# Infinite-type loxodromic isometries of the relative arc graph

Carolyn Abbott  Department of Mathematics
Brandeis University
Waltham, MA 02453<EMAIL_ADDRESS>,  Nicholas Miller  Department
of Mathematics
University of California, Berkeley
Berkeley, CA 94720<EMAIL_ADDRESS>and  Priyam Patel  Department of
Mathematics
University of Utah
Salt Lake City, UT 84112<EMAIL_ADDRESS>
###### Abstract.

An infinite-type surface $\Sigma$ is of type $\mathcal{S}$ if it has an
isolated puncture $p$ and admits shift maps. This includes all infinite-type
surfaces with an isolated puncture outside of two sporadic classes. Given such
a surface, we construct an infinite family of intrinsically infinite-type
mapping classes that act loxodromically on the relative arc graph
$\mathcal{A}(\Sigma,p)$. J. Bavard produced such an element for the plane
minus a Cantor set, and our result gives the first examples of such mapping
classes for all other surfaces of type $\mathcal{S}$. The elements we
construct are the composition of three shift maps on $\Sigma$, and we give an
alternate characterization of these elements as a composition of a pseudo-
Anosov on a finite-type subsurface of $\Sigma$ and a standard shift map. We
then explicitly find their limit points on the boundary of
$\mathcal{A}(\Sigma,p)$ and their limiting geodesic laminations. Finally, we
show that these infinite-type elements can be used to prove that
$\mathrm{Map}(\Sigma,p)$ has an infinite-dimensional space of quasimorphisms.

###### Contents

  1. 1 Introduction
  2. 2 Background
    1. 2.1 Space of ends and classification of infinite-type surfaces
    2. 2.2 Mapping class groups and arc graphs
    3. 2.3 Metric spaces and loxodromic isometries
    4. 2.4 Shift maps and the biinfinite flute surface
  3. 3 Coding arcs and standard position
    1. 3.1 Coding arcs
    2. 3.2 Standard position
      1. 3.2.1 Segments without back loops
      2. 3.2.2 Segments with back loops.
    3. 3.3 Gaps in segments
    4. 3.4 Taking inverses of segments
  4. 4 Loop Theorem
    1. 4.1 Cascading cancellation
    2. 4.2 The Loop Theorem
    3. 4.3 Consequences of the Loop Theorem
  5. 5 Constructing the homeomorphisms
  6. 6 Arcs that start like $\alpha_{i}^{(n)}$
  7. 7 Highways in arcs
    1. 7.1 Preliminaries on highways
    2. 7.2 Highways for the arcs $\alpha_{i}^{(n)}$
  8. 8 Loxodromic elements for $\mathcal{A}(\Sigma,p)$
    1. 8.1 The proof of Theorem 1.1
    2. 8.2 Alternate description of $g_{n}$
    3. 8.3 Geodesic axes
    4. 8.4 Limit points of the $g_{n}$
      1. 8.4.1 The Gromov boundary of $\mathcal{A}(\Sigma,p)$
      2. 8.4.2 The limit set of $g_{n}$
  9. 9 Infinite-type quasimorphisms
    1. 9.1 An intersection pairing on symmetric arcs and first properties
    2. 9.2 Production of “infinite-type” quasimorphisms
  10. 10 Convergence to a geodesic lamination
    1. 10.1 Geodesic laminations
    2. 10.2 Construction of the train track $\Theta$
    3. 10.3 The simple closed curves $c_{i}$
    4. 10.4 Proof of Theorem 1.2

##  1\. Introduction

A surface $\Sigma$ is of finite-type if $\pi_{1}(\Sigma)$ is finitely
generated, and otherwise $\Sigma$ is of infinite-type. Recently, there has
been a surge of interest in infinite-type surfaces and their mapping class
groups $\mathrm{Map}(\Sigma)$, which arise naturally in a variety of contexts
in low-dimensional topology, dynamics, and even descriptive set theory. See
[4] for a survey of recent results on infinite-type mapping class groups.

