# Loxodromic elements in big mapping class groups via the Hooper-Thurston-
Veech construction

Israel Morales and Ferrán Valdez  Ferrán Valdez
Centro de Ciencias Matemáticas, UNAM, Campus Morelia, C.P. 58190, Morelia,
Michoacán, México<EMAIL_ADDRESS>Israel Morales
Centro de Ciencias Matemáticas, UNAM, Campus Morelia, C.P. 58190, Morelia,
Michoacán, México<EMAIL_ADDRESS>
###### Abstract.

Let $S$ be an infinite-type surface and $p\in S$. We show that the Thurston-
Veech construction for pseudo-Anosov elements, adapted for infinite-type
surfaces, produces infinitely many loxodromic elements for the action of
$\mathrm{Mod}(S;p)$ on the loop graph $L(S;p)$ that do not leave any finite-
type subsurface $S^{\prime}\subset S$ invariant. Moreover, in the language of
[BaWa18B], Thurston-Veech’s construction produces loxodromic elements of any
weight. As a consequence of Bavard and Walker’s work, any subgroup of
$\mathrm{Mod}(S;p)$ containing two "Thurston-Veech loxodromics" of different
weight has an infinite-dimensional space of non-trivial quasimorphisms.

##  1\. Introduction

Let $S$ be an orientable infinite-type surface, $p\in S$ a marked
point111Through this text we think of $p$ as either a marked point or a marked
puncture. and $\mathrm{Mod}(S;p)$ the quotient of ${\rm Homeo}^{+}(S;p)$ by
isotopies which fix $p$ for all times. This group is related to the (big)
mapping class group222$\mathrm{Mod}(S)$ denotes ${\rm Homeo}^{+}(S;p)$ modulo
isotopy. of $S$ via Birman’s exact sequence:

$1\longrightarrow\pi_{1}(S,p)\overset{Push}{\longrightarrow}\mathrm{Mod}(S;p)\overset{Forget}{\longrightarrow}\mathrm{Mod}(S)\longrightarrow
1.$

The group $\mathrm{Mod}(S;p)$ acts by isometries on the (Gromov-hyperbolic,
infinite-diameter) loop graph $L(S;p)$, see [BaWa18A] and [BaWa18B]. Up to
date the only known examples of loxodromic elements for this action are:

  1. (1)

The loxodromic element $h\in S=\mathbb{S}^{2}\setminus\\{p=\infty\cup\rm
Cantor\\}$ defined by J. Bavard in [Bavard16]. In this case
$\mathrm{Mod}(S;p)=\mathrm{Mod}(S)$, $h$ is not in the pure mapping class
group $\mathrm{PMod}(S)$, does not preserve any finite type subsurface and, in
the language of [BaWa18B], has weight 1.

  2. (2)

Elements defined by pseudo-Anosov homeomorphisms supported in a finite-type
subsurface $S^{\prime}\subset S$ containing $p$. All these live in
$\mathrm{PMod}(S,p)$. Moreover, for any given $m\in\mathbb{N}$, $S^{\prime}$
can be chosen so that the loxodromic in question has weight $m$.

In [BaWa18B], the authors remark that _it would be interesting to construct
examples of loxodromic elements of weight greater than 1 which do not preserve
any finite type subsurface (up to isotopy)_.

The purpose of this article is to show that such examples can be obtained by
adapting the Thurston-Veech construction for pseudo-Anosov elements (see
[Thurston88], [Veech89] or [FarbMArgalit12]) to the context of infinite-type
surfaces. This adaptation is an extension of Thurston and Veech’s ideas built
upon previous work by Hooper [Hooper15], hence we call it the Hooper-Thurston-
Veech construction. Roughly speaking, we show that if one takes as input an
appropiate pair of multicurves $\alpha$, $\beta$ whose union fills $S$, then
the subgroup of $\mathrm{Mod}(S,p)$ generated by the (right) multitwists
$T_{\alpha}$ and $T_{\beta}$ contains infinitely many loxodromic elements.
More precisely, our main result is:

######  Theorem 1.1.

Let $S$ be an orientable infinite-type surface, $p\in S$ a marked point and
$m\in\mathbb{N}$. Let $\alpha=\\{\alpha_{i}\\}_{i\in I}$ and
$\beta=\\{\beta_{j}\\}_{j\in J}$ be multicurves in minimal position whose
union fills $S$ and such that:

  1. (1)

the configuration graph333The vertices of this graph are $I\cup J$. There is
an edge between $i\in I$ and $j\in J$ for every point of intersection between
$\alpha_{i}$ and $\beta_{j}$. $\mathcal{G}(\alpha\cup\beta)$ is of finite
valence444Valence here means the supremum of $degree(v)$ where $v$ is a vertex
of the graph in question.,

  2. (2)

for some fixed $N\in\mathbb{N}$, every connected component $D$ of
$S\setminus\alpha\cup\beta$ is a polygon or a once-punctured polygon555The
boundary of any connected component of $S\setminus\alpha\cup\beta$ is formed
by arcs contained in the curves forming $\alpha\cup\beta$ and hence we can
think of them as (topological) polygons. with at most $N$ sides and

  3. (3)

the connected component of $S\setminus\alpha\cup\beta$ containing $p$ is a
$2m$-polygon.

If $T_{\alpha}$, $T_{\beta}\in\mathrm{Mod}(S;p)$ are the (right) multitwists
w.r.t. $\alpha$ and $\beta$ respectively then any $f\in\mathrm{Mod}(S;p)$ in
the positive semigroup generated by $T_{\alpha}$ and $T_{\beta}^{-1}$ given by
a word on which both generators appear is a loxodromic element of weight $m$
for the action of $\mathrm{Mod}(S;p)$ on the loop graph $L(S;p)$.

######  Remark 1.2.

During the 2019 AIM-Workshop on Surfaces of Infinite Type we learned that
Abbott, Miller and Patel have also a construction of loxodromic mapping
classes whose support is an infinite type surface [AbMiPa]. Their examples are
obtained via a composition of handle shifts and live in the complement of the
closure of the subgroup of $\mathrm{Mod}(S,p)$ defined by homeomorphisms with
compact support. In contrast, loxodromic elements given by Theorem 1.1 are
limits of compactly supported mapping classes.

As we explain in Section 2.3, the weight of a loxodromic element
$f\in\mathrm{Mod}(S;p)$ is defined by Bavard and Walker using a precise
description of the (Gromov) boundary $\partial L(S;p)$ of the loop graph. For
finite-type surfaces, if $f$ is a pseudo-Anosov having a singularity at $p$,
this number corresponds to the number of separatrices based at $p$ of an
$f$-invariant transverse measured foliation. This quantity is relevant
because, as shown in [BaWa18B] and using the language of Bestvina and Fujiwara
[BestvinaFujiwara02], loxodromic elements with different weights are
independent and anti-aligned. This has several consequences, for example the
work of Bavard-Walker [BaWa18A] gives for free the following:

######  Corollary 1.3.

Let $f,g$ be two loxodromic elements in $\mathrm{Mod}(S;p)$ as in Theorem 1.1
and suppose that their weights are $m\neq m^{\prime}$. Then any subgroup of
$\mathrm{Mod}(S;p)$ containing them has an infinite-dimensional space of non-
trivial quasimorphisms.

Applications for random-walks on the loop graph with respect to probability
distributions supported on countable subgroups of $\mathrm{Mod}(S;p)$ can be
easily deduced from recent work of Maher-Tiozzo [MaherTiozzo18].

On the other hand, recent work by Rasmussen [Rass19] implies that the mapping
classes given by Theorem 1.1 are not WWPD in the language of Bestvina-
Bromberg-Fujiwara [BeBroFu15].

_About the proof of Theorem 1.1_. As in the case of Thurston’s work, our proof
relies on the existence of a flat surface structure $M$ on $S$, having a
conical singularity at $p$, for which the Dehn-twists $T_{\alpha}$ and
$T_{\beta}$ are affine automorphisms. In the case of finite-type surfaces,
this structure is unique (up to scaling) and its existence is guaranteed by
the Perron-Frobenius theorem. For infinite-type surfaces the presence of such
a flat surface structure is guaranteed once one can find a positive
eigenfunction of the adjacency operator on the (infinite bipartite)
configuration graph $\mathcal{G}(\alpha\cup\beta)$. Luckly, the spectral
theory of infinite graphs in this context secures the existence of uncountably
many flat surface structures (which are not related by scaling) on which the
Dehn-twists $T_{\alpha}$ and $T_{\beta}$ are affine automorphisms. The main
difficulty we encounter is that the description of the Gromov boundary
$\partial L(S;p)$ needed to certify that $f$ is a loxodromic element depends
on a hyperbolic structure on $S$ which, a priori, is not quasi-isometric to
any of the aforementioned flat surface structures. To overcome this difficulty
we propose arguments which are mostly of topological nature.

We strongly believe that Theorem 1.1 does not describe all possible
loxodromics living in the group generated by $T_{\alpha}$ and $T_{\beta}$.

######  Question 1.4.

Let $\alpha$ and $\beta$ be as in Theorem 1.1. Is every element $f$ in the
group generated by $T_{\alpha}$ and $T_{\beta}$ given by a word on which both
generators appear loxodromic? In particular, is $T_{\alpha}T_{\beta}$
loxodromic?

We spend a considerable part of this text in the proof of the next result,
which guarantees that Theorem 1.1 is not vacuous.

######  Theorem 1.5.

Let $S$ be an infinite-type surface, $p\in S$ a marked point and
$m\in\mathbb{N}$. Then there exist two multicurves $\alpha$ and $\beta$ whose
union fills $S$ and such that:

  1. (1)

the configuration graph $\mathcal{G}(\alpha\cup\beta)$ is of finite valence,

  2. (2)

every connected component $D_{i}$ of $S\setminus\alpha\cup\beta$ which does
not contain the pont $p$ is a polygon or a once-punctured polygon with $n_{i}$
sides, where $n_{i}\leq N:=\max\\{8,m\\}$, and

  3. (3)

$p$ is contained in a connected component of $D_{j}$ of
$S\setminus\alpha\cup\beta$ whose boundary $\partial D_{j}$ is a $2m$-polygon.

A crucial part on the proof of this result is to find, for _any_ infinite-type
surface $S$, a simple way to portrait $S$. We call this a topological normal
form. Once this is achieved, we give a recipe to construct the curves $\alpha$
and $\beta$ explicitly.

On the other hand, we find phenomena proper to big mapping class groups:

######  Corollary 1.6.

Let $S$ be the Loch-Ness monster666$S$ has infinite genus and only one end.
and consider the action of $\mathrm{Mod}(S;p)$ on the loop graph. Then there
exist a sequence of loxodromic elements $(f_{n})$ in $\mathrm{Mod}(S;p)$ which
converge in the compact-open topology to a non-trivial elliptic element.

######  Theorem 1.7.

There exits a family of translation surface structures
$\\{M_{\lambda}\\}_{\lambda\in[2,+\infty]}$ on a Loch Ness monster $S$ and
$f\in\mathrm{Mod}(S)$ such that:

  * •

For every $k\in\mathbb{N}$, $f^{k}$ does not fix any isotopy class of
essential simple closed curve in $S$.

  * •

If $\tau_{\lambda}:\mathrm{Aff}(M_{\lambda})\hookrightarrow\mathrm{Mod}(S)$ is
the natural map sending an affine homeomorphism to is mapping class, then
$D\tau_{\lambda}^{-1}(f)\in\mathrm{PSL}(2,\mathbb{R})$ is parabolic if
$\lambda=2$ and hyperbolic for every $\lambda>2$.

Recall that for finite-type surfaces $S$ a class $f\in\mathrm{Mod}(S)$ such
that for every $k\in\mathbb{N}$, $f^{k}$ does not fix any isotopy class of
essential simple closed curve in $S$ is necessarily pseudo-Anosov. In
particular, the derivative of any affine representative $\phi\in f$ is
hyperbolic.

We want to stress that for many infinite-type surfaces $\mathrm{Mod}(S)$ does
not admit an action on a metric space with unbounded orbits. For a more
detailed discussion on this fact and the large scale geometry of big mapping
class groups we refer the reader to [DurhamFanoniVlamis18], [MahnRafi20] and
references therein.

_Outline_. Section 2 is devoted to preliminaries about the loop graph, its
boundary and infinite-type flat surfaces. In Section 3 we present the Hooper-
Thurston-Veech construction777More precisely, the construction here presented
is a particular case of a more general construction built upon work of P.
Hooper by the second author and V. Delecroix. The first version of this more
general construction had mistakes that were pointed out by the first author..
In Section 3 we also proof Theorem 1.7. Finally, Section 4 is devoted to the
proof of Theorems 1.1, 1.5 and Corollary 1.6 (in this order).

_Acknowledgements_. We are greatful to Vincent Delecroix, Alexander Rasmussen
and Patrick Hooper for providing crucial remarks that lead to the proof of
Theorem 1.1. We also want to thank Vincent Delecroix for letting us include a
particular case of the Hooper-Thurston-Veech’s construction and Figures 2, 3
and 4, which are work in collaboration with the second author (see [DHV19]).
We are greatful to Juliette Bavard and Alden Walker for taking the time to
explain the details of their work, and to Chris Leininger for answering all
our questions. We want to thank the following institutions and grants for
their hospitality and financial support: Max-Planck Institut für Mathematik,
American Institut of Mathematics, PAPIIT IN102018, UNAM, PASDA de la DGAPA,
CONACYT and Fondo CONACYT-Ciencia Básica’s project 283960.

##  2\. Preliminaries

###  2.1. Infinite-type surfaces

Any orientable (topological) surface $S$ with empty boundary is determiner up
to homeomorphism by its genus (possibly infinite) and a pair of nested,
totally disconnected, separable, metrizable topological spaces
$\mathrm{Ends}_{\infty}(S)\subset\mathrm{Ends}(S)$ called the space of ends
accumulated by genus and the space of ends of $S$, respectively. Any such pair
of nested topological spaces occurs as the space of ends of some orientable
surface, see [Richards63]. On the other hand, $S\cup\mathrm{Ends}(S)$ can be
endowed with a natural topology which makes the correspoding space compact.
This space is called the Freudenthal end-point compactification of $S$, see
[Raymond60].

A surface $S$ is of finite (topological) type if its fundamental group is
finitely generated. In any other case we say that $S$ is an _infinite-type_
surface. All surfaces considered henceforth are orientable.

###  2.2. Multicurves

Let $S$ be an infinite-type surface. A collection of essential curves $l$ in
$S$ is locally finite if for every $x\in S$ there exist a neighborhood $U$ of
$x$ which intersects a finitely many elements of $l$. As surfaces are second-
countable topological spaces, any locally finite collection of essential
curves is countable.

A _multicurve_ in $S$ is a locally finite, pairwise disjoint, and pairwise
non-isotopic collection of essential curves in $S$.

