Moreover
the sequence of Hopf differentials $q_{i,n}$ of $h_{i,n}$ converges to the
Hopf differential $q_{i}$ of $h_{i}$ (here take the quotient so that $q_{i}$
is a holomorphic quadratic differential on $X$ and not $\widetilde{X}$).
Finally, the differential $q_{i}$ is the unique holomorphic quadratic
differential on $X$ whose horizontal foliation is Whitehead equivalent to
$\mathcal{F}_{i}$. Likewise the sequence of harmonic maps
$h^{\prime}_{i,n}:\widetilde{X}\to\widetilde{X^{\prime}}_{i,n}/{d_{n}}$
converges to the harmonic map $h^{\prime}:\widetilde{X}\to T^{\prime}_{i}$,
whose Hopf differential is $q^{\prime}_{i}$ is the limit of the Hopf
differentials $q_{i,n}^{\prime}$ of $h^{\prime}_{i,n}$ and has horizontal
foliation $\mathcal{F}^{\prime}_{i}$ corresponding to the lamination
$\lambda_{i}^{\prime}$. Notice, in addition, that
$(q_{1,n},q_{2,n})=\Phi(x_{n})/\|\Phi(x_{n})\|$ and similarly
$(q_{1,n}^{\prime},q_{2,n}^{\prime})=\Phi(x_{n}^{\prime})/\|\Phi(x_{n}^{\prime})\|$.
It follows that the limits of $\beta(\Phi(x_{n}))$ and
$\beta(\Phi(x_{n}^{\prime}))$ as $n\to+\infty$ exist and coincide with
$(q_{1},q_{2})$ and $(q_{1}^{\prime},q_{2}^{\prime})$. As the distance
functions $d_{i}$ and $d_{i}^{\prime}$ on $T_{i}$ and $T_{i}^{\prime}$ satisfy
$d_{i}=k\cdot d_{i}^{\prime}$, then, by homogeneity of the Hopf differential,
one has $q_{i}=k\cdot q^{\prime}_{i}$. Since the pairs $(q_{1},q_{2})$ and
$(q_{1}^{\prime},q_{2}^{\prime})$ both have unit norm, we conclude that $k=1$
and the limits of $\beta(\Phi(x_{n}))$ and $\beta(\Phi(x_{n}^{\prime}))$ as
$n\to+\infty$ are equal.

Continuity follows almost immediately: the map $\beta\circ\Phi$ is continuous
on the interior and extends continuously to the boundary by a diagonal
argument. Indeed, we can approximate a sequence along the boundary by
sequences in the interior.

Bijectivity of $\psi$ on the interior also follows by [Wol89] and Lemma 4.2.
On the boundary, given $q=(q_{1},q_{2})$ with $\|q\|=1$, if $X_{i,t}$ is the
hyperbolic surface corresponding to the rays $tq_{i}$ in Wolf’s
parameterization of $\mathrm{Teich}(S)$, we have that
$\beta(\Phi(X_{1,t},X_{2,t}))\to q$ as $t\to\infty$, thus $\psi$ is surjective
on the boundary. Since the limit of $\beta(\Phi(x_{n}))$ along diverging
sequences in $x_{n}\in\mathrm{Teich}(S)\times\mathrm{Teich}(S)$ only depends
on the projective class of the limit of $x_{n}$ and not on the particular
sequence, we deduce that $\psi$ is injective on the boundary, because every
point in $\mathbb{P}(\mathcal{ML}(S)\times\mathcal{ML}(S))$ can be obtained as
a limit along a ray defined above and the limit of $\beta\circ\Phi$ along
distinct rays is different.

It remains to prove $\psi^{-1}$ is continuous. We can actually write the
inverse explicitly:

$\psi^{-1}(q_{1},q_{2})=\begin{cases}\Phi^{-1}(\beta^{-1}(q_{1},q_{2}))\ \ \ \
\ \ \ \ \ \ \text{if $\|(q_{1},q_{2})\|<1$}\\\ [\lambda_{1},\lambda_{2}]\ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{if
$\|(q_{1},q_{2})\|=1$}\end{cases}$

where $\lambda_{i}$ is the measured lamination corresponding to the horizontal
foliation of $q_{i}$. Continuity of $\psi^{-1}$ on $\mathrm{BPQD}(X)$ is then
a consequence of Lemma 4.2 and Wolf’s parameterization. Continuity on the
boundary follows from Hubbard-Masur theorem [HM79]. In general, if
$q_{n}=(q_{1,n},q_{2,n})\in\mathrm{BPQD}(X)$ converges to
$(q_{1},q_{2})\in\partial\overline{\mathrm{BPQD}(X)}$, then there is a
sequence of scaling factors $c_{n}$ such that the pair of hyperbolic surfaces
$x_{n}=\psi^{-1}(q_{1,n},q_{2,n})$ rescaled by $c_{n}$ converge to real trees
$T_{1},T_{2}$ dual to measured laminations $\lambda_{1},\lambda_{2}$. We need
to show that $\psi^{-1}(q_{1},q_{2})$ is equal to $[\lambda_{1},\lambda_{2}]$.
Assume not, then we would have, by injectivity and continuity of $\psi$,

$\displaystyle(q_{1},q_{2})$
$\displaystyle=\psi(\psi^{-1}(q_{1},q_{2}))\neq\psi([\lambda_{1},\lambda_{2}])=\lim_{n\to+\infty}\psi(x_{n})=\lim_{n\to+\infty}(q_{1,n},q_{2,n})$

which contradicts the assumption on $(q_{1,n},q_{2,n})$.

Finally, we remark the compactification in [Ouy19] is independent of the
choice of a base point, so that the role of the base point $(S,J)$ is merely
an auxiliary one. This completes the proof of the theorem. ∎

##  5\. Fixed point for the mapping class group action

In this section, we study the action of the mapping class group
$\mathrm{MCG}(S)$ on the compactification
$\mathfrak{B}=\overline{\mathrm{Max}(S)}$ constructed in Theorem 4.1. We wish
to study the fixed points of this action. We will need the following
observations.

######  Lemma 5.1.

The action of the mapping class group on $\mathrm{Max}(S)$ extends
continuously to the closure $\mathfrak{B}=\overline{\text{Max}(S)}$.

######  Corollary 5.2.

For every $\phi\in\mathrm{MCG}(S)$, there exists $x\in\mathfrak{B}$ such that
$\phi(x)=x$.

The first main goal of this section is to analyze these fixed points via the
celebrated Nielsen-Thurston classification, which we recall for future
reference.