For finite-type surfaces $\Sigma$, Nielsen and Thurston [16, 22] give a
powerful classification of the elements of $\mathrm{Map}(\Sigma)$: every
element is periodic, reducible, or pseudo-Anosov. The action of
$\mathrm{Map}(\Sigma)$ by isometries on the (infinite-diameter and hyperbolic)
curve graph $\mathcal{C}(\Sigma)$ captures a coarser classification of the
elements of $\mathrm{Map}(\Sigma)$ since elements are either elliptic or
loxodromic. These two classifications, which are both interesting in their own
right, have a strong relationship; the loxodromic elements are exactly the
pseudo-Anosovs. In this way, the most interesting and complex mapping classes
correspond to the dynamically richest actions.

The situation for infinite-type surfaces is more complicated for a few
reasons. First, the exact analog of the Nielsen–Thurston classification is no
longer valid in this setting since some elements are neither periodic,
reducible, nor pseudo-Anosov in the traditional sense. Second, the curve graph
of an infinite-type surface has finite diameter unlike for finite-type
surfaces. This paper is motivated by one of the biggest open problems for
infinite-type surfaces, which is to give an analog of the Nielsen–Thurston
classification for infinite-type mapping classes. We work towards this goal by
studying the action of $\mathrm{Map}(\Sigma)$ on a different hyperbolic graph.

When $\Sigma$ is an infinite-type surface with at least one isolated puncture
$p$, the relative arc graph, $\mathcal{A}(\Sigma,p)$, plays the role of
$\mathcal{C}(\Sigma)$ and is defined as follows: the vertices correspond to
isotopy classes of simple arcs that begin and end at $p$ and edges connect
vertices for arcs admitting disjoint representatives. The subgroup
$\mathrm{Map}(\Sigma,p)$ of $\mathrm{Map}(\Sigma)$ that fixes the isolated
puncture $p$ acts on $\mathcal{A}(\Sigma,p)$ by isometries. This graph was
first defined by D. Calegari [11], who initiated its study by asking whether,
for the plane minus a Cantor set, this graph was infinite diameter and whether
any element of $\mathrm{Map}(\Sigma,p)$ acted loxodromically. In [5], J.
Bavard carried out Caelgari’s program for the plane minus a Cantor set and,
for that surface, showed that $\mathcal{A}(\Sigma,p)$ is both infinite-
diameter and hyperbolic. Aramayona–Fossas–Parlier [2] then showed that these
properties for $\mathcal{A}(\Sigma,p)$ hold more generally for any infinite-
type surface with at least one isolated puncture.

Given that the trichotomy of the Nielsen–Thurston classification does not
exactly hold for infinite-type surfaces, it is necessary to redefine
reducible, and therefore irreducible, mapping classes in this setting. One of
the most promising ways to motivate a new definition is to classify the
elements of infinite-type mapping class groups that are loxodromic with
respect to the action of $\mathrm{Map}(\Sigma)$ on a hyperbolic graph since
these elements correspond to infinite-order irreducibles in the finite-type
setting. In order to classify these elements, we must first construct them.

When $\Sigma$ is the sphere minus a Cantor set with an isolated puncture $p$
(i.e., $\Sigma$ is the plane minus a Cantor set), Bavard [5] constructed an
_intrinsically infinite-type_ mapping class that is loxodromic with respect to
the action of $\mathrm{Map}(\Sigma)$ on $\mathcal{A}(\Sigma,p)$, and for
several years, this was the only known such example. In this paper, we give a
new construction of mapping classes that are loxodromic with respect to the
action of $\mathrm{Map}(\Sigma,p)$ on the relative arc graph
$\mathcal{A}(\Sigma,p)$ for a large class of infinite-type surfaces.

######  Theorem 1.1.

For any surface $\Sigma$ of type $\mathcal{S}$, there is an infinite family of
intrinsically infinite-type homeomorphisms $\\{g_{n}\\}_{n\in\mathbb{N}}$ in
$\mathrm{Map}(\Sigma,p)$ such that each $g_{n}$ is loxodromic with respect to
the action of $\mathrm{Map}(\Sigma,p)$ on $\mathcal{A}(\Sigma,p)$.

\begin{overpic}[width=289.07999pt,trim=144.54pt 422.77994pt 90.3375pt
10.84006pt,clip={true},totalheight=202.3529pt]{shiftinaway1} \end{overpic}
Figure 1. A handleshift on an infinite-type surface (above) and a shift map on
an infinite-type surface (below).