Let $\alpha$ be a multicurve in $S$. We say that $\alpha$ _bounds a
subsurface_ $\Sigma$ of $S$, if the elements of $\alpha$ are exactly all the
boundary curves of the closure of $\Sigma$ in $S$. Also, we say that $\Sigma$
is _induced_ by $\alpha$ if there exists a subset
$\alpha^{\prime}\subset\alpha$ that bounds $\Sigma$ and there are no elements
of $\alpha\smallsetminus\alpha^{\prime}$ in its interior.

A multicurve $\alpha$ in $S$ is of _finite type_ if every connected component
of $S\smallsetminus\alpha$ is a finite-type surface.

Finite multicurves (that is, with formed by a finite number of curves) are not
necessarily of finite type. On the other hand, there are infinite multicurves
which are not of finite type, _e.g._ the blue ("vertical") curves in the
right-hand side of Figure 1.

Let $\alpha$ and $\beta$ be two multicurves in $S$. We say that $\alpha$ and
$\beta$ are in _minimal position_ if for every $\alpha_{i}\in\alpha$ and
$\beta_{j}\in\beta$, $|\alpha_{i}\cap\beta_{j}|$ realizes the minimal number
of (geometric) intersection points between a representative in the free
isotopy class of $\alpha_{i}$ and a representative in the homotopy class of
$\beta_{j}$. For every pair of multicurves one can find representatives in
their isotopy classes which are in minimal position.

Let $\alpha$ and $\beta$ be two multicurves in $S$ in minimal position. We say
that $\alpha$ and $\beta$ _fill_ $S$ if every connected component of
$S\smallsetminus\left(\alpha\cup\beta\right)$ is either a disk or a once-
punctured disk.

######  Remark 2.1.

Let $\alpha$ and $\beta$ be multicurves. Then:

  1. (1)

If $\alpha$ and $\beta$ are of finite type and fill $S$, then every
complementary connected component of $\alpha\cup\beta$ in $S$ is a polygon
with finitely many sides. The converse is not true, see the left-hand side of
Figure 1.

  2. (2)

There are pair of multicurves $\alpha$ and $\beta$ so that
$S\smallsetminus\left(\alpha\cup\beta\right)$ has a connected component that
is a polygon with infinitely many sides, see the right-hand side of Figure 1.

Figure 1.

###  2.3. The loop graph and its boundary

In [BaWa18A] and [BaWa18B], Bavard and Walker introduced the loop graph
$L(S;p)$ and prove that it is hyperbolic graph on which $\mathrm{Mod}(S,p)$
acts by isometries. Moreover, they made a precise description of the Gromov
boundary of $L(S;p)$ in terms of (hyperbolic) geodesics on the Poincaré disk.
We recall the general aspects of their work in what follows. The exposition
follows largely [BaWa18A], [BaWa18B] and [Rass19].

_The loop graph_. Let $S$ be an infinite type surface and $p\in S$. In what
follows we think of $p$ as a marked puncture in $S$. The isotopy class of a
topological embedding of $\gamma:(0,1)\hookrightarrow S$ is said to be a
_loop_ if it can be continuously extended to the end-point Freudenthal
compactification $S\cup\mathrm{Ends}(S)$ of $S$ with $\gamma(0)=\gamma(1)=p$.
On the other hand, if the continuous extension of $\gamma$ satifies that
$\gamma(0)=p$ and $\gamma(1)\in\mathrm{Ends}(P)\setminus\\{p\\}$ we call it a
short ray. The loop graph has as vertex set isotopy classes (relative to $p$)
of loops and adjacency is defined by disjointness (modulo isotopy) and it is
hyperbolic w.r.t to the combinatorial distance, see [BaWa18B]. In order to
describe the Gromov boundary of $L(S;p)$ we need to introduce the completed
ray graph.

_Long rays and the completed ray graph_. From now on we fix a hyperbolic
metric $\mu$ on $S$ of the first kind888That is, the Fuchsian group appearing
in the uniformization $\mathbb{D}\to S$ has as limit set the whole boundary of
the Poincaré disk. for which the marked point $p$ is a cusp. Every short ray
or loop has a unique geodesic representative in its isotopy class. We denote
by $\pi:\hat{S}\to S$ the infinite cyclic cover of $S$ defined by the
(conjugacy class of) cyclic subgroup of $\pi_{1}(S,\cdot)$ generated by a
simple loop around the cusp $p$ and call it _the conical cover of $S$_. The
surface $\hat{S}$ is conformally equivalent to a punctured disc and its unique
cusp $\hat{p}$ projects to $p$. We denote by $\partial\hat{S}$ the Gromov
boundary $\partial\hat{S}$ from which $\hat{p}$ has been removed. This cover
is usefull because for every geodesic representative of a short ray or loop in
$S$ there is a unique lift to $\hat{S}$ which is a geodesic with one end in
$\hat{p}$ and the other in $\partial\hat{S}$.

A long ray on S is a _simple_ bi-infinite geodesic of the form
$\pi(\hat{\delta})$, where $\hat{\delta}\subset\hat{S}$ is a geodesic from
$\hat{p}$ to $\hat{S}$, which is not a short ray or a loop. By definition,
each long ray limits to p at one end and does not limit to any point of
$\mathrm{Ends}(S)$ on the other end. The vertices of the completed ray graph
$\mathcal{R}(S;p)$ are isotopy classes of loops and short rays, and long rays.
Two vertices are adjacent if their geodesic representatives in $(S,\mu)$ are
disjoint. As before, we consider the combinatorial metric on
$\mathcal{R}(S;p)$ defined by declaring that each edge has length 1.

######  Theorem 2.2.

[BaWa18B] The completed ray graph $\mathcal{R}(S;p)$ is disconnected. There
exist a component containg all loops and short rays, which is of infinite
diameter and quasi-isometric to the loop graph. All other connected components
are (possibly infinite) cliques and each of them is formed exclusively by long
rays.

The component of $\mathcal{R}(S;p)$ containg all loops and short rays is
called the _main component_ of the completed ray graph. Long rays not
contained in the main component are called _high-filling_ rays and they each
one of they form cliques formed exclusively of high-filling rays.

_The Gromov boundary of the loop graph_. Let us denote by $\mathcal{H}(S,p)$
the set of all high-filling rays in $\mathcal{R}(S;p)$. Bavard and Walker
endow $\mathcal{H}(S,p)$ with a topology. This topology is based on the notion
of two rays $k$-beggining like each other, see Section 4.1 and Definition
5.2.4 in [BaWa18B]. On the other hand, they define a
$\mathrm{Mod}(S;p)$-action on $\mathcal{H}(S;p)$ by homeomorphisms. We sketch
this action in what follows. First they show that endpoints of lifts of loops
and short rays to the conical cover $\hat{S}$ are dense in $\partial\hat{S}$.
Using this, and the fact that mapping classes in $\mathrm{Mod}(S;p)$ permute
loops and short rays, they prove that any $\phi\in\mathrm{Mod}(S;p)$ lifts to
a homeomorphism of $\hat{S}$ which admits a unique continuous extension to a
homeomorphism of $\partial\hat{S}$. Finally, they show that this extension
preserves the subset of $\partial\hat{S}$ formed by endpoints of high-filling
rays, hence inducing the aforementioned action by homeomorphisms.

######  Theorem 2.3.

Let $\mathcal{E}(S;p)=\mathcal{H}(S;p)/\sim$, where $\sim$ identifies all
high-filling rays in the same clique, and endow this set with the quotient
topology. Then there exists a $\mathrm{Mod}(S;p)$-equivariant homeomorphism
$F:\mathcal{E}(S;p)\to\partial L(S;p)$, where $\partial L(S;p)$ is the Gromov
boundary of the loop graph.

In consequence any loxodromic element $\phi\in\mathrm{Mod}(S;p)$ fixes two
cliques of high-filling rays $\mathcal{C}^{-}(\phi)$ and
$\mathcal{C}^{+}(\phi)$.

######  Theorem 2.4 ([BaWa18B], Theorem 7.1.1).

The cliques $\mathcal{C}^{-}(\phi)$ and $\mathcal{C}^{+}(\phi)$ are finite and
of the same cardinality.

This allows to define the _weight_ of a loxodromic element $\phi$ as the
cardinality of either $\mathcal{C}^{-}(\phi)$ or $\mathcal{C}^{+}(\phi)$. As
said in the introduction, the importance of the weight of a loxodromic
elements is given by the following fact:

######  Lemma 2.5 ([BaWa18B], Lemma 9.2.7).

Let $g,h\in\mathrm{Mod}(S;p)$ be two loxodromic elements with different
weights. Then in the language of Bestvina-Fujiwara [BestvinaFujiwara02], $g$
and $h$ are independent and anti-aligned.

###  2.4. Flat surfaces

In this section we recall only basic concepts about flat surfaces needed for
the rest of the paper. It is important to remark that most of the flat
surfaces considered in this text are of infinite type. For a detailed
discussion on infinite-type flat surfaces we refer the reader to [DHV19].

We use $x,y$ for the standard coordinates in $\mathbb{R}^{2}$, $z=x+iy$ the
corresponding number in $\mathbb{C}$ and $(r,\theta)$ for polar coordinates
$x=r\cos(\theta)$ and $y=r\sin(\theta)$ (or $z=r\exp(i\theta)$). The Euclidean
metric $dx^{2}+dy^{2}$ can also be written as $(dr)^{2}+(rd\theta)^{2}$.

Let $S$ be an orientable surface and $g$ is a metric defined on the complement
of a discrete set $\Sigma\subset S$. A point $p\in S$ is called a _conical
singularity of angle $\pi n$_ for some $n\in\mathbb{N}$ if there exists an
open neighbourhood $U$ of $p$ such that $(U,g)$ is isometric to $(V,g_{n})$,
where $V\subset\mathbb{C}^{*}$ is a (punctured) neighborhood of the origin and
$g_{n}=(dr)^{2}+(nrd\theta)^{2}$. If $n=2$ we call $p$ a _regular point_. In
general, regular points are not considered as singularities, though as we see
in the proof of Theorem 1.1 sometimes it is convenient to think of them as
marked points.

A _flat surface structure_ on a topological surface $S$ is the maximal atlas
$\mathcal{T}=\\{\phi_{i}:U_{i}\to\mathbb{C}\\}$ where
$(U_{i})_{i\in\mathbb{N}}$ forms an open covering of $S$, each $\phi_{i}$ is a
homeomorphism from $U_{i}$ to $\phi(U_{i})$ and for each $i,j$ the transition
map $\phi_{j}\circ\phi_{i}^{-1}:\phi_{i}(U_{i}\cap U_{j})\to\phi_{j}(U_{i}\cap
U_{j})$ is a translation in $\mathbb{C}$ or a map of the form999These kind of
maps are also called _half-translations_ and for this reason flat surfaces
containing half-translation are also known as half-translation surfaces.
$z\to-z+\lambda$ for some $\lambda\in\mathbb{C}$.

######  Definition 2.6.

A _flat surface_ is a pair $M=(S,\mathcal{T})$ made of a connected topological
surface $S$ and a flat surface structure $\mathcal{T}$ on $S\setminus\Sigma$,
where:

  1. (1)

$\Sigma$ is a discrete subset of $S$ and

  2. (2)

every $z\in\Sigma$ is a conical singularity.

If the transition functions of $\mathcal{T}$ are all translations we call the
pair $(S,\mathcal{T})$ a translation surface.

######  Remark 2.7.

In the preceding definition $S$ can be of infinite topological type and
$\Sigma$ can be infinite. All points in $M\setminus\Sigma$ are regular. Every
flat surface carries a flat metric given by pulling back the Euclidean metric
in $\mathbb{C}$. We denote by $\widehat{M}$ the corresponding metric
completion and ${\rm Sing}(M)\subset\widehat{M}$ the set of non-regular
points, which can be thought as singularities of the flat metric. We stress
that the structure of $M$ near a non-regular point is not well understood in
full generality, see [BowmanValdez13] and [Randecker18].

Every flat surface $M$ which is not a translation surface has a (ramified)
double covering $\pi:\widetilde{M}\to M$ such that $\widetilde{M}$ is a
translation surface whose atlas is obtained by pulling back via $\pi$ the flat
surface structure of $M$. This is called the orientation double covering. If
$z_{0}\in M$ is a conical singularity of angle $n\pi$ then, if $n$ is even,
$\pi^{-1}(z_{0})$ is formed by two conical singularities of total angle
$n\pi$; whereas if $n$ is odd, $\pi^{-1}(z_{0})$ is a conical singularity of
total angle $2n\pi$. Hence the branching points of the orientation double
covering are the conical singularities in $M$ of angle $n\pi$, with $n$ odd.
This will be used in the proof of Theorem 1.1.

On the other hand, flat surfaces can be defined using the language of complex
geometry in terms of Riemann surfaces and quadratic differentials or by
glueing (possibly infinite) families of polygons along their edges. A detailed
discussion on these other definitions can be found in the first Chapter of
[DHV19].

Affine maps. A map $f\in Homeo(M)$ with $f(\Sigma)\subset\Sigma$ is called an
_affine automorphisms_ if the restriction $f:M\setminus\Sigma\to
M\setminus\Sigma$ to flat charts is an $\mathbb{R}$-affine map. We denote by
$\mathrm{Aff}(M)$ the group of affine homeomorphisms of $M$ and by
$\mathrm{Aff}^{+}(M)$ the subgroup of $\mathrm{Aff}(M)$ made of orientation
preserving affine automorphisms (_i.e_ their linear part has positive
determinant). Remark that the derivative $Df$ of an element
$f\in\mathrm{Aff}(M)$ is an element of ${\mathrm{GL}}_{2}(\mathbb{R})/\pm Id$.

Translation flows. For each direction $\theta\in\mathbb{R}/2\pi\mathbb{Z}$ we
have a well-defined translation flow
$F_{\mathbb{C},\theta}^{t}:\mathbb{C}\to\mathbb{C}$ given by
$F_{\mathbb{C},\theta}^{t}(z)=z+te^{i\theta}$. This flow defines a constant
vector field $X_{\mathbb{C},\theta}(z):=\frac{\partial
F_{\mathbb{C},\theta}^{t}}{\partial t}|_{t=0}(z)$. Now let $M$ be a
translation surface and $X_{M,\theta}$ the vector field on
$\widehat{M}\setminus{\rm Sing}(M)$ obtained by pulling back
$X_{\mathbb{C},\theta}$ using the charts of the structure. For every $z\in
M\setminus\mathrm{Sing}(M)$ let us denote by $\gamma_{z}:I\to M$, where
$I\subset\mathbb{R}$ is an interval containing zero, the maximal integral
curve of $X_{M,\theta}$ with initial condition $\gamma_{z}(0)=z$. We define
$F_{M,\theta}^{t}(z):=\gamma_{z}(t)$ and call it the _translation flow_ on $M$
in direction $\theta$. Let us remark that formally speaking $F^{t}_{M,\theta}$
is not a flow because the trajectory of the curve $\gamma_{z}(t)$ may reach a
point of $\mathrm{Sing}(M)$ in finite time. A trajectory of the translation
flow whose maximal domain of definition is a bounded interval (on both sides)
is called a _saddle connection_. When there is no need to distinguish
translation flows in different translation surfaces we abbreviate
$F_{M,\theta}^{t}$ and $X_{M,\theta}$ by $F_{\theta}^{t}$ and $X_{\theta}$
respectively. If $M$ is a flat surface but not a translation surface the pull
back of the vector field $e^{i\theta}$ to $M$ does not define a global vector
field on $M$ but it does define a _direction field_. In both cases integral
curves defines a foliation $\mathcal{F}_{\theta}$ on
$M\setminus\mathrm{Sing}(M)$.