######  Theorem 5.3 (Nielsen-Thurston classification, see [FM11, Chapter 13]).

Any diffeomorphism $\phi$ on $S$ is isotopic to a map $\phi^{\prime}$
satisfying one of the following mutually exclusive conditions:

  1. (1)

(periodic) $\phi^{\prime}$ is of finite order;

  2. (2)

(reducible) $\phi^{\prime}$ is not periodic, and there is a nonempty set
$\\{c_{1},\dots,c_{r}\\}$ of isotopy classes of essential pairwise disjoint
simple closed curves in $S$ such that
$\\{\phi^{\prime}(c_{i})\\}_{i=1}^{r}=\\{c_{i}\\}_{i=1}^{r}$;

  3. (3)

(pseudo-Anosov/pA) there exists $\lambda>1$ and two transverse measured
foliations $\mathcal{F}$ and $\mathcal{F}^{\prime}$ such that

$\phi^{\prime}(\mathcal{F})=\lambda\mathcal{F}\text{ and
}\phi^{\prime}(\mathcal{F}^{\prime})=\frac{1}{\lambda}\mathcal{F}^{\prime}.$

######  Remark 5.4.

Note that our definition of reducible mapping class is non-standard as we
assume that if $\phi$ is reducible, then it is not periodic. We do so to
improve our exposition. The set $\\{c_{1},\dots,c_{r}\\}$ in item (2) is a
reduction system of $\phi$. The canonical reduction system
$\\{\bar{c}_{1},\ldots,\bar{c}_{k}\\}$ of $\phi$ reducible is the intersection
of all the maximal (with respect to inclusion) reduction systems.
Equivalently, each $\bar{c}_{j}$ is part of a reduction system and if
$i(\bar{c}_{j},c)\neq 0$ and $n\neq 0$, then $\phi^{n}(c)\neq c$.

We are ready to characterize the fixed points of a mapping class acting on
$\mathfrak{B}$ and establish Proposition 1.1 from the introduction.

######  Proposition 5.5.

Suppose $\phi\in\mathrm{MCG}(S)$ and $\phi(x)=x$ for some
$x=(x_{1},x_{2})\in\mathfrak{B}$.

  1. (1)

If $\phi$ is periodic, then $x_{1}$ and $x_{2}$ are any two points fixed by
$\phi$ in the Thurston compactification of Teichmüller space such that
$(x_{1},x_{2})\in\mathfrak{B}$.

  2. (2)

If $\phi$ is pA, then $(x_{1},x_{2})\in\partial\mathfrak{B}$ and $x_{1}=0$, or
$x_{2}=0$ or $x_{1}=x_{2}$.

###### Proof.

  1. (1)

If $\phi$ fixes $x$ projectively, there exists $\alpha>0$ such that
$\phi(x_{1},x_{2})=(\alpha x_{1},\alpha x_{2})$. Since $\phi$ is periodic, we
can check that $\alpha=1$.

  2. (2)

Since $\phi$ fixes the projective class of $(x_{1},x_{2})$, there exists
$\alpha>0$ such that $\phi(x_{1},x_{2})=(\alpha x_{1},\alpha x_{2})$. On the
other hand, since $\phi$ is pseudo-Anosov, there exist two measured
laminations $y_{1}$ and $y_{2}$ and $\lambda>1$ such that $\phi(y_{1})=\lambda
y_{1}$ and $\phi(y_{2})=\frac{1}{\lambda}y_{2}$. Since $\phi$ does not fix any
other projective class of measured laminations [FLP21, Corollary 12.4], it
follows that $x_{i}=0$, $y_{1}$ or $y_{2}$ for $i=1,2$. We claim that
$x\neq(y_{1},y_{2})$ (and, symmetrically, $x\neq(y_{2},y_{1})$). Otherwise,
because $i(y_{1},y_{2})\neq 0$

$\lambda\cdot i(y_{1},y_{2})=i(\phi(y_{1}),y_{2})=\alpha\cdot
i(y_{1},y_{2})=i(y_{1},\phi(y_{2}))=\frac{1}{\lambda}\cdot i(y_{1},y_{2})$

which is a contradiction. ∎

There is a natural continuous projection map
$\pi\colon\mathfrak{B}\to\overline{\mathrm{Ind}(S)}$ defined as follows. For
$x\in\text{Max}(S)$, consider the corresponding equivariant minimal Lagrangian
$\widetilde{\Sigma}_{x}$. Then, $\pi(x)$ is the induced metric on
$\widetilde{\Sigma}_{x}$. Otherwise, if $x\in\partial\mathfrak{B}$, consider
the core of the tree corresponding to
$x\in\mathbb{P}(\mathcal{ML}(S)\times\mathcal{ML}(S))$: its length spectrum
coincides with that of a mixed structure $\mu$ on $S$. Set $\pi(x)=\mu$. This
projection $\pi$ is continuous then by [Ouy19, Theorem 6.13]. We consider the
corresponding action of $\mathrm{MCG}(S)$ on $\overline{\mathrm{Ind}(S)}$
given by push-forward.

######  Lemma 5.6.

The action of $\mathrm{MCG}(S)$ on $\mathfrak{B}$ and
$\overline{\mathrm{Ind}(S)}$ commute. In other words, for every
$\phi\in\mathrm{MCG}(S)$

$\pi\circ\phi=\phi\circ\pi.$

###### Proof.

If $x=(x_{1},x_{2})$ is in the interior of $\mathfrak{B}$, then
$\pi(\phi(x))=\phi(\pi(x))$ because $\phi(\pi(x))$ has the same length
spectrum as the induced metric on the minimal Lagrangian associated to
$\phi(x_{1})$ and $\phi(x_{2})$. Suppose $x\in\partial\mathfrak{B}$ and
consider a sequence $(x_{n})_{n\in\mathbb{N}}\subset\mathrm{Max}(S)$ such that
$x_{n}\to x$. Since, $\pi(\phi(x_{n}))=\phi(\pi(x_{n}))$ for all
$n\in\mathbb{N}$, the result follows by continuity of $\phi$ and $\pi$. ∎

We are now ready to establish the main theorems of this section. In
particular, Theorem 5.7 below is Theorem B from the introduction.

######  Theorem 5.7.

Assume $\phi\in\mathrm{MCG}(S)$ fixes
$\mu\in\partial\overline{\mathrm{Ind}(S)}$.

  1. (1)

If $\mu$ is purely flat, then $\phi$ is periodic.

  2. (2)

If $\mu$ is properly mixed, then $\phi$ is not pA.