Each mapping class in our construction is the composition of three
homeomorphisms called shift maps. Shift maps are generalizations of the
handleshift homeomorphisms constructed by the third author and N. Vlamis in
[17] (see Figure 1 for examples of both). Roughly, an infinite-type surface
$\Sigma$ with an isolated puncture $p$ is of type $\mathcal{S}$ if it there is
a proper embedding of the biinfinite flute surface containing $p$ into
$\Sigma$ such that certain shift maps on the flute surface induce shift maps
on $\Sigma$. See Section 2.4 for more details and Figure 3 for some examples
of surfaces of type $\mathcal{S}$. In Lemma 2.7, we show that a surface with
an isolated puncture is of type $\mathcal{S}$ if and only if it admits shift
maps. This set of surfaces consists of all infinite-type surfaces with an
isolated puncture except a flute surface with finite (possibly zero) genus and
a fluted Loch Ness monster. We call these two classes sporadic surfaces in
this context. See Figure 2 for examples of sporadic surfaces and Lemma 2.9 for
a proof of this fact. Since sporadic surfaces are exactly the small class of
surfaces with an isolated puncture that do not admit shift maps, different
methods will need to be developed in order to prove an analogue of Theorem 1.1
for these surfaces. It would be interesting to understand how elements of the
mapping class groups of sporadic surfaces act on the relative arc graph.

\begin{overpic}[trim=90.3375pt 542.025pt 18.06749pt
108.405pt,clip={true},width=289.07999pt]{sporadicsurf} \end{overpic} Figure 2.
Examples of sporadic infinite-type surfaces that are not of type
$\mathcal{S}$. The first is a flute with finite genus, the other three are
fluted Loch Ness Monster surfaces. \begin{overpic}[trim=130.08621pt
252.94499pt 83.11005pt
10.84006pt,clip={true},totalheight=265.58916pt,width=289.07999pt]{typeSsurf1}
\put(37.0,92.0){$p$} \put(37.0,58.0){$p$} \put(37.0,20.0){$p$} \end{overpic}
Figure 3. Examples of surfaces of type $\mathcal{S}$, the first of which is
the biinfinite flute surface $S$ itself.

The handleshift homeomorphisms mentioned above have proven to be crucial in
understanding various aspects of infinite-type mapping class groups. For
example, it is shown in [17] that they are needed to topologically generate
the pure mapping class group whenever $\Sigma$ has at least two non-planar
ends, and in [3] they are used to show that the pure mapping class groups of
such surfaces surject onto $\mathbb{Z}$. With this paper, we emphasize the
importance of more general shift maps to the theory of infinite-type mapping
class groups. Inspired by Bavard’s work in [5], we choose the shift maps in
our construction carefully so that their composition mimics some of the
behavior of pseudo-Anosov maps in the finite-type setting. In fact, we show
that there is an alternate description of our homeomorphisms as the
composition of a pseudo-Anosov homeomorphism on a finite-type subsurface and a
standard shift map on $\Sigma$ in Theorem 8.5. Additionally, in Section 10, we
use the work of D. Saric [21] to prove the following theorem regarding
geodesic laminations for the mapping classes constructed in Theorem 1.1.

######  Theorem 1.2.

If $\Sigma$ is a surface of type $\mathcal{S}$ equipped with its conformal
hyperbolic metric that is equal to its convex core, then there exists a simple
closed curve $c_{0}$ on $\Sigma$ such that the sequence
$(g_{n}^{i}(c_{0}))_{i\in\mathbb{N}}$ converges to a geodesic lamination on
$\Sigma$.

In particular, we produce a train track on $\Sigma$ and show the geodesic
lamination from this theorem is weakly carried by this train track.