######  Definition 2.8 (Cylinders and strips).

A _horizontal cylinder_ $C_{c,I}$ is a translation surface of the form
$([0,c]\times I)/\sim$, where $I\subset\mathbb{R}$ is open (but not
necessarily bounded), connected, and where $(0,s)$ is identified with $(c,s)$
for all $s\in I$. The numbers $c$ and $h=|I|$ are called the _circumference_
and _height_ of the cylinder respectively. The _modulus_ of $C_{c,I}$ is the
number $\frac{h}{c}$.

A _horizontal strip_ $C_{\infty,I}$ is a translation surface of the form
$\mathbb{R}\times I$, where $I$ is a bounded open interval. Analogously, the
height of the horizontal strip is $h=|I|$.

An open subset of a translation surface $M$ is called a _cylinder_
(respectively a _strip_) in direction $\theta$ (or parallel to $\theta$) if it
is isomorphic, as a translation surface, to $e^{-i\theta}C_{c,I}$ (respect. to
$e^{-i\theta}C_{\infty,I}$).

One can think of strips as cylinders of infinite circumference and finite
height. For flat surfaces which are not translation surfaces the definition of
cylinder still makes sense, though its direction is well defined only up to
change of sign.

######  Definition 2.9.

Let $M$ be a flat surface and $\theta\in\mathbb{R}/2\pi\mathbb{Z}$ a fixed
direction. A collection of maximal cylinders $\\{C_{i}\\}_{i\in I}$ parallel
to $\theta$ such that $\cup_{i\in I}C_{i}$ is dense in $M$ is called a
_cylinder decomposition_ in direction $\theta$.

##  3\. The Hooper-Thurston-Veech construction

The main result of this section is a generalization of the Thurston-Veech
construction for infinite-type surfaces. The key ingredient for this
generalization is the following:

######  Lemma 3.1.

Let $M$ be a flat surface for which there is a cylinder decomposition in the
horizontal direction. Suppose that every maximal cylinder in this
decomposition has modulus equal to $\frac{1}{\lambda}$ for some $\lambda>0$.
Then there exists a unique affine automorphism $\phi_{h}$ which fixes the
boundaries of the cylinders and whose derivative (in
$\mathrm{PSL}(2,\mathbb{R})$) is given by the matrix
$\left(\begin{smallmatrix}1&\lambda\\\ 0&1\end{smallmatrix}\right)$. Moreover,
the automorphism $\phi_{h}$ acts as a Dehn twist along the core curve of each
cylinder.

In general, if $M$ is a flat surface having a cylinder decomposition in
direction $\theta\in\mathbb{R}/2\pi\mathbb{Z}$ for which every cylinder has
modulus equal to $\frac{1}{\lambda}$ for some $\lambda\in\mathbb{R}^{*}$, one
can apply to $M$ the rotation $R_{\theta}\in\mathrm{SO}(2,\mathbb{R})$ that
takes $\theta$ to the horizontal direction and apply the preceding lemma. For
example, if $\theta=\frac{\pi}{2}$, then there exists a unique affine
automorphism $\varphi_{v}$ which fixes the boundaries of the vertical
cylinders, acting on each cylinder as a Dehn twists and with derivative (in
$\mathrm{PSL}(2,\mathbb{R})$) is given by the matrix
$\left(\begin{smallmatrix}1&0\\\ -\lambda&1\end{smallmatrix}\right)$. In
particular, if $M$ is a flat surface having cylinder decompositions in the
horizontal and vertical directions such that each cylinder involved have
modulus $\frac{1}{\lambda}$, then $\mathrm{Aff}(M)$ has two affine multitwists
$\phi_{h}$ and $\phi_{v}$ with
$D\phi_{h}=\left(\begin{smallmatrix}1&\lambda\\\ 0&1\end{smallmatrix}\right)$
and $D\phi_{v}=\left(\begin{smallmatrix}1&0\\\
-\lambda&1\end{smallmatrix}\right)$ in $\mathrm{PSL}(2,\mathbb{R})$.

Let us recall now the Thurston-Veech construction (see [Farb-Margalit],
Theorem 14.1):

######  Theorem 3.2 (Thurston-Veech construction).

Let $\alpha=\\{\alpha_{i}\\}_{i=1}^{n}$ and $\beta=\\{\beta_{j}\\}_{j=1}^{m}$
be two multicurves filling a finite type surface $S$. Then there exists
$\lambda=\lambda(a,b)\in\mathbb{R}^{*}$ and a representation $\rho:\langle
T_{\alpha},T_{\beta}\rangle\to\mathrm{PSL}(2,\mathbb{R})$ given by:

$T_{\alpha}\to\begin{pmatrix}1&\lambda\\\ 0&1\end{pmatrix},\hskip
14.22636ptT_{\beta}\to=\begin{pmatrix}1&0\\\ -\lambda&1\end{pmatrix}.$

Moreover, an element $f\in\langle T_{\alpha},T_{\beta}\rangle$ is periodic,
reducible or pseudo-Anosov according to whether $\rho(f)$ is elliptic,
parabolic or hyperbolic.

The proof of Theorem 3.2 uses Lemma 3.1. More precisely, one needs to find a
flat surface structure on $S$ which admits horizontal and vertical cylinder
decompositions $\\{H_{i}\\}_{i=1}^{n}$ and $\\{V_{j}\\}_{j=1}^{m}$ such that
each cylinders has modulus equal to $\frac{1}{\lambda}$ and for which
$\alpha_{i}$ and $\beta_{j}$ are the core curves of $H_{i}$ and $V_{j}$ for
each $i=1,\ldots,n$ and $j=1,\ldots,m$, respectively. By the Perron-Frobenius
theorem, such a flat structure always exists and is unique up to scaling.

######  Definition 3.3.

Let $\alpha=\cup_{i\in I}\alpha_{i}$ and $\beta=\cup_{j\in J}\beta_{j}$ be two
multicurves in a topological surface $S$ (in minimal position, not necessarily
filling). The _configuration graph_ of the pair $(\alpha,\beta)$ is the
bipartite graph $\mathcal{G}(\alpha\cup\beta)$ whose vertex set is $I\sqcup J$
and where there is an edge between two vertices $i\in I$ and $j\in J$ for
every intersection point between $\alpha_{i}$ and $\beta_{j}$.

Cylinder decompositions, bipartite graphs and harmonic functions. Let $M$ be a
flat surface having horizontal and vertical cylinder decompositions
$\mathcal{H}=\\{H_{i}\\}_{i\in I}$ and $\mathcal{V}=\\{V_{j}\\}_{j\in J}$ such
that each cylinder has modulus $\frac{1}{\lambda}$ for some $\lambda>0$. For
every $i\in I$ let $\alpha_{i}$ be the core curve of $H_{i}$ and for every
$j\in J$ let $\beta_{j}$ be the core curve of $V_{j}$. Then
$\alpha=\\{\alpha_{i}\\}_{i\in I}$ and $\beta=\\{\beta_{j}\\}_{j\in J}$ are
multicurves whose union fills $M$. Let $\textbf{h}:I\cup J\to\mathbb{R}_{>0}$
be the function which to an index associates the height of the corresponding
cylinder. Then $A\textbf{h}=\lambda\textbf{h}$ where $A$ is the adjacency
operator of the graph $\mathcal{G}(\alpha\cup\beta)$, that is:

(1) $(A\textbf{h})(v):=\sum_{w\sim v}\textbf{h}(w)$

where the sum above is taken over over edges $\\{v,w\\}$ having $v$ as one
endpoint, that is, the summand $\textbf{h}(w)$ appears as many times as there
are edges between the vertices $v$ and $w$.

######  Definition 3.4.

Let $\mathcal{G}=(V,E)$ be a graph with vertices of finite degree and
$A:V^{\mathbb{R}}\to\mathbb{R}$ as in (1). A function
$\textbf{h}:V\to\mathbb{R}$ that satisfies $A\textbf{h}=\lambda\textbf{h}$ is
called a _$\lambda$ -harmonic function_ of $\mathcal{G}$.

In summary: the existence of a horizontal and a vertical cylinder
decomposition where each cylinders has modulus $\frac{1}{\lambda}$ implies the
existence of a _positive_ $\lambda$-harmonic function of configuration graph
of the multicurves given by the core curves of the cylinders in the
decomposition.

The idea to generalize Thurston-Veech’s construction for infinite-type
surfaces is to reverse this process: given a pair of multicurves $\alpha$ and
$\beta$ whose union fills $S$, every positive $\lambda$-harmonic function of
$\mathcal{G}(\alpha\cup\beta)$ can be used to construct construct horizontal
and vertical cylinder decompositions of $S$ where all cylinders have the same
modulus.

######  Theorem 3.5 (Hooper-Thurson-Veech construction).

Let $S$ be an infinite-type surface. Suppose that there exist two multicurves
$\alpha=\\{\alpha_{i}\\}_{i\in I}$ and $\beta=\\{\beta_{j}\\}_{j\in J}$
filling $S$ such that:

  1. (1)

there is an uniform upper bound on the degree of the vertices of the
configuration graph $\mathcal{G}(\alpha\cup\beta)$ and

  2. (2)

every component of the complement of $S\setminus\alpha\cup\beta$ is a polygon
with a finite number of sides101010That is, each component of
$S\setminus\alpha\cup\beta$ is a disc whose boundary consists of finitely man
subarcs of curves in $\alpha\cup\beta$..

Then there exists $\lambda_{0}\geq 2$ such that for every
$\lambda\geq\lambda_{0}$ there exists a positive $\lambda$-harmonic function h
on $\mathcal{G}(\alpha\cup\beta)$ which defines a flat surface structure
$M=M(\alpha,\beta,\textbf{h})$ on $S$ admiting horizontal and vertical
cylinder decompositions $\mathcal{H}=\\{H_{i}\\}_{i\in I}$ and
$\mathcal{V}=\\{V_{j}\\}_{j\in J}$ where each cylinder has modulus
$\frac{1}{\lambda}$. Moreover, for every $i\in I$ and $j\in J$ the core curves
of $H_{i}$ and $V_{j}$ are $\alpha_{i}$ and $\beta_{j}$, respectively. In
particular, we have (right) multitwists $T_{\alpha}$ and $T_{\beta}$ in
$\mathrm{Aff}(M)$ which fix the boundary of each cylinder in $\mathcal{H}$ and
$\mathcal{V}$, respectively. For each $\lambda\geq\lambda_{0}$, the
derivatives of these multitwists define a representation $\rho:\langle
T_{\alpha},T_{\beta}\rangle\to\mathrm{PSL}(2,\mathbb{R})$ given by:

$T_{\alpha}\to\begin{pmatrix}1&\lambda\\\ 0&1\end{pmatrix},\hskip
14.22636ptT_{\beta}\to\begin{pmatrix}1&0\\\ -\lambda&1\end{pmatrix}.$

######  Remark 3.6.

Theorem 3.5 is a particular case of a more general version of Hooper-Thurston-
Veech’s construction due to V. Delecroix and the second author whose final
form was achieved after discussions with the first author, see [DHV19]. We do
not need this more general version for the proof of our main results. On the
other hand, the second assumption on the multicurves in Theorem 3.5 makes the
proof simpler that in the general case and for this reason we decided to
sketch it. Many of the key ideas in the proof of the result above and its
general version appear already in the work of P. Hooper [Hooper15]. The main
difference is that P. Hooper starts with a bipartite infinite graph with
uniformly bounded valence and then, using a positive harmonic function $h$ on
that graph, constructs a translation surface. We take as input an infinite-
type topological surface $S$ and a pair of filling multicurves to construct a
flat surface structure on $S$, which is a not _a priori_ a translation surface
structure.

Proof of Theorem 3.5. The union $\alpha\cup\beta$ of the multicurves $\alpha$
and $\beta$ defines a graph embedded in $S$: the vertices are points in
$\bigcup_{(i,j)\in I\times J}\alpha_{i}\cap\beta_{j}$ and edges are the
segments forming the connected components of
$\alpha\cup\beta\setminus\bigcup_{(i,j)\in I\times J}\alpha_{i}\cap\beta_{j}$.
Abusing notation we write $\alpha\cup\beta$ to refer to this graph. It is
important not to confuse the (geometric) graph $\alpha\cup\beta$ with the
(abstract) configuration graph $\mathcal{G}(\alpha\cup\beta)$. To define the
flat structure $M$ on $S$ we consider a dual graph $(\alpha\cup\beta)^{*}$
defined as follows. If $S$ had no punctures then $(\alpha\cup\beta)^{*}$ is
just the dual graph of $\alpha\cup\beta$. If $S$ has punctures111111We think
of punctures also as isolated ends or points at infinity. we make the
following convention to define the vertices of $(\alpha\cup\beta)^{*}$: for
every connected component $D$ of $S\setminus\alpha\cup\beta$ homeomorphic to a
disc choose a unique point $v_{D}$ inside the connected component. If $D$ is a
punctured disc, then choose $v_{D}$ to be the puncture.

Figure 2. The graph $(\alpha\cup\beta)^{*}$.

The points $v_{D}$ chosen above are the vertices of $(\alpha\cup\beta)^{*}$.
Vertices in this graph are joined by an edge in $S$ if the closures of the
corresponding connected components of $S\setminus\alpha\cup\beta$ share an
edge of $\alpha\cup\beta$. Edges are chosen to be pairwise disjoint. Remark
that $(\alpha\cup\beta)^{*}$ might have loops. See Figure 2.

Given that $\alpha\cup\beta$ fills, every connected component
$S\setminus(\alpha\cup\beta)^{*}$ is a topological quadrilateral which
contains a unique vertex of $\alpha\cup\beta$. Hence there is a well defined
bijection between edges in the abstract graph $\mathcal{G}(\alpha\cup\beta)$
and the set of these quadrilaterals. This way, for every edge $e\in
E(\mathcal{G}(\alpha\cup\beta))$ we denote by $R_{e}$ the closure in $S$ of
the corresponding topological quadrilateral with the convention to add to
$R_{e}$ vertices $v_{D}$ corresponding to punctures in $S$.

Note that there are only two sides of $R_{e}$ intersecting the multicurve
$\alpha$, which henceforth are called _vertical sides_. The other two sides
are in consequence called _horizontal_ , see Figure 3.