###### Proof.

For item (1), let us first assume by contradiction $\phi$ is pA. Since $\phi$
fixes $\mu$ projectively, there exist $\alpha>0$ such that
$\phi(\mu)=\alpha\mu$ and up to replacing $\phi$ with $\phi^{-1}$, we can
assume $\alpha\geq 1$. Now, there exists a measured foliation $\mathcal{F}$
and $\lambda>1$ such that $\phi(\mathcal{F})=\lambda\mathcal{F}$. Recall that
$\mu$ corresponds to a pair of filling measured foliations, thus $0\neq
i(\mu,\mathcal{F})$. On the other hand,

$0\neq
i(\mu,\mathcal{F})=\frac{1}{\alpha}i(\phi(\mu),\mathcal{F})=\frac{1}{\alpha}i(\mu,\phi^{-1}(\mathcal{F}))=\frac{1}{\alpha\lambda}i(\mu,\mathcal{F})$

and we achieve a contradiction.

On the other hand, suppose by contradiction that $\mu$ is purely flat and
$\phi$ is reducible. Then, there is a subsurface $T\subset S$ and $N\geq 0$
such that $\big{(}\phi^{N}\big{)}_{\big{|}_{T}}\colon T\to T$ and
$\big{(}\phi^{N}\big{)}_{\big{|}_{T}}$ is pA. Applying the above argument to
$\big{(}\phi^{N}\big{)}_{\big{|}_{T}}$ we achieve again a contradiction.

We establish item (2). Suppose $\mu$ is properly mixed, i.e. $\mu$ is not flat
but it has at least one flat piece. We can decompose $S$ as

$\left(\\{S_{\alpha}\\}_{\alpha\in\mathcal{A}},\\{d_{\beta}\\}_{\beta\in\mathcal{B}},\\{\mu_{\alpha}\\}_{\alpha\in\mathcal{A}}\right)$

where $\mu_{\alpha}$ is a flat structure or a (possibly-zero) laminar
structure on $S_{\alpha}$ and $d_{\beta}$ is a maximal collection of closed
geodesics so that

$i(d_{\beta},d_{\beta^{\prime}})=0\quad\text{and}\quad i(d_{\beta},\mu)=0$

for all $\beta,\beta^{\prime}\in\mathcal{B}$ and for every $c$ that intersects
some $d_{\beta}$ transversely, $i(c,\mu)>0$. Note that there exists a unique
set $\\{d_{\beta}\\}_{\beta\in\mathcal{B}}$ with these properties (see
[BIPP17, Theorem 1.1]).

######  Claim 5.8.

$\phi$ fixes the set $\\{d_{\beta}\\}_{\beta\in\mathcal{B}}$.

###### Proof of Claim.

Observe that

$i(\phi(d_{\beta}),\phi(d_{\beta^{\prime}}))=i(d_{\beta},d_{\beta^{\prime}})=0\quad\text{and}\quad
i(\mu,\phi(d_{\beta}))=i(\phi^{-1}(\mu),d_{\beta})=0.$

If $c$ is a curve that intersects $\phi(d_{\beta})$ transversely, then

$i(\phi^{-1}(c),d_{\beta})=i(c,\phi(d_{\beta}))>0\quad\text{and}\quad
i(c,\mu)=i(\phi^{-1}(c),\phi^{-1}(\mu))>0.$

Thus, by uniqueness,
$\\{\phi(d_{\beta})\\}_{\beta\in\mathcal{B}}=\\{d_{\beta}\\}_{\beta\in\mathcal{B}}$.
∎

Thanks to the previous claim, we know that there exists $N>0$ such that for
all $\alpha$ and $\beta$

$\phi^{N}(d_{\beta})=d_{\beta},\text{ and }\phi^{N}(S_{\alpha})=S_{\alpha}.$

If $\phi^{N}$ is periodic, we are done. So let us assume that $\phi^{N}$ is
not periodic. Then $\phi^{N}$ cannot be pA because $\mu$ is properly mixed, so
there exists $\alpha$ such that $\mu_{\alpha}$ is flat and we can apply the
argument in item (2) to $S_{\alpha}$. Therefore, $\phi^{N}$ is reducible. ∎

######  Remark 5.9.

For an explicit example of $\mu$ purely flat and $\phi$ periodic such that
$\phi(\mu)=\mu$, consider a singular flat metric on a surface of genus two
obtained by doubling a singular flat metric on a torus with boundary.

######  Theorem 5.10.

Suppose $\phi\in\mathrm{MCG}(S)$ is reducible and fixes
$\mu\in\partial\overline{\mathrm{Ind}(S)}$ which is properly mixed. Let
$S=(S_{\alpha},\\{d_{\beta}\\}_{\beta\in\mathcal{B}},\mu_{\alpha})$ be the
subdivision of $S$ induced by $\mu$.

  1. (1)

If for some $N>0$,
$\psi_{\alpha}=\big{(}\phi^{N}\big{)}_{\big{|}_{S_{\alpha}}}\colon
S_{\alpha}\to S_{\alpha}$ is pA, then $\mu_{\alpha}=0$.

  2. (2)

If $\mu_{\alpha}\neq 0$ for all $\alpha\in\mathcal{A}$, then the canonical
reduction system of $\phi$ is contained in
$\\{d_{\beta}\\}_{\beta\in\mathcal{B}}$.

###### Proof.

By hypotheses, we can decompose $S$ as
$\left(\\{S_{\alpha}\\}_{\alpha\in\mathcal{A}},\\{d_{\beta}\\}_{\beta\in\mathcal{B}},\\{\mu_{\alpha}\\}_{\alpha\in\mathcal{A}}\right)$.
By Claim 5.8, there exists $N>0$ such that $\phi^{N}$ fixes $d_{\beta}$ for
all $\beta\in\mathcal{B}$ and $\phi^{N}(S_{\alpha})=S_{\alpha}$. Set
$\psi_{\alpha}=(\phi^{N})_{\big{|}S_{\alpha}}\colon S_{\alpha}\to S_{\alpha}$.

In order to prove item (1), we need to consider three cases.

  * a.

If $\mu_{\alpha}=0$, then $\psi_{\alpha}$ can be any element in
$\text{MCG}(S_{\alpha})$.

  * b.

If $(S_{\alpha},\mu_{\alpha})$ is purely flat (there exists at least one
$\alpha$ for which this happens), then $\psi_{\alpha}$ can only be periodic by
Theorem 5.7 item (1).

  * c.