We emphasize that the elements arising from our construction are of
intrinsically infinite-type, that is, they do not lie in the closure of the
compactly supported mapping class group $\overline{\mathrm{Map}_{c}(\Sigma)}$,
where the closure is taken with respect to the compact-open topology on
$\mathrm{Map}(\Sigma)$. These are the first such examples for all surfaces of
type $\mathcal{S}$ outside of the plane minus a Cantor set. Additionally, we
emphasize that our construction does not rely on, and is not a generalization
of, one of the few known methods for constructing pseudo-Anosov mapping
classes for finite-type surfaces.

The most obvious candidates for mapping classes that are loxodromic with
respect to the action on $\mathcal{A}(\Sigma,p)$ are those that are pseudo-
Anosov on a finite-type subsurface $\Sigma^{\prime}\subset\Sigma$ containing
the special puncture $p$, that extend via the identity map to the rest of
$\Sigma$ (these are compactly supported mapping classes). In [7], Bavard and
Walker prove that these types of mapping classes do indeed act loxodromically
on a graph that is quasi-isometric to $\mathcal{A}(\Sigma,p)$. In that paper
they point out that, though their class of examples is interesting, it will be
even more important to construct mapping classes of intrinsically infinite-
type that act loxodromically on $\mathcal{A}(\Sigma,p)$; this remark was one
of the main points of inspiration for writing this paper. The intrinsically
infinite-type elements of $\mathrm{Map}(\Sigma)$ are more mysterious since
tools from finite-type surface theory do not directly generalize when studying
these elements.

######  Remark 1.3.

Morales and Valdez [15] have also produced non-compactly supported elements
that are loxodromic, but their elements are in the closure of the compactly
supported mapping class group. Their method is a generalization of the
Thurston–Veech construction of pseudo-Anosovs in the finite-type setting.

Aside from the motivation provided by a Nielsen–Thurston classification for
infinite-type mapping classes, Bestvina–Fujiwara [9] show that constructing
elements of $\mathrm{Map}(\Sigma)$ that act loxodromically on hyperbolic
graphs can be used to understand the second bounded cohomology
$H^{2}_{b}(\mathrm{Map}(\Sigma),\mathbb{R})$ of $\mathrm{Map}(\Sigma)$. In
particular, they show that, for a compact surface $\Sigma$, there exist
elements acting loxodromically on $\mathcal{C}(\Sigma)$ that are weakly
properly discontinuous (WPD). These elements are used to prove that the space
of quasimorphisms of $\mathrm{Map}(\Sigma)$ is infinite-dimensional, which is
sufficient to conclude that $H^{2}_{b}(\mathrm{Map}(\Sigma),\mathbb{R})$ is,
as well.

Along these lines, M. Bestvina asked the following question at the AIM
workshop on infinite-type surfaces [1, Problem 4.7]: “For
$\Sigma=\mathbb{R}^{2}-C$ (where $C$ is a Cantor set) is it true that every
subgroup of $\mathrm{Map}(\Sigma)$ has either infinite-dimensional space of
quasimorphisms or is amenable?” More generally, we would like to characterize
the infinite-type mapping classes that can be used to produce quasimorphisms
of $\mathrm{Map}(\Sigma)$. In Section 9, we show that the elements constructed
in Theorem 1.1 can be used to give a new proof of the following theorem,
originally due to Bavard [5] in the case of a plane minus a Cantor set and
Bavard and Walker [7] in the general case.

######  Theorem 1.4.

Let $\Sigma$ be a surface of type $\mathcal{S}$. The space of non-trivial
quasimorphisms on $\mathrm{Map}(\Sigma,p)$ is infinite dimensional.

In [7], Bavard and Walker use a weaker condition on loxodromic isometries
introduced by Bestvina–Bromberg–Fujiwara [8], called WWPD, to show that
homeomorphisms that are pseudo-Anosov on finite-type subsurfaces
$\Sigma^{\prime}$ and extend via the identity to the rest of $\Sigma$ can be
used to produce quasimorphisms of $\mathrm{Map}(\Sigma)$. A. Rasmussen shows
in [19] that for a surface $\Sigma$ with an isolated puncture $p$, an element
of $\mathrm{Map}(\Sigma,p)$ is WWPD with respect to the action on
$\mathcal{A}(\Sigma,p)$ if and only if it stabilizes a finite-type subsurface
$\Sigma^{\prime}$ containing the puncture $p$ and restricts to a pseudo-Anosov
on $\Sigma^{\prime}$. The elements we construct in Theorem 1.1 do not fix any
finite-type subsurface and thus are not WWPD. Despite this, we are still able
to build non-trivial quasimorphisms using a criterion of Bestvina and Fujiwara
[9] and an approach similar to that of Bavard in [5] which involves defining
an intersection pairings on a specific class of arcs on $\Sigma$. Our
construction gives subgroups of $\mathrm{Map}(\Sigma)$ that do not contain
WWPD elements but do have an infinite dimensional space of quasimorphisms.