We now build a flat surface structure on $S$ by declaring the topological
quadrilaterals $R_{e}$ of the dual graph $(\alpha\cup\beta)^{*}$ to be
Euclidean rectangles. Given that there is a uniform upper bound on the degree
of the vertices of the configuration graph $\mathcal{G}(\alpha\cup\beta)$
there exists121212For a more detailed discussion on $\lambda$-harmonic
functions we recommend Appendix C in [Hooper15] and reference therein.
$\lambda_{0}\geq 2$ such that for every $\lambda\geq\lambda_{0}$ there exists
a positive $\lambda$-harmonic function
$\textbf{h}:\mathcal{G}(\alpha\cup\beta)\to\mathbb{R}_{>0}$. We use this
function to define compatible heights of horizontal and vertical cylinders.
More precisely, let us define the maps:

$p_{\alpha}:E(\mathcal{G}(\alpha\cup\beta))\to
V(\mathcal{G}(\alpha\cup\beta))\qquad\text{and}\qquad
p_{\beta}:E(\mathcal{G}(\alpha\cup\beta))\to V(\mathcal{G}(\alpha\cup\beta))$

which to an edge $e$ of the configuration graph $\mathcal{G}(\alpha\cup\beta)$
associate its endpoints $p_{\alpha}(e)$ in $I$ and $p_{\beta}(e)$ in $J$. The
desired flat structure is defined by declaring131313For a formal description
on how to identify $R_{e}$ with $[0,\textbf{h}\circ
p_{\beta}(e)]\times[0,\textbf{h}\circ p_{\alpha}(e)]$ we refer the reader to
[DHV19]. $R_{e}$ to be the rectangle $[0,\textbf{h}\circ
p_{\beta}(e)]\times[0,\textbf{h}\circ p_{\alpha}(e)]$, see Figure 3.

Figure 3. Transforming topological rectangles into Euclidean rectangles.

We denote the resulting flat surface $M(\alpha,\beta,\textbf{h})$. Remark that
by contruction a vertex $v_{D}$ of the dual graph $(\alpha\cup\beta)^{*}$ of
valence $k$ defines a conical singularity of angle $\frac{\pi k}{2}$ in the
metric completion of $M(\alpha,\beta,\textbf{h})$. Given that $k$ is always an
even number we have that $M(\alpha,\beta,\textbf{h})$ is a half-translation
surface (_i.e._ given by a quadratic differential) when $k=2(2n-1)$ for some
$n\in\mathbb{Z}_{\geq 1}$.

Now, for every $i\in I$, the curve $\alpha_{i}$ is the core curve of the
horizontal cylinder $H_{i}:=\cup_{e\in p_{\alpha}^{-1}(i)}R_{e}$. Because h is
$\lambda$-harmonic we have

$\sum_{e\in p_{\alpha}^{-1}(i)}\textbf{h}(p_{\beta}(e))=\sum_{j\sim
i}\textbf{h}(j)=\lambda\textbf{h}(i).$

This equations say that the circumference $\sum_{e\in
p_{\alpha}^{-1}(i)}\textbf{h}(p_{\beta}(e))$ of $H_{i}$ is $\lambda$ times its
height $\textbf{h}(i)$, hence the modulus of $H_{i}$ is equal to
$\frac{1}{\lambda}$. The same computation with $\beta_{j}$ shows that the
vertical cylinders $V_{j}:=\cup_{e\in p_{\beta}^{-1}(j)}R_{e}$ have core curve
$\beta_{j}$ and modulus $\frac{1}{\lambda}$.∎

######  Remark 3.7.

As said before, Hooper-Thurston-Veech construction can be applied to more
general pairs of multicurves $\alpha,\beta$. Consider for example the case in
the Loch Ness monster depicted in Figure 4: here the graph
$\mathcal{G}(\alpha\cup\beta)$ has finite valence but there exist four
connected componets $\\{C_{i}\\}_{i=1}^{4}$ of $S\setminus\alpha\cup\beta$
which are infinite polygons, that is, whose boundary is formed by infinitely
many segments belonging to curves in $\alpha$ and in $\beta$. In this
situation the convention is to consider vertices in the dual graph
$(\alpha\cup\beta)^{*}$ of infinite degree as points at infinity (that is, not
in $S$). With this convention the Hooper-Thuston-Veech construction produces a
translation surface structure on $S$, because each $\partial C_{i}$ is
connected. In Figure 4 we illustrate the case of a $2$-harmonic function; the
resulting flat surface is a translation surface known as the infinite
staircase.

(a) Two oriented multicurves $\alpha$ (in blue) and $\beta$ (in red) in the
Loch Ness monster for which the Hooper-Thurston-Veech construction produces
the infinite staircase.

(b) The graph $\mathcal{G}(\alpha\cup\beta)$

Figure 4. The infinite staircase as a Hooper-Thurson-Veech surface.

_Proof of Theorem 1.7_. Let $\lambda\geq 2$ and consider the subgroup of
$\mathrm{SL}(2,\mathbb{R})$:

(2) $G_{\lambda}:=\langle\left(\begin{smallmatrix}1&\lambda\\\
0&1\end{smallmatrix}\right),\left(\begin{smallmatrix}1&0\\\
-\lambda&1\end{smallmatrix}\right)\rangle$

This group is free and its elements are matrices of the form
$\left(\begin{smallmatrix}1+k_{11}\lambda^{2}&k_{12}\lambda\\\
k_{21}\lambda&1+k_{22}\lambda^{2}\end{smallmatrix}\right)$,
$k_{ij}\in\mathbb{Z}$, such that the determinant is 1 and
$|\frac{1+k_{11}\lambda^{2}}{k_{12}\lambda}|$ does not belong to the interval
$(t^{-1},t)$, where $t=\frac{1}{2}(\lambda+\sqrt{\lambda^{2}-4})$, see
[Brenner55]. On the other hand, since

$\\{-\frac{\lambda}{2}<\Im(z)\leq\frac{\lambda}{2}\\}\cap\\{|z+\frac{1}{2\lambda}|>\frac{1}{2\lambda}\\}\cap\\{|z-\frac{1}{2\lambda}|\geq\frac{1}{2\lambda}\\}\subset\mathbb{H}^{2}$

is a fundamental domain in the hyperbolic plane for $G_{\lambda}$, this group
has no elliptic elements. Morevoer, if $\lambda>2$ there are only two
conjugacy classes of parabolics (correspoding to the generators of
$G_{\lambda}$) and if $\lambda=2$ then $\left(\begin{smallmatrix}1&\lambda\\\
0&1\end{smallmatrix}\right)\left(\begin{smallmatrix}1&0\\\
-\lambda&1\end{smallmatrix}\right)=\left(\begin{smallmatrix}1-\lambda^{2}&\lambda\\\
-\lambda&1\end{smallmatrix}\right)$ and
$\left(\begin{smallmatrix}1&-\lambda\\\
0&1\end{smallmatrix}\right)\left(\begin{smallmatrix}1&0\\\
\lambda&1\end{smallmatrix}\right)=\left(\begin{smallmatrix}1-\lambda^{2}&-\lambda\\\
\lambda&1\end{smallmatrix}\right)$ determine, together with the generators of
$G_{\lambda}$, the only 4 conjugacy classes of parabolics in $G_{\lambda}$.
Remark that $\left(\begin{smallmatrix}1-\lambda^{2}&\lambda\\\
-\lambda&1\end{smallmatrix}\right)$ and
$\left(\begin{smallmatrix}1-\lambda^{2}&-\lambda\\\
\lambda&1\end{smallmatrix}\right)$ are hyperbolic if $\lambda>2$.

If $\alpha$ and $\beta$ are the multicurves depicted in Figure 4 (A) on the
Loch Ness monster then $\mathcal{G}(\alpha\cup\beta)$ is the infinite
bipartite graph on Figure 4 (B). Let us index the vertices of this graph by
the integers as in the Figure so that $\textbf{h}_{2}(n)=1$, for all $\hskip
2.84526ptn\in\mathbb{Z}$, is a positive $2$-harmonic function on
$\mathcal{G}(\alpha\cup\beta)$. If $\lambda>2$ and
$r_{+}=\frac{\lambda+\sqrt{\lambda^{2}-4}}{2}$ the positive function
$\textbf{h}_{\lambda}(n)=r_{+}^{n},\hskip 2.84526ptn\in\mathbb{Z}$ is
$\lambda$-harmonic on $\mathcal{G}(\alpha\cup\beta)$. The desired family of
translation surfaces $\\{M_{\lambda}\\}_{\lambda\in[2,+\infty)}$, is obtained
by applying Hooper-Thuston-Veech’s construction to the multicurves $\alpha$,
$\beta$ and the family of positive $\lambda$-harmonic functions
$\\{\textbf{h}_{\lambda}\\}_{\lambda\in[2,+\infty)}$. The desired class
$f\in\mathrm{Mod}(S)$ is given by the product of (right) multitwist
$T_{\alpha}T_{\beta}$. No positive power of $f$ fixes an isotopy class of
simple closed curve in $S$ because on one hand if $\lambda=2$ the translation
flow on eigendirection of the parabolic matrix $D\tau_{2}^{-1}(f)$ decomposes
$M_{2}$ in two strips and, on the other if $\lambda>2$, then
$D\tau_{\lambda}^{-1}(f)$ is hyperbolic. ∎

######  Remark 3.8.

There are only 3 infinite graphs which admit $2$-harmonic functions, these are
depicted in Figure 5 together with their correspoding positive $2$-harmonic
functions (which are unique up to rescaling). None of them comes from a pair
of multicurves satifying (2) in Theorem 3.5.

Figure 5. Graphs with $2$-harmonic functions.

Renormalizable directions. The main results on Hooper’s work [Hooper15] deal
with the dynamical properties of the translation flow in _renormalizable
directions_.

######  Definition 3.9.

Consider the action of $G_{\lambda}$ as defined in (2) by homographies on the
real projective line $\mathbb{RP}^{1}$. We say that a direction
$\theta\in\mathbb{R}/2\pi\mathbb{Z}$ is $\lambda$-renormalizable if its
projectivization lies in the limit set of $G_{\lambda}$ and is not an
eigendirection of any matrix conjugated in $G_{\lambda}$ to a matrix of the
form:

$\begin{pmatrix}1&\lambda\\\ 0&1\end{pmatrix},\hskip
5.69054pt\begin{pmatrix}1&0\\\ \lambda&1\end{pmatrix}\hskip 5.69054ptor\hskip
5.69054pt\begin{pmatrix}1&0\\\
-\lambda&1\end{pmatrix}\cdot\begin{pmatrix}1&\lambda\\\ 0&1\end{pmatrix}$

We use two of Hooper’s results in the proof of Theorem 3.5. Recall that in
Hooper’s work one takes as input an infinite bipartite graph and a positive
$\lambda$-harmonic function on this graph to produce a translation surface.

######  Theorem 3.10 (Theorem 6.2, [Hooper15]).

Let $M$ be a translation surface obtained from an infinite bipartite graph as
in [_Ibid_.] using a positive $\lambda$-harmonic function and let $\theta$ be
a $\lambda$-renormalizable direction. Then the translation flow
$F_{\theta}^{t}$ on $M$ does not have saddle connections.

######  Theorem 3.11 (Theorem 6.4, [Hooper15]).

Let $M$ be a translation surface obtained from an infinite bipartite graph as
in [_Ibid_.] using a positive $\lambda$-harmonic function and let $\theta$ be
a $\lambda$-renormalizable direction. Then the translation flow
$F_{\theta}^{t}$ is conservative, that is, given $A\subset M$ of positive
measure and any $T>0$, for Lebesgue almost every $x\in M$ there is a $t>T$
such that $F_{\theta}^{t}(x)\in A$

##  4\. Proof of results

###  4.1. Proof of Theorem 1.1

The proof is divided in two parts. In the first part we use the Hooper-
Thurston-Veech construction (see Section 3) to find two transverse measured
$f$-invariant foliations $\mathcal{F}^{u}$ and $\mathcal{F}^{s}$ on $S$ for
which $p$ is a singular point and for which each foliation has $m$
separatrices based at $p$. We prove that each separatrix based at $p$ is dense
in $S$. Then, we consider a hyperbolic metric on $S$ of the first kind
(allowing us to talk about the completed ray graph $R(S;p)$). We stretch each
separatrix of $\mathcal{F}^{u}$ and $\mathcal{F}^{s}$ based at $p$ to a
geodesic with respect to this metric. This defines two sets $\Gamma^{+}$ and
$\Gamma^{-}$ of geodesics, each having cardinality $m$. In the second part of
the proof, we show that $\Gamma^{+}$ and $\Gamma^{-}$ are the only cliques of
high-filling rays fixed by $f$ in the Gromov boundary of the loop graph.

_Flat structures_. We use the Hooper-Thurston-Veech construction (Section 3)
for this part of the proof. Let $\alpha$ and $\beta$ be two multicurves
satisfying the hypothesis of Theorem 1.1. Fix
$\mathbf{h}:\mathcal{G}(\alpha\cup\beta)\to\mathbb{R}_{>0}$ a positive
$\lambda$-harmonic function on the configuration graph
$\mathcal{G}(\alpha\cup\beta)$. Let $M=M(\alpha,\beta,\mathbf{h})$ be the flat
structure on $S$ given by the Hooper-Thurston-Veech construction and

$\rho:\langle T_{\alpha},T_{\beta}\rangle\to\mathrm{PSL}(2,\mathbb{R})$

the corresponding presentation. Here, we have chosen $p$ as one of the
vertices of the dual graph $(\alpha\cup\beta)^{*}$ (see the proof of Theorem
3.5) and therefore it makes sense to consider the classes that the affine
multitwists $T_{\alpha}$, $T_{\beta}$ define in $\mathrm{Mod}(S;p)$. We abuse
notation and denote also by $T_{\alpha}$, $T_{\beta}$ these classes.

The eigenspaces of the hyperbolic matrix $\rho(f)$ define two transverse
($f$-invariant) measured foliations $(\mathcal{F}^{u},\mu_{u})$ and
$(\mathcal{F}^{s},\mu_{s})$ (for unstable and stable, respectively) on $M$ .
Moreover, we have that
$f\cdot(\mathcal{F}^{u},\mu_{u})=(\mathcal{F}^{u},\eta\mu_{u})$ and
$f\cdot(\mathcal{F}^{s},\mu_{s})=(\mathcal{F}^{s},\eta^{-1}\mu_{u})$, where
$\eta>1$ is (up to sign) an eingenvalue of $\rho(f)$. For simplicity we
abbreviate the notation for these foliations by $\mathcal{F}^{u}$ and
$\mathcal{F}^{s}$.