Suppose $(S_{\alpha},\mu_{\alpha})$ is purely laminar. Since $\mu$ has a flat
piece $\mu_{\beta}$, we know that $\psi_{\beta}$ is periodic and hence it
fixes $\mu_{\beta}$ (not just projectively). We deduce that
$\phi^{N}(\mu)=\mu$, otherwise we could find $z\neq 1$ such that
$\psi_{\alpha}(\mu_{\alpha})=z\mu_{\alpha}$, but then $\phi^{N}$ would not fix
$\mu$ projectively. We can now conclude that $\psi_{\alpha}$ cannot be pA.
This is because if $c$ is a curve such that $i(\mu_{\alpha},c)>0$, then

$i(\mu_{\alpha},c)=i(\psi_{\alpha}^{-1}(\mu_{\alpha}),c)=i(\mu_{\alpha},\psi_{\alpha}(c))$

but $i(\mu_{\alpha},\psi_{\alpha}(c))\neq i(\mu_{\alpha},c)$ because
$\psi_{\alpha}$ would change the length of curves transverse to
$\mu_{\alpha}$.

This completes the proof of item (1).

We wish to prove that the canonical reduction system
$\\{\bar{c}_{1},\ldots,\bar{c}_{k}\\}$ of $\phi$ is a subset of
$\\{d_{\beta}\\}_{\beta\in\mathcal{B}}$ under the additional assumption that
$\mu_{\alpha}\neq 0$ for all $\alpha\in\mathcal{A}$. First, observe that by
Claim 5.8 $\\{d_{\beta}\\}_{\beta\in\mathcal{B}}$ is contained in a maximal
reduction system for $\phi$. In particular $i(\bar{c}_{j},d_{\beta})=0$ for
all $j$ and $\beta$. Moreover, since $\mu$ is properly mixed, there exists
$\beta$ such that $\mu_{\beta}$ is flat, hence $\psi$ fixes $\mu$, not just
its projective class, as observed before.

Assume, by contradiction
$\bar{c}_{j}\not\in\\{d_{\beta}\\}_{\beta\in\mathcal{B}}$. Suppose
$\bar{c}_{j}$ is contained in a purely flat piece $(S_{\alpha},\mu_{\alpha})$.
Then, by Theorem 5.7 necessarily $\psi_{\alpha}$ is periodic. But this
contradicts the property that if $i(\bar{c}_{j},c)\neq 0$ and $n\neq 0$, then
$\phi^{n}(c)\neq c$ since there exists $m$ such that $\psi_{\alpha}^{m}$ is
the identity. Therefore $\bar{c}_{j}$ is contained in a purely laminar piece
$\mu_{\alpha}$.

By definition of $\\{d_{\beta}\\}_{\beta\in\mathcal{B}}$, $\psi_{\alpha}$
fixes a measured lamination $\mathcal{F}$ of full support in $S_{\alpha}$. If
$\psi_{\alpha}$ is pA, then this contradicts item (1). Assume that
$\psi_{\alpha}$ is periodic, so that there exists $m>0$ such that
$\psi_{\alpha}^{m}$ is the identity. Then, we achieve again a contradiction
because there would exist $c$ such that $i(\bar{c}_{j},c)\neq 0$ but
$\phi^{m}(c)=c$.

Finally, assume that $\psi_{\alpha}$ is reducible. We first observe that there
needs to exist a subsurface $S_{\alpha,\beta}$ such that an opportune power of
$\psi_{\alpha}$ restricted to $S_{\alpha,\beta}$ is pA. Otherwise, there
exists $N\in\mathbb{Z}_{\geq 0}$ such that $\psi_{\alpha}^{N}=\text{id}$ and
we obtain a contradiction by considering any closed curve $c\in S_{\alpha}$
which intersects $\bar{c}_{j}$ transversely. Now, let $S_{\alpha,\beta}$ be a
subsurface of $S_{\alpha}$ and $\psi_{\alpha,\beta}$ a power of
$\psi_{\alpha}$ such that $\psi_{\alpha,\beta}\big{|}_{S_{\alpha,\beta}}$ is
pA. Let $\mathcal{F}^{s}$ be the unstable measured foliation of
$\psi_{\alpha,\beta}$ with stretch factor $z>1$. But then, as we observed
earlier, $\psi_{\alpha}(\mathcal{F})$ fixes the measured lamination (not just
its projective class) and, since $\mathcal{F}$ has full support, there exists
an integer $M>0$ such that

$0\neq
i(\mathcal{F},\mathcal{F}^{s})=i(\psi_{\alpha}^{M}(\mathcal{F}),\mathcal{F}^{s})=i(\mathcal{F},\psi_{\alpha}^{-M}(\mathcal{F}^{s}))=\frac{1}{z}i(\mathcal{F},\mathcal{F}^{s}),$

which gives a contradiction. ∎

##  6\. $\overline{\mathfrak{a}^{+}}$-valued measured laminations and mixed
structures

In this final section we introduce Weyl chamber valued measured laminations
and use them to refine the notion of mixed structures on a closed surface
defined in [DLR10], and generalized to higher order differentials in [OT20,
OT21]. We show that the core of the product of two trees dual to measured
laminations is dual to such a mixed structure, thus giving a new
interpretation of the boundary objects in our compactification of
$\mathrm{Max}(S)$.

Let $\mathfrak{g}$ be a real semisimple Lie algebra. The choice of a maximal
compact subalgebra $\mathfrak{k}$ induces an orthogonal decomposition of
$\mathfrak{g}$ for the Killing form:

$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{m}\ .$

A Cartan subalgebra $\mathfrak{a}\subset\mathfrak{g}$ is a maximal abelian
subspace of $\mathfrak{m}$. This induces a decomposition of $\mathfrak{g}$ in
$\mathrm{ad}(\mathfrak{a})$-eigenspaces

$\mathfrak{g}=\mathfrak{g}_{0}\bigoplus_{\alpha\in\Sigma}\mathfrak{g}_{\alpha}\
.$

Elements of
$\Sigma\subset\mathfrak{a}^{*}=\operatorname{Hom}(\mathfrak{a},\mathbb{R})$
are called restricted roots of $\mathfrak{a}$ in $\mathfrak{g}$. Here we can
extract a subset $\Delta$ of _simple_ roots with the property that any
$\alpha\in\Sigma$ can be expressed as linear combinations of simple roots with
coefficients all of the same sign. This distinguishes, thus, a subset of
positive roots that we denote by $\Sigma^{+}\subset\Sigma$. The closed
positive Weyl chamber of $\mathfrak{a}$ associated to $\Sigma^{+}$ is then the
cone

$\overline{\mathfrak{a}^{+}}=\\{X\in\mathfrak{a}\ |\ \alpha(X)\geq 0\ \
\forall\alpha\in\Sigma^{+}\\}\ .$

We also denote by $W$ the Weyl group of $\mathfrak{g}$, i.e.
$W=N(\mathfrak{a})/\mathfrak{a}$.