Plan of the paper: In order to prove Theorem 1.1, we explicitly compute the
images of a particular arc on $\Sigma$ under iterates of each homeomorphism
$g_{n}$ and prove that these images form a quasi-geodesic axis for the action
of $\langle g_{n}\rangle$ on $\mathcal{A}(\Sigma,p)$. Though some of the
methods in our paper are inspired by Bavard’s work in [5], we note that there
are a variety of additional challenges in proving Theorem 1.1 for such a wide
class of surfaces. In fact, we first prove the theorem for the biinfinite
flute surface $S$ and then use the fact that the inclusion of
$\mathcal{A}(S,p)$ into $\mathcal{A}(\Sigma,p)$ is a $(2,0)$–quasi-isometric
embedding (see Lemma 2.10) to extend the theorem to all surfaces of type
$\mathcal{S}$. One of the first challenges in proving Theorem 1.1 is
rigorously coding arcs on $S$, which we do in Section 3.1, in order to
quantify how long two arcs on $S$ fellow travel. We then introduce standard
position for an arc on $S$ in Section 3.2 so that we can use the code for an
arc to find its image under our shift maps in a well-defined way. Most
importantly, we must understand when segments of arcs become trivial under our
shift maps, and in Section 4.1 we introduce a kind of cancellation in the
image of the code for a segment which we call cascading cancellation. This
kind of cancellation will cause technical problems throughout the paper and
much of Section 6 is devoted to understanding how to control it.

The rest of Section 4 is devoted to proving Theorem 4.8 (The Loop Theorem)
which answers the question of when a segment in an arc becomes trivial under
our shift maps. We define the homeomorphisms $g_{n}$ of Theorem 1.1 in Section
5, show that we have “starts like” functions in Section 6, and show that we
have highways in Section 7. Finally, we prove Theorem 1.1 in Section 8,
introduce an intersection pairing for arcs and prove Theorem 1.4 in Section 9,
and prove the convergence to a geodesic lamination from Theorem 1.2 in Section
10.

Acknowledgements: The authors are grateful to Chris Leininger for pointing out
the work of Verberne and to Ekta Patel for helping us with computational work
related to this project. The authors would also like to thank Alden Walker and
Nick Vlamis for helpful discussions as well as Elizabeth Field for helpful
comments on an earlier draft. The first author was partially supported by NSF
DMS–1803368 and DMS–2106906, the second author was partially supported by NSF
DMS–1408458, and the third author was partially supported by NSF DMS–1840190
and DMS–2046889. The authors would also like to thank the American Institute
of Mathematics and the organizers of the Infinite-type Surfaces Workshop that
was held there, during which some of this work was completed.

##  2\. Background

###  2.1. Space of ends and classification of infinite-type surfaces

Central to the classification of infinite-type surfaces is the definition of
the space of ends $E(\Sigma)$ of an infinite-type surface $\Sigma$.
Informally, an end of $\Sigma$ is a way to escape or go off to infinity in
$\Sigma$. More formally we have:

######  Definition 2.1.

An exiting sequence in $\Sigma$ is a sequence $\\{U_{n}\\}_{n\in\mathbb{N}}$
of connected open subsets of $\Sigma$ satisfying:

  1. (1)

$U_{n}\subset U_{m}$ whenever $m<n$;

  2. (2)

$U_{n}$ is not relatively compact for any $n\in\mathbb{N}$, that is, the
closure of $U_{n}$ in $\Sigma$ is not compact;

  3. (3)

the boundary of $U_{n}$ is compact for each $n\in\mathbb{N}$; and

  4. (4)

any relatively compact subset of $\Sigma$ is disjoint from all but finitely
many of the $U_{n}$’s.