_The set $\mathfrak{V}$_. Recall that $M=M(\alpha,\beta,\textbf{h})$ is
constructed by glueing a family of rectangles $\\{R_{e}\\}_{e\in E}$, where
$E=E(\mathcal{G}(\alpha\cup\beta))$ is the set of edges of the configuration
graph $\mathcal{G}(\alpha\cup\beta)$, along their edges using translations and
half-translations. By the way the Hooper-Thurston-Veech construction is
carried out, sometimes the corners of these rectangles are not part of the
surface $S$: this is the case when there are connected components of
$S\setminus\alpha\cup\beta$ which are punctured discs. However, every corner
of a rectangle $\\{R_{e}\\}_{e\in E}$ belongs to the metric completion
$\widehat{M}$ of $M$ (w.r.t. the natural flat metric). We define
$\mathfrak{V}\subset\widehat{M}$ to be the set of points that are corners of
rectangles in $\\{R_{e}\\}_{e\in E}$ (after glueings). Remark that since all
connected components of $S\setminus\alpha\cup\beta$ are (topological) polygons
with an uniformly bounded number of sides, points in $\mathfrak{V}$ are
regular points or conical singularities of $\widehat{M}$ whose total angle is
uniformly bounded. Moreover the set $\mathrm{Fix}(f)$ of fixed points of the
continuous extension of $f$ to $\widehat{M}$ contains $\mathfrak{V}$. Indeed,
if $\mathcal{H}=\\{H_{i}\\}$ and $\mathcal{V}=\\{V_{j}\\}$ denote the
horizontal and vertical (maximal) cylinder decompositions of $M$, then
$\mathfrak{V}=\cup_{i\neq j}(\partial H_{i}\cap\partial V_{j})$, where the
boundary of each cylinder is taken in the metric completion $\widehat{M}$. The
claim follows from the fact that for every $i\in I$ and $j\in J$, $T_{\alpha}$
fixes $\partial H_{i}$ and $T_{\beta}$ fixes $\partial V_{j}$.

For each $q\in\mathfrak{V}$ we denote by $\mathrm{Sep}_{q}(*)$ the set of
leaves of $*\in\\{\mathcal{F}^{u},\mathcal{F}^{s}\\}$ based at $q$. We call
such a leaf a _separatrix based at $q$_. Remark that if the total angle of the
flat structure $M$ around $q$ is $k\pi$ then
$|\mathrm{Sep}_{q}(\mathcal{F}^{u})|=|\mathrm{Sep}_{q}(\mathcal{F}^{s})|=k$.
The following fact is essential for the second part of the proof.

######  Proposition 4.1.

Let $q\in\mathfrak{V}$. Then any separatrix in
$\mathrm{Sep}_{q}(\mathcal{F}^{u})\cup\mathrm{Sep}_{q}(\mathcal{F}^{s})$ is
dense in $M$.

###### Proof.

We consider first the case when $M$ is a translation surface. At the end of
the proof we deal with the case when $M$ is a half-translation surface.

We show that any separatrix in $\mathrm{Sep}_{q}(\mathcal{F}^{u})$ is dense.
The arguments for separatrices in $\mathrm{Sep}_{q}(\mathcal{F}^{s})$ are
analogous.

_Claim_ : $\cup_{q\in\mathfrak{V}}\mathrm{Sep}_{q}(\mathcal{F}^{u})$, the
union of all separatrices of $\mathcal{F}^{u}$, is dense in $M$. To prove this
claim we strongly use the work of Hooper [Hooper15]141414Hooper only deals
with the case when $M$ is a translation surface. This is why when $M$ is a
half-translation surface we consider its orientation double cover.. In
particular, we use the fact that leaves in $\mathcal{F}^{u}$ are parallel to a
renormalizable direction, see Definition 3.9. We proceed by contradiction by
assuming that the complement of the closure of $\cup_{q\in
V}\mathrm{Sep}_{q}(\mathcal{F}^{u})$ in $\widehat{M}$ is non-empty. Let $U$ be
a connected component of this complement. Then $U$ is
$\mathcal{F}^{u}$-invariant. If $U$ contains a closed leaf of
$\mathcal{F}^{u}$ then it has to be a cylinder, but this cannot happen because
there are no saddle connections parallel to renormalizable directions, see
Theorem 6.2 in [Hooper15]. Then $U$ contains a transversal to
$\mathcal{F}^{u}$ to which leaves never return. In other words, $U$ contains
an infinite strip, _i.e._ a set which (up to rotation) is isometric to
$(a,b)\times\mathbb{R}$ for some $a<b$. This is impossible since the
translation flow on $M$ in a renormalizable direction is conservative151515The
translation flow $F_{\theta}^{t}$ is called conservative if given $A\subset M$
of positive measure and any $T>0$, for Lebesgue almost every $x\in M$ there is
a $t>T$ such that $F_{\theta}^{t}(x)\in A$., see Theorem 6.4 in [Hooper15].
The claim follows.

We strongly recommend that the reader uses Figure 6 as a guide for the next
paragraphs.

Figure 6.

Henceforth if $\gamma$ is a separatrix of $\mathcal{F}^{u}$, we denote by
$\gamma(t)$, $t>0$ the parametrization for which $\lim_{t\to
0}\gamma(t)\in\mathfrak{V}$ and such that $|\gamma^{\prime}|=1$ (w.r.t. to the
flat metric on $M$).

For each horizontal cylinder $H_{k}$ in $M$ and
$\xi\in\mathfrak{V}\cap\partial H_{k}$ we denote by
$\gamma^{u}_{\xi,H_{k}}\subset M$ (respectively $\gamma^{s}_{\xi,H_{k}}$) the
unique separatrix of $\mathcal{F}^{u}$ (respect. of $\mathcal{F}^{s}$) based
at $\xi$ within $H_{k}$, that is, for which $\gamma^{u}_{\xi,H_{k}}(t)\in
H_{k}$ for all $t$ in a small neighbourhood of $0$. For a vertical cylinder
$V_{l}$, $\gamma^{u}_{\xi,V_{l}}$ and $\gamma^{s}_{\xi,V_{l}}$ are defined in
a similar way. Let $\mathfrak{V}^{b}(H_{k})$ and $\mathfrak{V}^{t}(H_{k})$
denote the points in $\mathfrak{V}\cap H_{k}$ in the bottom and in the
top161616We pull back the standard orientation of the Euclidean plane to $M$
to make sense of the east-west and bottom-top sides of a cylinder. connected
component of $\partial H_{k}$ respectively; and for any vertical cylinder
$V_{l}$ let $\mathfrak{V}^{e}(V_{l})$ and $\mathfrak{V}^{w}(V_{l})$ denote the
points in $\mathfrak{V}\cap V_{l}$ in the east and west connected component of
$\partial V_{l}$ respectively.

Without loss of generality we suppose that $q\in V^{b}(H_{k})\cap
V^{e}(V_{l})$. We denote by $\omega(\gamma_{q,H_{k}}^{u})$ the $\omega$-limit
set of $\gamma_{q,H_{k}}^{u}$.

_Claim_ : the union of all separatrices of $\mathcal{F}^{u}$ based at points
in $\partial H_{k}\cup\partial V_{l}$ within $H_{k}$ and $V_{l}$ respectively

(3) $\left(\bigcup_{\xi\in\mathfrak{V}\cap\partial
H_{k}}\gamma^{u}_{\xi,H_{k}}\right)\cup\left(\bigcup_{\xi\in\mathfrak{V}\cap\partial
V_{l}}\gamma^{u}_{\xi,V_{l}}\right)$

is contained in $\omega(\gamma_{q,H_{k}}^{u})$.

_Proof of claim_. Remark that since $H_{k}$ is tiled by rectangles
corresponding to points of intersection of the core curve $\alpha_{k}$ with
curves in $\beta$, $|\mathfrak{V}^{b}(H_{k})|=|\mathfrak{V}^{t}(H_{k})|$ and
for each $\xi\in\mathfrak{V}^{b}(H_{k})$ there is exactly one point in
$\mathfrak{V}^{t}(H_{k})$ just above. Hence, using the east-west orientation
of $H_{k}$, we can order the elements of
$\mathfrak{V}^{b}(H_{k})\cup\mathfrak{V}^{t}(H_{k})$ cyclically: we write
$\mathfrak{V}^{b}(H_{k})=\\{q_{j}^{b}\\}_{j\in\mathbb{Z}/N\mathbb{Z}}$,
$\mathfrak{V}^{t}(H_{k})=\\{q_{j}^{t}\\}_{j\in\mathbb{Z}/N\mathbb{Z}}$ for
some $N\geq 1$. The sets
$\mathfrak{V}^{e}(V_{l})=\\{q_{j}^{e}\\}_{j\in\mathbb{Z}/M\mathbb{Z}}$,
$\mathfrak{V}^{w}(V_{l})=\\{q_{j}^{w}\\}_{j\in\mathbb{Z}/M\mathbb{Z}}$, for
some $M\geq 1$, are defined in a similar way.

We suppose that the labeling is such that above $q_{j}^{b}$ lies $q_{j}^{t}$
for all $j\in\mathbb{Z}/N\mathbb{Z}$, and that $q=q_{0}^{b}=q_{0}^{e}$. Recall
that $DT_{\alpha}=\left(\begin{smallmatrix}1&\lambda\\\
0&1\end{smallmatrix}\right)$, $DT_{\beta}^{-1}=\left(\begin{smallmatrix}1&0\\\
\lambda&1\end{smallmatrix}\right)$, hence $Df=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)$, with $a,b,c,d\in\mathbb{R}_{>0}$. In particular
$Df$ sends the positive quadrant $\mathbb{R}_{x\geq 0,y\geq 0}$ into itself.
If we suppose, without loss of generality, that the unstable eingenspace of
$Df$ (without its zero) lies in the interior of $\mathbb{R}_{x\geq 0,y\geq
0}\cup\mathbb{R}_{x\leq 0,y\leq 0}$, then the stable eigenspace of $Df$
(without its zero) has to lie in the interior of $\mathbb{R}_{x\geq 0,y\leq
0}\cup\mathbb{R}_{x\leq 0,y\geq 0}$. Hence, for every
$j\in\mathbb{Z}/N\mathbb{Z}$ we have that $\gamma^{u}_{q_{j}^{b},H_{k}}$
intersects $\gamma^{s}_{\xi,H_{k}}$, for every
$\xi\in\\{q_{j}^{t},q_{j+1}^{b}\\}$ and
$\gamma^{s}_{q_{j+1}^{t},V_{l^{\prime}}}$, where $V_{l^{\prime}}$ is the
vertical cylinder intersecting $H_{k}$ and having
$\\{q_{j}^{b},q_{j}^{t},q_{j+1}^{b},q_{j+1}^{t}\\}$ in its
boundary171717Remark that in $M$ these points need not to be all different
from each other. For example $q_{j}^{b}=q_{j}^{t}$ and
$q_{j+1}^{b}=q_{j+1}^{t}$ if the core curve of $V_{l^{\prime}}$ only
intersects the core curve of $H_{k}$. In any case the claims remain valid..
From Figure 6 we can see that some of these points of intersection are
actually in $H_{k}\cup V_{l^{\prime}}$. By applying repeatedly $f$ to all
these points of intersection of separatrices we obtain that
$\xi\in\omega(\gamma_{q_{j}^{b}}^{u},H_{k})$ for every
$j\in\mathbb{Z}/N\mathbb{Z}$ and
$\xi\in\\{q_{j+1}^{b},q_{j}^{t},q_{j+1}^{t}\\}$. This implies that
$\gamma^{u}_{\xi,H_{k}}\subset\omega(\gamma_{q_{j}^{b},H_{k}}^{u})$ for every
$j\in\mathbb{Z}/N\mathbb{Z}$ and
$\xi\in\\{q_{j+1}^{b},q_{j}^{t},q_{j+1}^{t}\\}$. In particular, we get that
$\omega(\gamma_{q=q_{0}^{b},H_{k}}^{u})$ contains $\gamma^{u}_{\xi,H_{k}}$ for
every $\xi\in\\{q_{1}^{b},q_{1}^{t},q_{0}^{t}\\}$. As a consequence, we have
that $\omega(\gamma_{q,H_{k}}^{u})$ contains181818Here we are using the
following general principle: if $\gamma_{1},\gamma_{2}$ are trayectories of a
vector field on a translation surface and $\gamma_{1}$ is contained in
$\omega(\gamma_{2})$, then $\omega(\gamma_{1})\subset\omega(\gamma_{2})$.
$\omega(\gamma^{u}_{q_{1}^{b},H_{k}})$, which in turn contains
$\\{\gamma^{u}_{q_{2}^{b},H_{k}},\gamma^{u}_{q_{1}^{t},H_{k}},\gamma^{u}_{q_{2}^{t},H_{k}}\\}$.
Proceeding inductively we get that $\omega(\gamma_{q,H_{k}}^{u})$ contains

$\bigcup_{\xi\in\mathfrak{V}\cap\partial H_{k}}\gamma^{u}_{\xi,H_{k}}.$

The positivity of the matrix $Df$ and the fact that its unstable eigenspace
lies in $\mathbb{R}_{x\geq 0,y\geq 0}\cup\mathbb{R}_{x\leq 0,y\leq 0}$ also
imply that for every $j\in\mathbb{Z}/M\mathbb{Z}$ the separatrix
$\gamma_{q_{j}^{e},V_{l}}^{u}$ intersects $\gamma_{\xi,V_{l}}^{s}$ for
$\xi\in\\{q_{j}^{w},q_{j+1}^{e},q_{j+1}^{w}\\}$. From here on, the logic to
show that $\omega(\gamma_{q,H_{k}}^{u})$ contains
$\bigcup_{\xi\in\mathfrak{V}\cap\partial V_{l}}\gamma^{u}_{\xi,V_{l}}$ is the
same as the one presented in the preeceding paragraph and the claim follows.

The arguments in the proof of the preceding claim are local so they can be
used to show that:

  * •

For every $j\in\mathbb{Z}/N\mathbb{Z}$, the limit set
$\omega(\gamma_{q_{j}^{b},H_{k}}^{u})$ contains all separatrices:

$\left(\bigcup_{\xi\in\mathfrak{V}\cap\partial
H_{k}}\gamma^{u}_{\xi,H_{k}}\right)\cup\left(\bigcup_{\xi\in\mathfrak{V}\cap\partial
V_{l^{\prime}}}\gamma^{u}_{\xi,V_{l^{\prime}}}\right)$

where $V_{l^{\prime}}$ is such that $q_{j}^{b}=\partial H_{k}\cap\partial
V_{l^{\prime}}$.

  * •

For every $j\in\mathbb{Z}/M\mathbb{Z}$, the limit set
$\omega(\gamma_{q_{j}^{e},H_{k}}^{u})$ contains all separatrices:

$\left(\bigcup_{\xi\in\mathfrak{V}\cap\partial
V_{l}}\gamma^{u}_{\xi,V_{l}}\right)\cup\left(\bigcup_{\xi\in\mathfrak{V}\cap\partial
H_{k^{\prime}}}\gamma^{u}_{\xi,H_{k^{\prime}}}\right)$

where $H_{k^{\prime}}$ is a horizontal cylinder such that $q_{j}^{e}=\partial
V_{l}\cap\partial H_{k^{\prime}}$.