######  Definition 6.1.

An $\overline{\mathfrak{a}^{+}}$-valued measured lamination on a (not
necessarily closed) surface $S$ is a geodesic lamination $\lambda$ on $S$ that
supports a measure $\mu$ on transverse arcs that takes value in
$\overline{\mathfrak{a}^{+}}$ and satisfies the following properties:

  1. a)

$\mu(\gamma)\neq 0$, if $\gamma$ intersects $\lambda$ transversely;

  2. b)

if $\gamma$ and $\gamma^{\prime}$ are homotopic arcs transverse to $\lambda$
and there is a homotopy between them that preserves transversality at every
time, then $\mu(\gamma)=\mu(\gamma^{\prime})$;

  3. c)

$\mu$ is additive on concatenation of paths, i.e.
$\mu(\gamma\gamma^{\prime})=\mu(\gamma)+\mu(\gamma^{\prime})$ for all $\gamma$
and $\gamma^{\prime}$ transverse to $\lambda$ such that concatenation is
defined.

######  Remark 6.2.

If $\mathfrak{g}=\mathfrak{sl}(2,\mathbb{R})$, then we can identify the closed
positive Weyl chamber with $\mathbb{R}_{\geq 0}$. Thus, in this case,
Definition 6.1 recovers the standard notion of measured laminations.
Similarly, if
$\mathfrak{g}=\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})$,
then $\overline{\mathfrak{a}^{+}}$-valued laminations can be identified with
ordered pairs $(\lambda_{1},\lambda_{2})$, such that $\lambda_{1}$,
$\lambda_{2}$ and $\lambda_{1}+\lambda_{2}$ are measured laminations (i.e.
$\lambda_{1}$ and $\lambda_{2}$ are nowhere transverse).

We can also extend the classical notion of trees dual to a measured lamination
to this context.

######  Definition 6.3.

Let $(T,d)$ be an $\mathbb{R}$-tree acted upon by the fundamental group of
$S$. We say that the action of $\pi_{1}(S)$ is dual to an
$\overline{\mathfrak{a}^{+}}$-valued measured lamination $\mu$, if there is an
equivariant map $p:\widetilde{S}\rightarrow T$ and an
$\overline{\mathfrak{a}^{+}}$-valued distance $d_{\mathfrak{a}^{+}}:T\times
T\rightarrow\overline{\mathfrak{a}^{+}}$ such that

  1. a)

for all $x,y\in\widetilde{S}$, we have
$d_{\mathfrak{a}^{+}}(p(x),p(y))=\mu(\gamma)$ for some (hence any) arc
$\gamma:[0,1]\rightarrow\widetilde{S}$ transverse to the support of $\mu$ with
$\gamma(0)=x$ and $\gamma(1)=y$;

  2. b)

given a geodesic path $\gamma:[0,1]\rightarrow T$, we have
$d(\gamma(0),\gamma(1))\geq\|d_{\mathfrak{a}^{+}}(\gamma(0),\gamma(1))\|$.
Here $\|\cdot\|$ denotes the standard Euclidean norm of a vector in
$\overline{\mathfrak{a}^{+}}$

We now combine $\overline{\mathfrak{a}^{+}}$-valued measured laminations with
the classical notion of $1/k$-translation surfaces in order to define a hybrid
structure on $S$.

######  Definition 6.4.

Let $\overline{\mathfrak{a}^{+}}$ be a closed Weyl chamber and $k\geq 1$ an
integer. An $(\overline{\mathfrak{a}^{+}},k)$-mixed structure on a closed
surface $S$ is the datum of

  1. (1)

a collection of non-homotopically trivial, pairwise non-homotopic, disjoint
simple closed curves $\gamma_{1},\dots,\gamma_{n}$ on $S$;

  2. (2)

for each connected component $S^{\prime}$ of $S\setminus\cup_{j}\gamma_{j}$
either

     * •

an $\overline{\mathfrak{a}^{+}}$-valued measured lamination $\lambda$, where
we allow each $\gamma_{j}$ to be in the support; or

     * •

a meromorphic $k$-differential of finite area that endows $S^{\prime}$ with a
$1/k$-translation surface structure.

These $(\overline{\mathfrak{a}^{+}},k)$-mixed structures can be interpreted as
dual to the $(\mathfrak{a},W)$-complexes studied by Anne Parreau ([Par15]) in
the context of $\mathfrak{g}=\mathfrak{sl}(3,\mathbb{R})$. Let us recall
briefly how these complexes are defined and explain in which sense these
notions can be considered dual to each other.

Following [Par15], an $(\mathfrak{a},W)$-complex $K$ is the union of (possibly
degenerate) polygons in $\mathfrak{a}$ glued together along boundary segments
via elements of $W_{aff}=W\rtimes\mathbb{R}$. More precisely, there is a
family of affine simplices $P_{\mu}\subset\mathfrak{a}$ and injective maps
$\phi_{\mu}:P_{\mu}\rightarrow K$ such that if $K_{\mu}=\phi_{\mu}(P_{\mu})$
and $K_{\mu^{\prime}}=\phi_{\mu^{\prime}}(P_{\mu^{\prime}})$ have non-empty
intersection then there is $w_{\mu,\mu^{\prime}}\in W_{aff}$ such that
$\phi_{\mu}(x)=\phi_{\mu^{\prime}}(x^{\prime})$ if and only if
$x^{\prime}=w_{\mu,\mu^{\prime}}(x)$ and $P_{\mu}\cap
w_{\mu,\mu^{\prime}}^{-1}(P_{\mu^{\prime}})$ is a face in $P_{\mu}$. We only
consider connected and simply-connected $(\mathfrak{a},W)$-complexes acted
upon by $\pi_{1}(S)$. Note that, since the gluing maps between simplices are
Euclidean isometries, the Euclidean distance on $\mathfrak{a}$ induces a
distance on $K$. We will only work with $(\mathfrak{a},W)$-complexes whose
induced distance is CAT(0). Similarly, $K$ is also endowed with an
$\overline{\mathfrak{a}^{+}}$-valued distance inherited from $\mathfrak{a}$.

Examples of $(\mathfrak{a},W)$-complexes are subcomplexes of an Euclidean
building modelled on $W_{aff}$. We will see that cores of product of two trees
dual to measured laminations are indeed $(\mathfrak{a},W)$-complexes where
$\mathfrak{a}$ is the Cartan subalgebra of
$\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})$ and
$W=\\{\pm\mathrm{Id}\\}$.