Two exiting sequences $\\{U_{n}\\}_{n\in\mathbb{N}}$ and
$\\{V_{n}\\}_{n\in\mathbb{N}}$ are equivalent if for every $n\in\mathbb{N}$
there exists $m\in\mathbb{N}$ such that $U_{m}\subset V_{n}$ and $V_{m}\subset
U_{n}$. An end of $\Sigma$ is an equivalence class of exiting sequences.

The space of ends $E(\Sigma)$, or simply $E$, of $\Sigma$ is the set of ends
of $\Sigma$ equipped with a natural topology for which it is totally
disconnected, Hausdorff, second countable, and compact. In particular,
$E(\Sigma)$ is homeomorphic to a closed subset of the Cantor set. To describe
the topology, let $V$ be an open subset of $\Sigma$ with compact boundary,
define $\widehat{V}=\left\\{\left[\\{U_{n}\\}_{n\in\mathbb{N}}\right]\in
E:U_{n}\subset V\text{ for some }n\in\mathbb{N}\right\\}$ and let
$\mathcal{V}=\\{\widehat{V}:V\subset\Sigma\text{ is open with compact
boundary}\\}$. The set $E$ becomes a topological space by declaring
$\mathcal{V}$ a basis for the topology.

We note that ends can be isolated or not and can be planar (if there exists an
$i$ such that $U_{i}$ is homeomorphic to an open subset of the plane
$\mathbb{R}^{2}$) or nonplanar (if every $U_{i}$ has infinite genus). The set
of nonplanar ends of $\Sigma$ is a closed subspace of $E(\Sigma)$ and will be
denoted by $E_{g}(\Sigma)$.

Kerékjártó [24] and Richards [20] showed that the homeomorphism type of an
orientable infinite-type surface is determined by the quadruple

$(g,b,E_{g}(\Sigma),E(\Sigma))$

where $g\in\mathbb{Z}_{\geq 0}\cup\\{\infty\\}$ is the genus of $\Sigma$ and
$b\in\mathbb{Z}_{\geq 0}$ is the number of (compact) boundary components of
$\Sigma$.

Of particular interest to us is the infinite-type surface called the
biinfinite flute obtained from an infinite cylinder by deleting a countable
discrete sequence of points exiting both ends of the cylinder (see Figure 3).
By the classification theorem of Kerékjártó and Richards, this surface can
also be obtained from $\mathbb{S}^{2}$ by deleting $\\{x_{i}\\}$,
$\\{y_{i}\\}$, $x$, and $y$, where $\\{x_{i}\\}$ and $\\{y_{i}\\}$ are
countable discrete sequences of points converging to distinct points $x$ and
$y$, respectively. Note that $S$ has two special non-isolated ends.

###  2.2. Mapping class groups and arc graphs

The mapping class group, $\mathrm{Map}(\Sigma)$, of a surface $\Sigma$ is the
group of orientation-preserving homeomorphisms of $\Sigma$ up to isotopy. The
natural topology on any group of homeomorphisms is the compact-open topology
and $\mathrm{Map}(\Sigma)$ is endowed with the quotient topology with respect
to the compact-open topology on the space of homeomorphisms of $\Sigma$. When
$\Sigma$ is a finite-type surface, this topology agrees with the discrete
topology on $\mathrm{Map}(\Sigma)$, but when $\Sigma$ is of infinite type it
does not. There are several important subgroups of $\mathrm{Map}(\Sigma)$:
$\mathrm{Map}_{c}(\Sigma)$ is the subgroup consisting of mapping classes with
compact support, $\operatorname{PMap}(\Sigma)$ is the pure mapping class group
consisting of mapping classes which fix the set of ends pointwise,
$\overline{\mathrm{Map}_{c}(\Sigma)}<\operatorname{PMap}(\Sigma)$ is the
closure of the compactly supported mapping class group with respect to the
topology described above, and when $\Sigma$ has an isolated puncture $p$,
$\mathrm{Map}(\Sigma,p)$ is the subgroup of mapping classes that fix $p$.