If we now denote by $\alpha_{k}\in\alpha$ the core curve of $H_{k}$ then the
preceding discussion can be summarized as follows:
$\omega(\gamma_{q,H_{k}}^{u})$ contains all separatrices of $\mathcal{F}^{u}$
based at points in the boundary of cylinders (and stemming within those
cylinders) whose core curves belong to the link of $\alpha_{k}$ in the
configuration graph $\mathcal{G}(\alpha\cup\beta)$; moreover, if
$\beta_{l}\in{\rm link}(\alpha_{k})$ then $\omega(\gamma_{q,H_{k}}^{u})$
contains all separatrices of $\mathcal{F}^{u}$ based at points in the boundary
of cylinders (and stemming within those cylinders) whose core curves belong to
${\rm link}(\beta_{l})$. This way we can extend the arguments above to the
whole configuration graph $\mathcal{G}(\alpha\cup\beta)$ to conclude that
$\omega(\gamma_{q,H_{k}}^{u})$ contains
$\cup_{q\in\mathfrak{V}}\mathrm{Sep}_{q}(\mathcal{F}^{u})$. Since the later is
dense in $M$ we conclude that $\gamma_{q,H_{k}}^{u}$ is dense in $M$.

We now suppose that $M$ is a half-translation surface (_i.e._ given by a
quadratic differential). Let $\pi:\widetilde{M}\to M$ the orientation double
cover of $M$.

We claim that for every horizontal cylinder $H_{i}$ in $M$ the lift
$\pi^{-1}(H_{i})$ is formed by two disjoint isometric copies
$\widetilde{H}_{i_{1}}$, $\widetilde{H}_{i_{2}}$ of $H_{i}$ and these are
maximal horizontal cylinders in $\widetilde{M}$. Recall that if
$p\in\mathfrak{V}$ is a conic singularity of angle $n\pi$, then $\pi^{-1}(p)$
is formed by two conical singularities of angle $n\pi$ if $n$ is even, whereas
if $n$ is odd $\pi^{-1}(p)$ is a conical singularity of angle $2n\pi$. Given
that the multicurves $\alpha$ and $\beta$ are in minimal position, points in
$\mathfrak{V}\cap\partial H_{i}$ which are conical singularities of angle
$\pi$ are actually punctures of $S$. This implies that
$\widetilde{H}_{i_{1}}\cup\widetilde{H}_{i_{2}}$ cannot be merged within
$\widetilde{M}$ into a flat cylinder. The same conclusion holds when
$\mathfrak{V}\cap\partial H_{i}$ has conical singularities of angle different
from $\pi$ and the claim follows. Analogously, we have that for every vertical
cylinder $V_{j}$ in $M$ the lift $\pi^{-1}(V_{j})$ is formed by two disjoint
isometric copies $\widetilde{V}_{j_{1}}$, $\widetilde{V}_{j_{2}}$ of $V_{j}$
and these are maximal vertical cylinders in $\widetilde{M}$. The families
$\widetilde{\mathcal{H}}=\\{\widetilde{\mathcal{H}}_{i_{1}},\widetilde{\mathcal{H}}_{i_{2}}\\}$
and
$\widetilde{\mathcal{V}}=\\{\widetilde{\mathcal{V}}_{j_{1}},\widetilde{\mathcal{V}}_{j_{2}}\\}$
define horizontal and vertical (maximal) cylinder decompositions of
$\widetilde{M}$ respectively.

Let $\tilde{\alpha}$, $\tilde{\beta}$ denote the lifts to the orientation
double cover of $\alpha$ and $\beta$ respectively. Given that the moduli of
cylinders downstairs and upstairs is the same, we have a pair of affine
multitwists
$\widetilde{T_{\tilde{\alpha}}},\widetilde{T_{\tilde{\beta}}}\in\mathrm{Aff}(\widetilde{M})$
with $DT_{\alpha}=D\widetilde{T_{\tilde{\alpha}}}$ and
$DT_{\beta}=D\widetilde{T_{\tilde{\beta}}}$ in $\mathrm{PSL}(2,\mathbb{R})$.
If we rewrite the word defining $f$ replacing each appearence of $T_{\alpha}$
with $\widetilde{T_{\tilde{\alpha}}}$ and each appearence of $T_{\beta}^{-1}$
with $\widetilde{T_{\tilde{\beta}}}^{-1}$ the result is an affine multitwist
$\tilde{f}$ on $\widetilde{M}$ with $D\tilde{f}=Df$ in
$\mathrm{PSL}(2,\mathbb{R})$. The eigendirections of $\tilde{f}$ define a pair
of transverse $\tilde{f}$-invariant measured foliations
$\widetilde{\mathcal{F}}_{u}$ and $\widetilde{\mathcal{F}}_{s}$. Moreover, we
have that $\widetilde{\mathcal{F}}_{u}=\pi^{-1}(\mathcal{F}_{u})$ and
$\widetilde{\mathcal{F}}_{s}=\pi^{-1}(\mathcal{F}_{s})$ (_i.e._ the projection
$\pi$ sends leaves to leaves). Let
$\hat{\pi}:\widehat{\widetilde{M}}\to\widehat{M}$ be the continuous extension
of the projection $\pi$ to the metric completions of $M$ and $\widetilde{M}$
and define $\tilde{\mathfrak{V}}:=\hat{\pi}^{-1}(\mathfrak{V})$. Remark that
$\tilde{\mathfrak{V}}=\cup(\partial\widetilde{H_{i}}\cap\partial\widetilde{V_{j}})$,
where the boundaries of the cylinders are taken in $\widehat{\widetilde{M}}$.
As with $M$, for every $q\in\widetilde{\mathfrak{V}}$ we define
$\mathrm{Sep}_{q}(*)$ as the set of leaves of
$*\in\\{\widetilde{\mathcal{F}}^{u},\widetilde{\mathcal{F}}^{s}\\}$ based at
$q$. In this context, the proof of Proposition 4.1 for translation surfaces
then applies to $\widetilde{M}$ and we get the following:

######  Corollary 4.2.

Let $q\in\widetilde{\mathfrak{V}}$. Then any separatrix in
$\mathrm{Sep}_{q}(\widetilde{\mathcal{F}}^{u})\cup\mathrm{Sep}_{q}(\widetilde{\mathcal{F}}^{s})$
is dense in $\widetilde{M}$.

If separatrices are dense upstairs they are dense downstairs. This ends the
proof of Proposition 4.1.

∎

Let now $p\in S$ be the marked puncture and
$\mathrm{Sep}_{p}(\mathcal{F}^{u})=\\{\gamma_{1},\ldots,\gamma_{m}\\}$. We
denote by $S_{\mu}$ a fixed complete hyperbolic structure on $S$ ($\mu$ stands
for the metric) of the first kind and define the completed ray graph $R(S;p)$
with respect to $\mu$. Remark that in $S_{\mu}$ the point $p$ becomes a cusp,
_i.e._ a point at infinity. In what follows we associate to each $\gamma_{i}$
a simple geodesic in $S_{\mu}$ based at $p$. For elements in
$\mathrm{Sep}_{p}(\mathcal{F}^{s})$ the arguments are analogous. The ideas we
present are largely inspired by the work of P. Levitt [Levitt83].

Henceforth $\pi:\mathbb{D}\to S_{\mu}$ denotes the universal cover,
$\Gamma<{\rm PSL(2,\mathbb{R})}$ the Fuchsian group for which
$S_{\mu}=\mathbb{D}/\Gamma$, $\tilde{p}\in\partial\mathbb{D}$ a chosen point
in lift of the cusp $p$ to $\partial\mathbb{D}$ and
$\tilde{\gamma_{i}}=\tilde{\gamma_{i}}(\tilde{p})$ the unique lift of
$\gamma_{i}$ to $\mathbb{D}$ based at $\tilde{p}$.

_Claim:_ $\tilde{\gamma_{i}}$ converges to two distincs points in
$\partial\mathbb{D}$.

First remark that since $\gamma_{i}$ is not a loop, then it is sufficient to
show that $\tilde{\gamma_{i}}(t)$ converges to a point when considering the
parametrization that begins at $\tilde{p}$ and $t\to\infty$. Recall that in
$S$, the point $p$ is in a region bounded by a $2m$-polygon whose sides belong
to closed curves ${\alpha_{1},\ldots,\alpha_{m},\beta_{1},\ldots,\beta_{m}}$
in $\alpha\cup\beta$. Each of these curves is transverse to the leaves of
$\mathcal{F}^{u}$ and of $\mathcal{F}^{s}$. Up to making an isotopy, we can
suppose without loss of generality that the first element in
${\alpha_{1},\ldots,\alpha_{m},\beta_{1},\ldots,\beta_{m}}$ intersected by
$\gamma_{i}(t)$ (for the natural parametrization used in the proof of
Proposition 4.1) is $\alpha_{j}$. Given that $\alpha_{j}$ is transverse to
$\mathcal{F}^{u}$ and $\gamma_{i}$ is dense in $M$ we have that
$\gamma_{i}\cap\alpha_{j}$ is dense in $\alpha_{j}$. In consequence
$\tilde{\gamma_{i}}$ intersects $\pi^{-1}(\alpha_{j})$ infinitely often.
Remark that $\tilde{\gamma_{i}}$ intersects a connected component of
$\pi^{-1}(\alpha_{j})$ at most once. Indeed, if this was not the case there
would exists a disc $D$ embedded in $\mathbb{D}$ whose boundary $\partial D$
is formed by an arc in $\tilde{\gamma_{i}}$ and an arc contained in
$\pi^{-1}(\alpha_{j})$ transverse to
$\widetilde{\mathcal{F}^{u}}:=\pi^{-1}(\mathcal{F}^{u})$. This is impossible
because all singularities of $\widetilde{\mathcal{F}^{u}}$ are saddles (in
particular only a finite number of separatrices can steam from each one of
them) and there is a finite number of them inside $D$. Then, all limit points
in $\widetilde{\gamma_{i}}$ in $\mathbb{D}\cup\partial\mathbb{D}$ different
from $\tilde{p}$ are in the intersection of an infinite family of nest domains
$\mathbb{D}\cup\partial\mathbb{D}$ whose boundaries in $\mathbb{D}$ are
components of $\pi^{-1}(\alpha_{j})$. Moreover, this intersection is a single
point $q_{i}\in\partial\mathbb{D}$ because it has to be connected and the
endpoints in $\partial\mathbb{D}$ of components in $\pi^{-1}(\alpha_{j})$ are
dense in $\mathbb{D}$ since $\Gamma$ is Fuchsian of the first kind. This
finishes the proof of our claim above.

We define thus $\widetilde{\delta_{i}}=\widetilde{\delta_{i}}(\tilde{p})$ to
be the geodesic in $\mathbb{D}$ whose endpoints are $\tilde{p}$ and $q_{i}$ as
above and $\delta_{i}:=\pi(\widetilde{\delta_{i}})$. The geodesic $\delta_{i}$
is well defined: it does not depend on the lift of $\gamma_{i}$ based at
$\tilde{p}$ we have chosen and if we changed $\tilde{p}$ by some
$\tilde{p}^{\prime}=gp$ for some $g\in\Gamma$ then by continuity
$q_{i}^{\prime}=gq_{i}$. On the other hand $\delta_{i}$ is simple: if this was
not the case two components of $\pi^{-1}(\delta_{i})$ would intersect and this
implies that two components of $\pi^{-1}(\gamma_{i})$ intersect, which is
imposible since $\widetilde{\mathcal{F}^{u}}$ is a foliation. Remark that if
$\gamma_{i}\neq\gamma_{j}$ then $\delta_{i}$ and $\delta_{j}$ are disjoint. If
this was not the case then there would be a geodesic in $\pi^{-1}(\delta_{i})$
intersecting a geodesic in $\pi^{-1}(\delta_{j})$, but this would imply that a
connected component of $\pi^{-1}(\gamma_{i})$ intersects a connected component
of $\pi^{-1}(\gamma_{j})$, which is impossible since
$\widetilde{\mathcal{F}}^{u}$ is a foliation.

Hence we can associate to the set of separatrices
$\\{\gamma_{1},\ldots,\gamma_{m}\\}$ a set of pairwise distinct simple
geodesics $\\{\delta_{1},\ldots,\delta_{m}\\}$ based at $p$. Remark that by
construction this set is $f$-invariant. In what follows we show that
$\\{\delta_{1},\ldots,\delta_{m}\\}$ is a clique of high-filling rays. By
applying the same arguments to the separatrices of $\mathcal{F}^{u}$ based at
$p$ one obtains a different $f$-invariant clique of $m$ high-filling rays.
These correspond to the only two points in the Gromov boundary of the loop
graph $L(S;p)$ fixed by $f$.

Let $\tilde{\delta}$ be a geodesic in $\mathbb{D}$ based at $\tilde{p}$ such
that $\delta:=\pi(\tilde{\delta})$ is a simple geodesic in $S_{\mu}$ which
does not belong to $\\{\delta_{1},\ldots,\delta_{m}\\}$. We denote by $q$ the
endpoint of $\tilde{\delta}$ which is different from $\tilde{p}$. Since every
short ray or loop has a geodesic representative, it is sufficient to show that
for every $i=1,\ldots,m$ there exists a (geodesic) component of
$\pi^{-1}(\delta_{i})$ which intersects $\tilde{\delta}$. We recommend to use
Figure 7 as a guide for the next paragraph.

Figure 7.