######  Definition 6.5.

We say that an $(\mathfrak{a},W)$-complex $K$ acted upon by $\pi_{1}(S)$ is
dual to an $(\overline{\mathfrak{a}^{+}},k)$-mixed structure $\mu$ on $S$ if
we can decompose $K$ into a $1$-dimensional part $K_{1}$ and a $2$-dimensional
part $K_{2}$ such that

  * •

$K_{1}$ is the union of $\mathbb{R}$-trees dual to the laminar part of $\mu$;

  * •

$K_{2}$ is endowed with a $1/k$-translation surface structure isomorphic to
the universal cover of the flat parts of $\mu$.

Note that the $2$-dimensional part of an $(\mathfrak{a},W)$-complex can be
endowed with a $1/k$ translation surface structure only if $W$ contains the
subgroup generated by rotations of angle $\frac{2\pi}{k}$.

We believe that these mixed structures naturally appear in a harmonic map
compactification of the Hitchin and maximal components of the character
variety for real Lie groups $G$ of rank $2$. In this context, Labourie
([Lab17]), Collier ([Col16]) and Collier-Tholozan-Toulisse ([CTT19]) proved
that given a Hitchin or maximal representation $\rho:\pi_{1}(S)\rightarrow G$
there exists a unique $\rho$-equivariant minimal surface
$\widetilde{\Sigma}_{\rho}$ in $G/K$, where $K$ is a maximal compact subgroup
of $G$. One could then find a compactification of these components by studying
the limiting behaviour of $\widetilde{\Sigma}_{\rho_{n}}$ when $\rho_{n}$
leaves all compact sets in the character variety. Up to subsequences, and
after rescaling the metric on $G/K$ appropriately,
$\widetilde{\Sigma}_{\rho_{n}}$ should converge to a subcomplex
$\widetilde{\Sigma}_{\infty}\subset B$, where $B$ is a non-discrete Euclidean
building modelled on the affine Weyl-group of $G$. We conjecture that
$\Sigma_{\infty}$ is dual to a mixed structure as in Definition 6.5 where
$\overline{\mathfrak{a}^{+}}$ is a Cartan subalgebra of the Lie algebra of $G$
and $k$ depends on the particular group. More precisely, we conjecture the
following:

######  Conjecture 6.6.

  1. a)

Let $G$ be a real split semisimple Lie group of rank $2$. Then the boundary of
$\mathrm{Hit}(S,G)$ can be identified with the space of projective classes of
$(\overline{\mathfrak{a}^{+}},k)$-mixed structures where:

     * •

if $G=\mathrm{SL}(3,\mathbb{R})$, then $\mathfrak{a}=A_{2}$ and $k=3$;

     * •

if $G=\mathrm{Sp}(4,\mathbb{R})$, then $\mathfrak{a}=B_{2}$ and $k=4$;

     * •

if $G=G_{2}^{\mathbb{R}}$, then $\mathfrak{a}=G_{2}$ and $k=6$.

  2. b)

Let $G$ be a real semisimple Lie group of Hermitian type and rank $2$. Then
the boundary of $\mathrm{Max}(S,G)$ can be identified with the space of
projective classes of $(\overline{\mathfrak{a}^{+}},k)$-mixed structures
where:

     * •

if $G=\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$, then
$\mathfrak{a}=A_{1}\times A_{1}$ and $k=2$;

     * •

if $G=\mathrm{SO}(2,n)$ with $n\geq 3$, then $\mathfrak{a}=B_{2}$ and $k=4$.

In support of this conjecture, we show that the core of the product of two
trees dual to measured laminations is dual to an $(\overline{A_{1}^{+}\times
A_{1}^{+}},2)$-mixed structure and that we can identify
$\mathrm{Core}(\mathcal{T},\mathcal{T})$ with the space of such structures,
thus proving the conjecture for
$G=\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$. Moreover, in a
subsequent paper ([LTW21]), the third author, in collaboration with Loftin and
Wolf, will give further evidence towards Conjecture 6.6 by describing the
geometry of the harmonic maps to buildings arising from some diverging
sequences of $\mathrm{SL}(3,\mathbb{R})$-Hitchin representations. It would be
interesting to introduce a higher rank version of our vector valued mixed
structures, at least for the case of $\mathrm{SL}(d,\mathbb{R})$-Hitchin
components, and relate it to the subspaces of the Euclidean building studied
in [Le16] and [Mar19a].

######  Lemma 6.7.

Let $T_{1}$ and $T_{2}$ be real trees dual to measured laminations
$\lambda_{1}$ and $\lambda_{2}$ and let $C$ be the core of $T_{1}\times
T_{2}$. Then $C$ is an $(A_{1}\times A_{1},\\{\pm\text{Id}\\})$-complex dual
to an $(\overline{A_{1}^{+}\times A_{1}^{+}},2)$-mixed structure on $S$.

###### Proof.

We already saw in Section 3 that $C$ is the union of a $1$-dimensional
subcomplex $C_{1}$ and a $2$-dimensional subcomplex $C_{2}$ of $T_{1}\times
T_{2}$. Moreover, we showed that each connected component of $C_{2}$ is the
universal cover of a half-translation surface structure on a subsurface
$S^{\prime}$ of $S$, on which the laminations $\lambda_{1}$ and $\lambda_{2}$
fill. Thus, it only remains to show that each connected component
$C_{1}^{\prime}$ of $C_{1}$ is a tree dual to an $\overline{A_{1}^{+}\times
A_{1}^{+}}$-valued measured lamination.
Recall from Section 3 that $C_{1}^{\prime}$ is the image under the map $F$
defined in Equation (1) of a domain $\Omega\subset\mathbb{H}^{2}$ that can be
identified with the universal cover of a subsurface $S^{\prime}$ of $S$ on
which $\lambda_{1}$ and $\lambda_{2}$ are nowhere transverse. Moreover, we
observe that $C_{1}^{\prime}$ has a natural distance $d$ induced by the
ambient space

$d((x_{0},y_{0}),(x_{1},y_{1}))=\sqrt{d_{1}(x_{0},y_{0})^{2}+d_{2}(x_{1},y_{1})^{2}}$

and a natural $\overline{A_{1}^{+}\times A_{1}^{+}}$-valued distance $\vec{d}$
defined by

$\vec{d}((x_{0},y_{0}),(x_{1},y_{1}))=(d_{1}(x_{0},y_{0}),d_{2}(x_{1},y_{1}))\
.$