When $\Sigma$ is finite-type, $\mathrm{Map}(\Sigma)$ is algebraically
generated by finitely many Dehn twists [14]. Infinite-type mapping class
groups, sometimes called big mapping class groups, are uncountable groups, so
there is no countable algebraic generating set. However, one can consider
topological generating sets (countable dense subsets of
$\mathrm{Map}(\Sigma)$) and in [17], Vlamis and the third author prove that
for many infinite-type surfaces, Dehn twists are not sufficient in
topologically generating even $\operatorname{PMap}(\Sigma)$. They show that in
addition to Dehn twists, a new class of homeomorphisms called handleshifts
(defined in Section 2.4) are often needed to topologically generate
$\operatorname{PMap}(\Sigma)$. In a subsequent paper with Aramayona [3],
Vlamis and the third author give an algebraic description of
$\operatorname{PMap}(\Sigma)$ that will be relevant in Section 8. When
$\Sigma$ is an infinite-type surface with $n>1$ nonplanar ends, they prove
that
$\operatorname{PMap}(\Sigma)=\overline{\mathrm{Map}_{c}(\Sigma)}\rtimes\mathbb{Z}^{n-1}$,
where $\mathbb{Z}^{n-1}$ is generated by $n-1$ handleshifts with disjoint
support. In particular, when $\Sigma$ has exactly 2 nonplanar ends (for
example when $\Sigma$ is the ladder surface),
$\operatorname{PMap}(\Sigma)=\overline{\mathrm{Map}_{c}(\Sigma)}\rtimes\mathbb{Z}$
where $\mathbb{Z}=\langle H\rangle$ and $H$ is the standard handleshift,
shifting each genus of $\Sigma$ over to the right by one.

In this paper we are primarily concerned with mapping classes of intrinsically
infinite-type.

######  Definition 2.2.

An element $f\in\mathrm{Map}(\Sigma)$ is of intrinsically infinite-type if
$f\notin\overline{\mathrm{Map}_{c}(\Sigma)}$.

More specifically, we are interested in how such elements act on a particular
graph of arcs called the relative arc graph.

Let $\Sigma$ be a connected, orientable surface with empty boundary, and let
$\Pi\subset\Sigma$ be the set of punctures of $\Sigma$, which we assume to be
non-empty. In this subsection, it is convenient to regard $\Pi$ as a set of
marked points on $\Sigma$. By a proper arc on $\Sigma$ we mean a map
$\alpha\colon[0,1]\to\Sigma$ such that $\alpha^{-1}(\Pi)=\\{0,1\\}$. We often
conflate an arc with its image in $\Sigma$. An arc is simple if it is an
embedding when restricted to the open interval $(0,1)$.

The arc graph $\mathcal{A}(\Sigma)$ is the simplicial graph whose vertices are
isotopy classes of simple arcs on $\Sigma$, where we only consider isotopies
rel endpoints, and two (isotopy classes of) arcs are connected by an edge if
they can be realized disjointly away from $\Pi$. The mapping class group
$\mathrm{Map}(\Sigma)$ acts on $\mathcal{A}(\Sigma)$ by isometries. Hensel,
Przytycki, and Webb [12] show that when $\Sigma$ has finite-type, the graph
$\mathcal{A}(\Sigma)$ is infinite diameter and 7–hyperbolic. On the other
hand, when $\Sigma$ is infinite-type with infinitely many punctures, it is
straight-forward to see that $\mathcal{A}(\Sigma)$ has diameter 2, and so this
graph is not particularly useful for studying $\mathrm{Map}(\Sigma)$.

Assuming that $\Pi$ contains a non-empty set of isolated punctures, Aramayona,
Fossas, and Parlier [2] construct a particular subgraph of the arc graph which
has interesting geometry, even when $\Pi$ is infinite. We are interested in a
special case of this construction, involving a single isolated puncture $p$.

######  Definition 2.3.

The relative arc graph $\mathcal{A}(\Sigma,p)$ is the subgraph of
$\mathcal{A}(\Sigma)$ spanned by arcs which start and end at $p$. More
precisely, the vertices of $\mathcal{A}(\Sigma,p)$ are isotopy classes of arcs
on $\Sigma$ with endpoints on $p$, where we allow only isotopy rel endpoints.
There is an edge between two (isotopy classes of) arcs if they can be realized
disjointly away from $p$.