All components of $\pi^{-1}(\mathrm{Sep}_{p}(\mathcal{F}^{u}))$ with one
endpoint in $\tilde{p}$ are of the form $\\{g^{k}\tilde{\gamma}_{1},\ldots
g^{k}\tilde{\gamma}_{m}\\}_{k\in\mathbb{Z}}$, with $g\in\Gamma$ parabolic
fixing $\tilde{p}$. Hence, there exists a closed disc
$D\subset\mathbb{D}\cup\partial\mathbb{D}$ whose boundary is formed by
$\tilde{p}\cup\tilde{\gamma}_{k}\cup\tilde{\gamma}_{l}\cup A$, where $A$ is a
closed arc in $\partial\mathbb{D}$ containing $q$; $\tilde{\gamma}_{k}$,
$\tilde{\gamma}_{l}$ are (lifts of) separatrices and there is no element of
$\pi^{-1}(\mathrm{Sep}_{p}(\mathcal{F}^{u}))$ with one endpoint in $\tilde{p}$
in the interior of $D$. Remark that the endpoints of $A$ are $q_{k}$ and
$q_{l}$ (the endpoints of $\tilde{\gamma}_{k}$ and $\tilde{\gamma}_{l}$
respectively). Given that $p$ is a saddle-type singularity of the foliation
$\mathcal{F}^{u}$ there exists a neighbourhood of $\tilde{p}$ in $D$ which
contains a segment191919As a matter of fact this segment can be taked to live
in one of the (lifts of) the curves in $\alpha\cup\beta$ forming the boundary
of the disc in $S\setminus\alpha\cup\beta$ containing $p$. $\Sigma$ with one
endpoint in $\tilde{\gamma}_{k}$ and the other in $\tilde{\gamma}_{l}$, and
which is transverse to $\widetilde{\mathcal{F}^{u}}$ except at one point $\xi$
in its interior. Moreover, since $p$ is an isolated singularity of
$\mathcal{F}^{u}$, we can suppose that the closure of the connected component
of $D\setminus\Sigma$ in $\mathbb{D}$ having $\tilde{p}$ in its boundary does
not contain singular points of $\widetilde{\mathcal{F}^{u}}$ different from
$\tilde{p}$. The point $\xi$ divides $\Sigma$ in two connected components
$\Sigma_{L}$ and $\Sigma_{R}$. On the other hand $q$ divides $A$ in two
connected components $A_{L}$ and $A_{R}$. Now let $i=1,\ldots,m$ be fixed.
Since $\gamma_{i}$ is dense in $M$ we have that $\pi^{-1}(\gamma_{i})$ is
dense in $\mathbb{D}$ and in particular $\pi^{-1}(\gamma_{i})\cap\Sigma$ is
dense in $\Sigma$. Hence we can pick a leaf $\tilde{\gamma}_{i}^{\prime}$ in
$\pi^{-1}(\gamma_{i})$ passing through a point $\xi_{L}\in\Sigma_{L}$ and
suppose without loss of generality that one of its endpoints $q_{i}^{\prime}$
is in $A_{L}$. Then
$\tilde{\gamma}_{i}^{\prime}\cap\Sigma=\\{\xi_{L},\xi_{R}\\}$ with
$\xi_{R}\in\Sigma_{R}$. Again, given that $\pi^{-1}(\gamma_{i})\cap\Sigma$ is
dense in $\Sigma$, we can find a leaf
$\tilde{\gamma_{i}}^{\prime\prime}\in\pi^{-1}(\gamma_{i})$ which intersects
$\Sigma$ transversally at a point $\eta_{R}$ between $\xi_{R}$ and
$\tilde{\gamma}_{l}$, and which has an enpoint $t_{R}$ in $A_{R}$ arbitrarly
close to $q_{l}$. This is true because of the way $q_{l}$ was found: there is
a family of connected components of $\pi^{-1}(\alpha)$, for some closed curve
$\alpha$ in $M$ transverse to $\mathcal{F}^{u}$, bounding domains in
$\mathbb{D}\cup\partial\mathbb{D}$ whose intersection is $q_{l}$. Now, by the
way $\Sigma$ was chosen we have that
$\tilde{\gamma}_{i}^{\prime\prime}\cap\Sigma=\\{\eta_{L},\eta_{R}\\}$ with
$\eta_{L}\in\Sigma_{L}$. This implies that $\tilde{\gamma}_{i}^{\prime\prime}$
has and endpoint $t_{L}$ in $A_{L}$. Hence the geodesic
$\tilde{\delta}^{\prime}$ determined by the endpoints $t_{L}$ and $t_{R}$
intersects $\tilde{\delta}$ and $\delta_{i}=\pi(\tilde{\delta}^{\prime})$
intersects $\delta=\pi(\tilde{\delta})$.∎

######  Remark 4.3.

In the proof of Theorem 1.1 we made use of the fact that every short-ray or
loop has a geodesic representative, but this is not necessary. As a matter of
fact the following is true: if $\tilde{\delta}$ is any curve in $\mathbb{D}$
based at $\widetilde{p}$ whose extremities define two different points in
$\partial\mathbb{D}$ and such that $\pi(\tilde{\delta})=\delta$ is simple and
does not belong to the set of separatrices
$\\{\gamma_{1},\ldots,\gamma_{m}\\}$, then for any $j=1,\ldots,m$ the geodesic
$\delta_{j}$ intersects $\delta$.

On the other hand, in the proof of Theorem 1.1 the density of each separatrix
of $\mathcal{F}^{u}$ or $\mathcal{F}^{s}$ on the _whole_ surface $S$ is not
used. The proof remains valid if we only require separetrices of
$\mathcal{F}^{u}$ and $\mathcal{F}^{s}$ to be dense on a subsurface
$S^{\prime}\subset S$ of finite type with enough topology, _e.g._ such that
all curves defining the polygon on which $p$ lives are essential in
$S^{\prime}$. In particular we have the following:

######  Corollary 4.4.

Let $S^{\prime}\subset S$ be an essential subsurface of finite topological
type containing $p$ and $h\in\mathrm{Mod}(S^{\prime})$ a pseudo-Anosov element
for which $p$ is a $k$-prong for some $k\in\mathbb{N}$. Let $\hat{h}$ be the
extension (as the identity) of $h$ to $S$. Then $\hat{h}$ is a loxodromic
element of weight $k$. Moreover, the separatrices of the invariant transverse
measured foliations of $h$ based at $p$ define202020By a stretching process as
described in the proof of Theorem 1.1 the cliques of high-filling rays fixed
by $\hat{h}$.

This result already appears in the work of Bavard and Walker [BaWa18B], see
Lemma 7.2.1 and Theorem 8.3.1.

###  4.2. Proof of Theorem 1.5

####  4.2.1. Preliminaries

In this section we present, for each infinite type surface, a model that is
convenient for the proof of Theorem 1.5.

Normal forms for infinite-type surfaces. In what follows we detail how to
construct, for any infinite type surface $S$, a graph
$T(S)\subset\mathbb{H}^{3}$ having a regular neighbourhood whose boundary is
homeomorphic to $S$ (our choice of ambient space obeys illustrative purposes
only). There are many ways to construct such graph. The one we present is
intended to make the proof of Theorem 1.5 more transparent.

Let $2^{\mathbb{N}}:=\prod_{j\in\mathbb{N}}\\{0,1\\}_{j}$ be the Cantor set.
In general terms, the construction is as follows. We consider a rooted binary
tree $T2^{\mathbb{N}}$, a homeomorphism
$f:\prod_{j\in\mathbb{N}}\\{0,1\\}\to\mathrm{Ends}(T2^{\mathbb{N}})$ from the
standard binary Cantor set to the space of ends of this tree, and we choose a
topological embedding
$i:\mathrm{Ends}(S)\hookrightarrow\prod_{j\in\mathbb{N}}\\{0,1\\}$. We show
that there exists a subtree of $T2^{\mathbb{N}}$ whose space of ends is
precisely $f\circ i(\mathrm{Ends}(S))$. For our purposes it is important that
this subtree is _simple_ (see Definition 4.5 below). Then, if $S$ has genus,
we perform a surgery on vertices of the aforementioned subtree of of
$T2^{\mathbb{N}}$ belonging to rays starting at the root and having one end on
$f\circ i(\mathrm{Ends}_{\infty}(S))$.

The rooted binary tree. For every $n\in\mathbb{N}$ let
$2^{(n)}:=\prod_{i=1}^{n}\\{0,1\\}$ and $\pi_{i}:2^{(n)}\to\\{0,1\\}$ the
projection on to the $i^{th}$ coordinate. The rooted binary tree is the graph
$T2^{\mathbb{N}}$ whose vertex set $V(T2^{\mathbb{N}})$ is the union of the
simbol $\mathfrak{r}$ (this will be the root of the tree) with the set
$\\{D:D\in 2^{(n)}\hskip 2.84526pt\text{for some $n\in\mathbb{N}$}\\}$. The
edges $E(T2^{\mathbb{N}})$ are $\\{(\mathfrak{r},0)$, $(\mathfrak{r},1)\\}$
together with:

$\\{(D,D^{\prime}):D\in 2^{(n)}\hskip 2.84526ptD^{\prime}\in 2^{(n+1)}\hskip
2.84526pt\text{for some $n\in\mathbb{N}$},\hskip 2.84526pt\text{and}\hskip
2.84526pt\pi_{i}(D_{s})=\pi_{i}(D_{t})\hskip 2.84526pt\forall 1\leq i\leq
n\\}$

Henceforth $T2^{\mathbb{N}}$ is endowed with the combinatorial distance. For
every $\hat{x}=(x_{n})\in 2^{\mathbb{N}}$ we define
$r(\hat{x})=(\mathfrak{r},a_{1},\ldots,a_{n}):=(x_{1},\ldots,x_{n},\ldots)$ to
be the infinite geodesic ray in $T2^{N}$ starting from $\mathfrak{r}$ and
ending in $\mathrm{Ends}(T2^{\mathbb{N}})$. Then, the map

(4) $f:\prod_{i\in\mathbb{N}}\\{0,1\\}\to\mathrm{Ends}(T2^{\mathbb{N}})$

which associates to each infinite sequence $\hat{x}=(x_{n})_{n\in\mathbb{N}}$
the end $f(\hat{x})$ of $T2^{\mathbb{N}}$ defined by the infinite geodesic ray
$r(\hat{x})$ is a homeomorphism.

######  Definition 4.5.

Let $v$ and $v^{*}$ two different vertices in a subtree $\mathcal{T}$ of
$T2^{\mathbb{N}}$. If $v$ is contained in the geodesic which connects $v^{*}$
with $\mathfrak{r}$, then we say that $v^{*}$ is a _descendant_ of $v$. A
connected rooted subtree of $\mathcal{T}$ without leaves is _simple_ if all
descendants of a vertex $v\neq\mathfrak{r}$ of degree two, have also degree
two.

######  Lemma 4.6.

Let $F\subset\mathrm{Ends}(T2^{\mathbb{N}})$ be closed. Then there exists a
_simple_ subtree $T$ of $T2^{\mathbb{N}}$ rooted at $\mathfrak{r}$ such that
$F$ is homeomorphic to $\mathrm{Ends}(T)$.

We postpone the proof of this lemma to the end of the section.

######  Definition 4.7.

Given a subset $F$ of $\mathrm{Ends}(T2^{\mathbb{N}})$ we define
$T_{F}:=\bigcup_{\hat{x}\in f^{-1}(F)}r(\hat{x})\subseteq T2^{\mathbb{N}}$ and
call it the _tree induced by $F$_.

Surgery. Let $T$ be a subtree of $T2^{\mathbb{N}}$ rooted at $\mathfrak{r}$
and having no leaves different from this vertex, if any. Let $L$ be a subset
of the vertex set of $T$. We denote by $\Gamma_{T,L}$ the graph obtained from
$T$ and $L$ after performing the following operations on each vertex $v\in L$:

  1. (1)

If $v$ has degree 3 with adjacent descendants $v^{\prime},v^{\prime\prime}$ we
delete first the edges $\\{(v,v^{\prime}),(v,v^{\prime\prime},)\\}$. Then we
add to $L$ two vertices $v_{*}^{\prime},v_{*}^{\prime\prime}$ and the edges
$\\{(v,v_{*}^{\prime}),(v,v_{*}^{\prime\prime}),(v_{*}^{\prime},v^{\prime}),(v_{*}^{\prime\prime},v^{\prime\prime}),(v^{\prime}_{*},v_{*}^{\prime\prime})\\}$.

  2. (2)

If $v$ has degree 2 and $v^{\prime}$ is its adjacent descendant, we delete
first the edge $(v,v^{\prime})$. Then we add to $L$ two vertices
$v_{*}^{\prime},v_{*}^{\prime\prime}$ and the edges
$\\{(v,v_{*}^{\prime}),(v,v_{*}^{\prime\prime}),(v_{*}^{\prime},v^{\prime}),(v^{\prime}_{*},v_{*}^{\prime\prime})\\}$.

######  Definition 4.8.

Let $S$ be a surface of infinite type of genus
$g\in\mathbb{N}\cup\\{\infty\\}$ and
$\mathrm{Ends}_{\infty}(S)\subset\mathrm{Ends}(S)\subset 2^{\mathbb{N}}$ its
space of ends accumulated by genus and space of ends, respectively. We define
the graph $T(S)$ according to the following cases. In all of them we suppose
w.l.o.g that $T_{f(\mathrm{Ends}(S))}$ is simple.

  1. (1)

If $g=0$ let $T(S):=T_{f(\mathrm{Ends}(S))}$,

  2. (2)

if $g\in\mathbb{Z}_{>0}$ let $T(S):=\Gamma_{T,L}$ where
$T=T_{f(\mathrm{Ends}(S))}$ and $L={a_{1},a_{2},\ldots,a_{g}}$ and
$(\mathfrak{r},a_{1},\ldots,a_{g},\ldots)=r(\hat{x})$ for some
$\hat{x}\in\mathrm{Ends}(S)\subset 2^{\mathbb{N}}$, and

  3. (3)

if $g=\infty$ let $T(S):=\Gamma_{T,L}$ where $T=T_{f(\mathrm{Ends}(S))}$ and
$L$ is the set of vertices of the subtree
$T_{f(\mathrm{Ends}_{\infty}(S))}\subset T_{f(\mathrm{Ends}(S))}$.

By construction, there exists a geometric realization for $T(S)$ as a graph in
the plane $\\{(x,0,z)\in\mathbb{H}^{3}:z>0\\}$ in 3-dimensional hyperbolic
space, which we denote again by $T(S)$. Moreover there exists a closed regular
neighbourhood $N(T(S))$ so that $S$ is homeomorphic to $S^{\prime}=\partial
N(T(S))$, see Figure 8. Observe that $T(S)$ is a strong deformation retract of
$N(T(S))$. We identify $S$ with $S^{\prime}$, and we say that $S$ is in
_normal form_ , and $T(S)$ is the _underlying_ graph that induces $S$.

Figure 8. Embedding of the tree $T(S)$.

######  Remark 4.9.

In [BaWa18B], A. Walker and J. Bavard carry out a similar construction. For
this they introduce the notion of _rooted core tree_ $T$ from which they
construct a surface $\Sigma(T)$ homeomorphic to a given infinite type surface
$S$, see Lemma 2.3.1 in [BaWa18B]. The graph $T_{f(\mathrm{Ends}(S))}$ (see
Definition 4.8) is turned into a rooted core tree by declaring that the
vertices of $T_{f(\mathrm{Ends}_{\infty}(S))}$ are the marked vertices. The
main difference with the work of Walker and Bavard is that the normal form we
are proposing comes from a _simple_ tree. This property is strongly used in
the proof of Theorem 1.5.

_Proof of Lemma 4.6_. Let $T_{F}$ be the subtree of $T2^{\mathbb{N}}$ induced
by $F$. Let $V^{\prime}$ be the set of vertices of $T_{F}$ of degree 2,
different from $\mathfrak{r}$, having at least one descendant of degree 3.
Then $V^{\prime}=\sqcup_{i\in I}V^{\prime}_{i}$, where:

  1. (1)

$V_{i}^{\prime}$ is a subset of the vertices of a ray $r(\hat{x})$ for some
$\hat{x}\in f^{-1}(F)$,

  2. (2)

for every $i\in I$, one can label
$V^{\prime}_{i}=\\{a_{i,1},\ldots,a_{i,k_{i}}\\}$ so that $a_{i,l+1}$ is a
descendant of $a_{i,l}$ adjacent to $a_{i,l}$.

  3. (3)

for every $i\in I$, the vertex $A_{i}$ of $T_{F}$ adjacent to $a_{i,1}$ other
than $a_{i,2}$ is either the root $\mathfrak{r}$ or a vertex of degree 3.
Similarly, the vertex $B_{i}$ of $T_{F}$ adjacent to $a_{i,k_{i}}$ other than
$a_{i,k_{i}-1}$ is of degree 3.