We claim that $(C_{1}^{\prime},d)$ is an $\mathbb{R}$-tree dual to the
$\overline{A_{1}^{+}\times A_{1}^{+}}$-valued measured lamination
$\vec{\lambda}=(\lambda_{1},\lambda_{2})$ (see Remark 6.2). By Lemma 3.4,
$C_{1}^{\prime}$ can be identified with the $\mathbb{R}$-tree dual to the
measured lamination $\lambda_{0}=\lambda_{1}+\lambda_{2}$, if endowed with the
distance $d_{0}$ introduced in Section 3. In particular, there is a continuous
$\pi_{1}(S^{\prime})$-equivariant map $p:=p_{\lambda_{0}}:\Omega\rightarrow
C_{1}^{\prime}$. It follows immediately from the definitions and the fact that
$T_{1}$ and $T_{2}$ are dual to the laminations $\lambda_{1}$ and
$\lambda_{2}$ that for all $x,y\in\Omega^{\prime}$ we have

$\vec{d}(p(x),p(y))=\vec{\lambda}(\gamma)$

for all $\gamma:[0,1]\rightarrow\Omega$ transverse to the support of
$\lambda_{0}$ with $\gamma(0)=x$ and $\gamma(1)=y$.
Property b) in Definition 6.3 also holds. Indeed, a geodesic path
$\gamma=(\gamma_{1},\gamma_{2}):[0,1]\rightarrow C_{1}^{\prime}\subset
T_{1}\times T_{2}$, seen in the quadrant $\gamma_{1}\times\gamma_{2}$,
consists of a concatenation of horizontal, vertical or diagonal paths in which
the projections onto the two factors are always non-decreasing. Hence,

$d(\gamma(0),\gamma(1))\geq\|\vec{d}(\gamma(0),\gamma(1))\|\ ,$

and the proof is complete. ∎

######  Theorem 6.8.

The space of $(\overline{A_{1}^{+}\times A_{1}^{+}},2)$-mixed structures on
$S$ is homeomorphic to $\mathrm{Core}(\mathcal{T},\mathcal{T})$.

###### Proof.

Let $Y$ denote the set of $(\overline{A_{1}^{+}\times A_{1}^{+}},2)$-mixed
structures on $S$. We still need to define a topology on $Y$. We will
construct a bijection

$\varphi:Y\rightarrow\mathcal{ML}(S)\times\mathcal{ML}(S)$

with the property that for all $y\in Y$ the core of the product of trees
corresponding to $\varphi(y)$ is dual to the $(\overline{A_{1}^{+}\times
A_{1}^{+}},2)$-mixed structure $y$. We then give $Y$ the topology that makes
$\varphi$ a homeomorphism, thus proving the result.
Given $y\in Y$, let $\gamma_{1},\dots,\gamma_{n}$ be the simple closed curves
subdividing $S$ into its laminar and flat parts, as in Definition 6.4. Let
$S_{i}$ for $i=1,\dots,m$ denote the connected components of
$S\setminus\cup_{j}\gamma_{j}$. If $S_{i}$ is endowed with a half-translation
surface structure induced by a meromorphic quadratic differential $q_{i}$ of
finite area, then the horizontal and vertical foliations of $q_{i}$ determine
a pair of measured laminations $(\lambda^{i}_{1},\lambda^{i}_{2})$. Here we
are implicitly using the well-known homeomorphism between the space of
measured foliations arising this way and the space of measured laminations,
see for instance ([Lev83], [LM08]). On the other hand, by Remark 6.2, if
$S_{i}$ carries an $\overline{\mathfrak{a}^{+}}$-valued measured lamination,
then this is equivalent to a pair of measured laminations
$(\lambda_{1}^{i},\lambda_{2}^{i})$ possibly containing some boundary curves
$\gamma_{j}$ in their support. We can then associate to $y\in Y$ the pair of
measured laminations
$(\lambda_{1},\lambda_{2})\in\mathcal{ML}(S)\times\mathcal{ML}(S)$ defined as
$\lambda_{j}=\sum_{i}^{m}\lambda_{j}^{i}$ for $j=1,2$. Since the horizontal
and vertical measured foliations uniquely determine a meromorphic quadratic
differential of finite area ([GM91]), using Remark 6.2 and Lemma 3.3, it is
clear that $\varphi$ is a bijection.
Moreover, comparing the definition of the map $\varphi$ with Lemma 6.7, it is
easy to verify that the core of the product of trees dual to the pair
$\varphi(y)$ is dual to the $(\overline{A_{1}^{+}\times A_{1}^{+}},2)$-mixed
structure $y$ we started with. ∎