Replacing the finite simple path from $A_{i}$ to $B_{i}$ by an edge
$(A_{i},B_{i})$ does not modify the space of ends of $T_{F}$. By doing this
for every $i\in I$ we obtain a simple tree as desired. ∎

_Proof of Theorem 1.5_. This proof is divided in two parts. First we show the
existence of a pair of multicurves of finite type whose union fills $S$ and
which satisfy (1) and (2). In the second part we use these to construct the
desired multicurves $\alpha$ and $\beta$.

First part: Let $S$ be an infinite-type surface in its normal form, and
$T(S)\subset\mathbb{H}^{3}$ the underlying graph that induces $S$. We are
supposing that $T(S)$ is obtained after surgery from a simple tree as
described above. The idea here is to construct two disjoint collections $A$
(blue curves) and $B$ (red curves) of pairwise disjoint curves in $S$ such
that after forgetting the non-essential curves in $A\cup B$, we get the pair
of multicurves $\alpha$ and $\beta$ which satisfy (1) and (2) as in Theorem
1.5.

Let $T_{g}(S)$ be the full subgraph of $T(S)$ generated by all the vertices
which define a triangle in $T(S)$. Observe that, since $T(S)$ is constructed
by performing a surgery on a simple tree, the graph $T_{g}(S)$ is connected.

Let $T_{g}^{\prime}(S)$ be the subset obtained as the union of $T_{g}(S)$ with
all the edges in $T(S)$ adjacent to $T_{g}(S)$. As $T_{g}(S)$ is connected,
then $T_{g}^{\prime}(S)$ is also connected. Let $\Delta$ be a triangle in
$T_{g}(S)$, and $\Delta^{\prime}$ be the disjoint union of $\Delta$ with all
the edges in $T_{g}^{\prime}(S)$ adjacent to $\Delta$. We notice that
$\Delta^{\prime}$ is one of the following two possibilities: (1) the disjoint
union of $\Delta$ with exactly three edges adjacent to it, or (2) the disjoint
union of $\Delta$ with exactly two edges adjacent to it. For each case, we
choose blue a red curves in $S$ as indicated in the Figure 9.

Figure 9. Blue and red curves associated to the neighborhood $\Delta^{\prime}$
of a triangle $\Delta$.

For each edge $e$ in $T_{g}^{\prime}(S)$ which connects two triangles in
$T_{g}^{\prime}(S)$, we choose a blue curve in $S$ as indicated in Figure 10.

Figure 10.  Blue curve associated to an edge which connects two triangles.

We consider the following cases.

$\mathrm{Ends}(S)=\mathrm{Ends}_{\infty}(S)$. In this case $A$ and $B$ are the
multicurves formed by the blue and red curves as chosen above respectively.

$\mathrm{Ends}(S)\neq\mathrm{Ends}_{\infty}(S)$. Let $C$ be a connected
component of $T(S)-T_{g}^{\prime}(S)$. Given that $T(S)$ is obtained from a
simple tree, $C$ is a tree with infinitely many vertices. Let $v$ be the only
vertex in $C$ which is adjacent to an edge $e(v)$ in $T_{g}^{\prime}(S)$. If
$v$ has degree one in $C$, then every vertex of $C$ different from $v$ has
degree two because $T_{f(\mathrm{Ends}(S))}$ is a simple subtree of
$T2^{\mathbb{N}}$. In this case, we have that the subsurface $S(C)\subset S$
induced by $C$ is homeomorphic to a punctured disc. In particular, the red
curve in $S$ associated to the edge $e(v)$ chosen as depicted in Figure 9 is
not essential in $S$.

Suppose now that $v$ has degree two in $C$. We color with blue all the edges
in $C$ having vertices at combinatorial distances $k$ and $k+1$ from $v$ for
every even $k\in\mathbb{Z}_{\geq 0}$. We color all other edges in $C$ in red,
see the left-hand side in Figure 11. Let $e$ and $e^{\prime}$ be two edges in
$C$ of the same color and suppose that they shares a vertex $v$. Suppose that
all vertices of $e\cup e^{\prime}$ different from $v$ have degree three. If
$e$ and $e^{\prime}$ are marked with blue color (respectively red color), we
choose the red curve (respect. blue curve) in $S$ as in the right-hand side of
Figure 11.

Figure 11. (Left) Edges of $C$ colored with blue and red alternating in
levels. (Right) The corresponding curve in $S$ for the pair of edges $e$ and
$e^{\prime}$ marked with blue color.

For the edge $e(v)\in T^{\prime}_{g}(S)$, we choose the blue curve in $S$ as
in the left-hand side of Figure 12. Finally, for each edge $e$ of $C$, we take
a curve in $S$ with the same marked color of $e$ as is showed in the right-
hand side of Figure 12.

Figure 12. (Left) The corresponding curve for the $e$ which connects $F$ with
$T_{g}^{\prime}(S)$. (Right)

Blue (red) curve associated to an edge $e$ of $F$.

If $\mathrm{Ends}(S)$ has at most one isolated planar end here ends the
construction of the multicurves $A$ and $B$.

Now suppose that $\mathrm{Ends}(S)$ has more that one isolated planar end,
that is, $S$ has at least two punctures. Let $R$ be the full subgraph of
$T(S)$ generated by all the vertices of degree 3 in $T(S)$ together with the
root vertex $\mathfrak{r}$, and define $T_{g}^{\prime\prime}(S)$ as the full
subgraph of $T(S)$ generated by all the vertices in $T(S)$ at distance at most
1 from $R$. The graph $T_{g}^{\prime\prime}(S)$ is connected (again, because
$T_{f(\mathrm{Ends}(S))}$ is simple) and contains $T_{g}^{\prime}(S)$ . It
also has at least two leaves, i.e., vertices of degree one which we denote by
$v_{1}$ and $v_{2}$. If $v_{1}$ and $v_{2}$ are at distance 3 in
$T_{g}^{\prime\prime}(S)$ there exist a single edge $e$ in
$T_{g}^{\prime\prime}(S)$ whose adjacent vertices are at distance 1 from
$v_{1}$ or $v_{2}$. Let us suppose that this is not an edge of a triangle in
$T(S)$. Then $e$ is contained in a connected component of $T(S)\smallsetminus
T_{g}^{\prime}(S)$. In this case, if $e$ is marked with red color (blue
color), we choose the blue curve (red curve) in $S$ as is shown in Figure 13.
In all other cases we do nothing and this finishes the construction of the
multicurves $A$ and $B$.

Figure 13. The corresponding blue curve in $S$ for the red edge $e$.

We define $\alpha=\\{\alpha_{i}\\}_{i\in I}$ and $\beta=\\{\beta_{j}\\}_{j\in
J}$ as the set of essential curves in $A$ and $B$, respectively. By
construction, $\alpha$ and $\beta$ are multicurves of finite type in minimal
position and $\rm{i}(\alpha_{i},\beta_{j})\leq 2$ for every $i\in I$ and $j\in
J$. Upon investigation, one can observe that each connected component of
$S\setminus\alpha\cup\beta$ is a disc or a punctured disc whose boundary is
formed by 2, 4, 6 or 8 segments.

Second part: Let $m\in\mathbb{N}$. Take a finite multicurve $\delta$ in $S$
such that, if $Q$ is the connected component of $S\smallsetminus\delta$ which
contains $p$, we have that $Q\setminus p$ is homeomorphic to $S_{0}^{m+3}$,
i.e., a genus zero surfaces with $m+3$ punctures. In $Q$ we choose blue and
red curves to form a chain as in Figure 14 and color them in blue and red so
that no two curves of the same color intersect. We denote the blue and red
curves in $Q$ by $\alpha^{\prime}$ and $\beta^{\prime}$ respectively. Remark
that the connected component of
$Q\smallsetminus(\alpha^{\prime}\cup\beta^{\prime})$ containing the point $p$
is a $2m-$polygon.

Figure 14. The multicurves $\alpha^{\prime}$ (blue) and $\beta^{\prime}$
(red).

The idea is to extend $\alpha^{\prime}$ and $\beta^{\prime}$ to multicurves
$\alpha$ and $\beta$, respectively, which satisfy all the desired properties.
We consider two cases for the rest of the proof.

$m$ even. Without loss of generality we suppose that all punctures in $Q$
different from $p$ are encircled by elements of $\alpha^{\prime}$. Now, let
$F$ be a connected component of $S\smallsetminus\alpha^{\prime}$ not
containing the point $p$. Then the closure $\overline{F}$ of $F$ in $S$ is a
surface with $b>0$ boundary components. Moreover,
$\overline{F}\cap\beta^{\prime}$ is a finite collection of disjoint essential
arcs in $\overline{F}$ with end points in $\partial\overline{F}$, and the end
points of an arc in $\overline{F}\cap\beta^{\prime}$ are in a common connected
component of $\partial\overline{F}$. We denote by $\theta_{F}$ the collection
of arcs in $\overline{F}\cap\beta^{\prime}$, and by $\delta_{F}$ to be the set
of curves in $\delta$ contained in $F$.

Claim: There exists a pair of multicurves $\alpha_{F}^{\prime\prime}$ and
$\beta_{F}^{\prime\prime}$ whose union fills $F$, which satisfy (1) & (2) in
Theorem 1.5 and such that $\theta_{F}\cap\beta_{F}^{\prime\prime}=\emptyset$.

Remark that if we define
$\alpha:=\alpha^{\prime}\bigcup\left(\bigcup_{F\subset
S\setminus\alpha^{\prime}}\alpha_{F}^{\prime\prime}\right)$ and
$\beta:=\beta^{\prime}\bigcup\left(\bigcup_{F\subset
S\setminus\alpha^{\prime}}\beta_{F}^{\prime\prime}\right)$, then $\alpha$ and
$\beta$ are the desired pair of multicurves.

We divided the proof of our claim in two cases: $b=1$ and $b>1$.

Case $b=1$. If $F$ is a finite-type surface is it not difficult to find the
multicurves $\alpha_{F}^{\prime\prime}$ and $\beta_{F}^{\prime\prime}$. If $F$
is of infinite-type let $\alpha^{\prime\prime}_{F}$ and
$\beta^{\prime\prime}_{F}$ be the blue and red curves obtained from applying
the first part of the proof of Theorem 1.5 to $F$. Remark that by construction
all arcs in $\theta_{F}$ intersect only one curve in
$\alpha_{F}^{\prime\prime}\cup\beta_{F}^{\prime\prime}$, hence, up to
interchanging the colors of $\alpha^{\prime\prime}_{F}$ and
$\beta_{F}^{\prime\prime}$, we can get that
$\beta_{F}^{\prime\prime}\cap\theta_{F}=\emptyset$.

Case $b>1$. Again, the case when $F$ is a finite-type surface is left to the
reader. If $F$ is of infinite type, let $\gamma$ be the separating curve in
$\overline{F}$ which bounds a subsurface $W\subset\overline{F}$ of genus 0
with one puncture, $b$ boundary components and such that $\partial
W=\partial\overline{F}$ and write $\overline{F}\smallsetminus\gamma=W\sqcup
F_{1}$. Let $\theta_{F_{1}}$ to be the set of arcs given by
$\theta_{F}\cap\overline{F_{1}}$. Let $\eta_{1}$ and $\eta_{2}$ be two curves
in $F_{1}$ (not necessary essential) such that $\gamma,\eta_{1}$ and
$\eta_{2}$ bounds a pair of pants $P$ in $F_{1}$. If an element of
$\theta_{F}$ intersects $\eta_{1}\cup\eta_{2}$ then we replace it with one
that doesn’t and which is disjoint from all other arcs in $\theta_{F}$. Up to
making these replacements, we can assume that $\theta_{F}$ does not intersects
$\eta_{1}\cup\eta_{2}$, see Figure 15. Hence,
$\theta_{F_{1}}\subseteq\overline{P}$. As $\overline{F_{1}}$ has one boundary
component, by the case $b=1$ above, there exist a pair of multicurves
$\alpha_{F_{1}}^{\prime\prime}$ and $\beta_{F_{1}}^{\prime\prime}$ which fills
$F_{1}$ and such that
$\theta_{F_{1}}\cap\beta_{F_{1}}^{\prime\prime}=\emptyset$. Define then
$\alpha_{F}^{{}^{\prime\prime}}:=\\{\gamma\\}\cup\alpha_{F_{1}}^{{}^{\prime\prime}}$
and $\beta_{F}^{{}^{\prime\prime}}:=\beta_{F_{2}}^{{}^{\prime\prime}}$.

Figure 15. Case $b>1$.

m odd. Without loss of generality we suppose that $\alpha^{\prime}$ encircles
all punctures in $Q$ different from $p$ except one. We add the curves
$\alpha_{1}$ and $\beta_{1}$ to $\alpha^{\prime}$ and $\beta^{\prime}$ as
depicted in Figure 16 respectively. Then we consider each connected component
$F$ of $S\setminus\alpha^{\prime}$ and proceed as in the preceding case. ∎

Figure 16. Case $m$ odd.

######  Remark 4.10.

If $\alpha$ and $\beta$ are multicurves as constructed in the proof of Theorem
1.5, then $S\setminus\alpha\cup\beta$ is a family of polygons, each of which
has either 2, 4, 6 or 8 sides. Hence, if $M=M(\alpha,\beta,\textbf{h})$ is
given by the Hooper-Thurston-Veech construction, then the set $\mathfrak{V}$
defined in the proof of Theorem 1.1 is formed by regular points and conic
singularities of total angle $\pi$, $3\pi$ or $4\pi$.

###  4.3. Proof of Corollary 1.6

Our arguments use Figure 17. Let $\alpha$ and $\beta$ be the multicurves in
blue and red illustrated in the figure; the union of these fills the surface
$S$ in question (a Loch Ness monster). Let us write
$\beta=\beta^{\prime}\sqcup\\{a_{i}\\}_{i\in\mathbb{N}}\sqcup\\{b_{i}\\}_{i\in\mathbb{N}}$,
and for each $n\in\mathbb{N}$ let
$\beta_{n}:=\beta^{\prime}\sqcup\\{a_{i}\\}_{i\geq n}\sqcup\\{b_{i}\\}_{i\geq
n}$. Theorem 1.1 implies that $f_{n}:=T_{\alpha}\circ T_{\beta_{n}}^{-1}$ acts
loxodromically on the loop graph $L(S;p)$ and hence it acts loxodromically on
the main component of the completed ray graph $\mathcal{R}(S;p)$ (see Theorem
2.2). Remark that $f_{n}$ converges to $f=T_{\alpha}\circ
T_{\beta^{\prime}}^{-1}$ in the compact-open topology. On the other hand, $f$
fixes the short rays $l$ and $l^{\prime}$ and hence it acts elliptically on
both $\mathcal{R}(S;p)$ and $L(S;p)$. ∎

######  Remark 4.11.

Using techniques similar to the ones presented in the proof of Theorem 1.5 one
can construct explicit sequences $(f_{n})$ as above for any infinite-type
surface $S$.

Figure 17.

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