## References

  * [AC19] Daniele Alessandrini and Brian Collier.  The geometry of maximal components of the ${\rm Sp}(4,\mathbb{R})$ character variety.  Geom. Topol., 23(3):1251–1337, 2019.
  * [Ale08] Daniele Alessandrini.  Tropicalization of group representations.  Algebr. Geom. Topol., 8(1):279–307, 2008.
  * [Bes88] Mladen Bestvina.  Degenerations of the hyperbolic space.  Duke Mathematical Journal, 56(1):143–161, 1988.
  * [BIPP17] Marc Burger, Alessandra Iozzi, Anne Parreau, and Maria Beatrice Pozzetti.  A structure theorem for geodesic currents and length spectrum compactifications.  arXiv:1710.07060, 2017.
  * [BIPP19] Marc Burger, Alessandra Iozzi, Anne Parreau, and Maria Beatrice Pozzetti.  Currents, systoles, and compactifications of character varieties.  To appear in Proc. Lond. Math. Soc. arXiv:1902.07680, 2019.
  * [BIPP21a] Marc Burger, Alessandra Iozzi, Anne Parreau, and Maria Beatrice Pozzetti.  Positive crossratios, barycenters, trees and applications to maximal representations.  arXiv:2103.17161, 2021.
  * [BIPP21b] Marc Burger, Alessandra Iozzi, Anne Parreau, and Maria Beatrice Pozzetti.  The real spectrum compactification of character varieties: characterizations and applications.  C. R. Math. Acad. Sci. Paris, 359:439–463, 2021.
  * [BIPP21c] Marc Burger, Alessandra Iozzi, Anne Parreau, and Maria Beatrice Pozzetti.  Weyl chamber length compactification of $\mathrm{PSL}(2,\mathbb{R})\times\mathrm{PSL}(2,\mathbb{R})$ maximal character variety.  2021\.
  * [Bon88] Francis Bonahon.  The geometry of Teichmüller space via geodesic currents.  Invent. Math., 92(1):139–162, 1988.
  * [BP17] Marc Burger and Maria Beatrice Pozzetti.  Maximal representations, non-Archimedean Siegel spaces, and buildings.  Geom. Topol., 21(6):3539–3599, 2017.
  * [Bru88] G. W. Brumfiel.  The real spectrum compactification of Teichmüller space.  In Geometry of group representations (Boulder, CO, 1987), volume 74 of Contemp. Math., pages 51–75. Amer. Math. Soc., Providence, RI, 1988.
  * [Col16] Brian Collier.  Maximal Sp(4,R) surface group representations, minimal immersions and cyclic surfaces.  Geometriae Dedicata, 180(1):241–285, 2016.
  * [CTT19] Brian Collier, Nicolas Tholozan, and Jérémy Toulisse.  The geometry of maximal representations of surface groups into SO(2,n).  Duke Mathematical Journal, 168(15):2873–2949, 2019.
  * [DLR10] Moon Duchin, Christopher J. Leininger, and Kasra Rafi.  Length spectra and degeneration of flat metrics.  Invent. Math., 182(2):231–277, 2010.
  * [ES64] J. Eells and J.H. Sampson.  Harmonic mappings of Riemannian manifolds.  Amer. J. Math, 86:109–159, 1964.
  * [FG06] Vladimir Fock and Alexander Goncharov.  Moduli spaces of local systems and higher Teichmüller theory.  Publ. Math. Inst. Hautes Études Sci., (103):1–211, 2006.
  * [FLP21] Albert Fathi, François Laudenbach, and Valentin Poénaru.  Thurston’s Work on Surfaces (MN-48), volume 48.  Princeton University Press, 2021.
  * [FM11] Benson Farb and Dan Margalit.  A primer on mapping class groups (pms-49).  Princeton university press, 2011.
  * [GL17] Vincent Guirardel and Gilbert Levitt.  JSJ decompositions of groups.  Astérisque, (395):vii+165, 2017.
  * [GM91] Frederick P. Gardiner and Howard Masur.  Extremal length geometry of Teichmüller space.  Complex Variables, Theory and Application: An International Journal, 16(2-3):209–237, 1991.
  * [Gol88] William M. Goldman.  Topological components of spaces of representations.  Invent. Math., 93(3):557–607, 1988.
  * [Gui05] Vincent Guirardel.  Coeur et nombre d’intersection pour les actions de groupes sur les arbres.  Annales Scientifiques de l’École Normale Supérieure, 38(6):847–888, 2005.
  * [Har67] Philip Hartman.  On homotopic harmonic maps.  Canadian journal of mathematics, 19:673–687, 1967.
  * [HM79] John Hubbard and Howard Masur.  Quadratic differentials and foliations.  Acta Mathematica, 142(1):221–274, 1979.
  * [KS97] Nicholas J. Korevaar and Richard M. Schoen.  Global existence theorems for harmonic maps to non-locally compact spaces.  Comm. Anal. Geom., 5(2):333–387, 1997.
  * [Lab17] François Labourie.  Cyclic surfaces and Hitchin components in rank 2.  Ann. of Math. (2), 185(1):1–58, 2017.
  * [Le16] Ian Le.  Higher laminations and affine buildings.  Geom. Topol., 20(3):1673–1735, 2016.
  * [Lev83] Gilbert Levitt.  Foliations and laminations on hyperbolic surfaces.  Topology, 22:119–135, 1983.
  * [LM08] Elon Lindenstrauss and Mirzakhani Maryam.  Ergodic theory of the space of measured laminations.  Int. Math. Res. Not., 2008(9):1–48, 2008.
  * [Lof01] John C. Loftin.  Affine spheres and convex $\mathbb{RP}^{n}$-manifolds.  Amer. J. Math., 123(2):255–274, 2001.
  * [LTW21] John Loftin, Andrea Tamburelli, and Michael Wolf.  Limits of cubic differentials and buildings.  In preparation, 2021.
  * [Mar19a] Giuseppe Martone.  Positive configurations of flags in a building and limits of positive representations.  Math. Z., 293(3-4):1337–1368, 2019.
  * [Mar19b] Giuseppe Martone.  Sequences of Hitchin representations of tree-type.  arXiv:1910.13585, 2019.
  * [Mes07] Geoffrey Mess.  Lorentz spacetimes of constant curvature.  Geom. Dedicata, 126:3–45, 2007.
  * [MO93] J. Morgan and J. Otal.  Relative growth rates of closed geodesics on a surface under varying hyperbolic structures.  Commentarii Mathematici Helvetici, 68:171–208, 1993.
  * [MS84] John W Morgan and Peter B Shalen.  Valuations, trees, and degenerations of hyperbolic structures, I.  Annals of Mathematics, 120(3):401–476, 1984.
  * [OT20] Charles Ouyang and Andrea Tamburelli.  Length spectrum compactification of the SO(2,3) Hitchin component.  arXiv:2010.03499, 2020.
  * [OT21] Charles Ouyang and Andrea Tamburelli.  Limits of Blaschke metrics.  Duke Math. J., 170(8):1683–1722, 2021.
  * [Ota90] Jean-Pierre Otal.  Le spectre marqué des longueurs des surfaces à courbure négative.  Ann. of Math. (2), 131(1):151–162, 1990.
  * [Ouy19] Charles Ouyang.  High energy harmonic maps and degeneration of minimal surfaces.  arXiv:1910.06999, 2019.
  * [Par12] Anne Parreau.  Compactification d’espaces de représentations de groupes de type fini.  Math. Z., 272(1-2):51–86, 2012.
  * [Par15] Anne Parreau.  Invariant subspaces for some surface groups acting on A2-Euclidean buildings.  arXiv:1504.03775 to appear in TAMS, 2015.
  * [Pau88] Frédéric Paulin.  Topologie de Gromov équivariante, structures hyperboliques et arbres réels.  Invent. Math., 94(1):53–80, 1988.
  * [Sch93] Richard M. Schoen.  The role of harmonic mappings in rigidity and deformation problems.  In Complex geometry (Osaka, 1990), volume 143 of Lecture Notes in Pure and Appl. Math., pages 179–200. Dekker, New York, 1993.
  * [Sko96] Richard K Skora.  Splittings of surfaces.  Journal of the American Mathematical Society, pages 605–616, 1996\.
  * [Wol89] Michael Wolf.  The Teichmüller theory of harmonic maps.  J. Differential Geom., 29(2):449–479, 1989.
  * [Wol95] Michael Wolf.  Harmonic maps from surfaces to R-trees.  Mathematische Zeitschrift, 218(1):577–593, 1995.

Department of Mathematics, University of Michigan, Ann Arbor
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Department of Mathematics and Statistics, University of Massachusetts, Amherst
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Department of Mathematics, Rice University
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