Aramayona, Fossas, and Parlier show that $\mathcal{A}(\Sigma,p)$ is connected,
has infinite diameter, and is $7$–hyperbolic (see [2, Theorem 1.1]). While
$\mathrm{Map}(\Sigma)$ does not necessarily act on $\mathcal{A}(\Sigma,p)$,
the subgroup $\mathrm{Map}(\Sigma,p)$ that fixes the puncture $p$ does act by
isometries on this graph. When $\Sigma$ has only one isolated puncture $p$,
$\mathrm{Map}(\Sigma)=\mathrm{Map}(\Sigma,p)$.

###  2.3. Metric spaces and loxodromic isometries

We now introduce some basics of metric spaces and isometries of a hyperbolic
metric space. Given a metric space $X$, we denote by $d_{X}$ the distance
function on $X$. A map $f\colon X\to Y$ between metric spaces $X$ and $Y$ is a
$(K,C)$–quasi-isometric embedding if there is are constants $K\geq 1$, $C\geq
0$ such that for all $x,y\in X$,

$\frac{1}{K}d_{X}(x,y)-C\leq d_{Y}(f(x),f(y))\leq Kd_{X}(x,y)+C.$

A geodesic in $X$ is an isometric embedding of an interval into $X$ and a
$(K,C)$–quasi-geodesic in $X$ is a $(K,C)$–quasi-isometric embedding of an
interval into $X$. We call the constants $K,C$ the quality of the quasi-
geodesic. By an abuse of notation, we often conflate a (quasi-)geodesic and
its image in $X$.

######  Definition 2.4.

Given an action by isometries of a group $G$ on a hyperbolic space $X$, an
element $g\in G$ is elliptic if it has bounded orbits; loxodromic if the map
$\mathbb{Z}\to X$ given by $n\mapsto g^{n}x_{0}$ for some (equivalently, any)
$x_{0}\in X$ is a quasi-isometric embedding; and parabolic otherwise.

Any bi-infinite quasi-geodesic in $X$ which is preserved by a loxodromic
isometry $g\in G$ is called an axis of $g$. An axis always exists; for any
$x_{0}\in X$, the set $\\{g^{n}x_{0}\mid n\in\mathbb{Z}\\}$ is a (discrete)
quasi-geodesic preserved by $g$. If $X$ is a geodesic metric space, in the
sense that there exists a geodesic connecting any two points of $X$, then we
may construct a continuous quasi-geodesic axis as follows. Fix a geodesic
$[x_{0},gx_{0}]$ from $x_{0}$ to $gx_{0}$. Then $g$ stabilizes the path formed
by concatenating the geodesics $g^{n}[x_{0},gx_{0}]$; this path is a quasi-
geodesic axis of $g$ in $X$. Varying the point $x_{0}$ will change the quality
of the quasi-geodesic. Let $g^{+}=\lim_{n\to\infty}g^{n}x_{0}$ and
$g^{-}=\lim_{n\to-\infty}g^{n}x_{0}$ be points in the Gromov boundary
$\partial X$ of $X$. The limit set of $\langle g\rangle$ is the subset
$\\{g^{+},g^{-}\\}\subseteq\partial X$; this set is fixed pointwise by $g$. It
is straightforward to show that the limit set $\\{g^{+},g^{-}\\}$ does not
depend on the choice of $x_{0}\in X$.

###  2.4. Shift maps and the biinfinite flute surface

A handleshift was first defined in [17] as follows. Consider the surface
$S^{\prime}$ defined by taking the strip $\mathbb{R}\times[-1,1]$, removing a
disk of radius $\frac{1}{2}$ with center $(n,0)$ for each $n\in\mathbb{Z}$,
and attaching a torus with one boundary component to the boundary of each such
disk. A handleshift on $S^{\prime}$ is the homeomorphism that acts like a
translation, sending $(x,y)$ in $S$ to $(x+1,y)$ and which tapers to the
identity on $\partial S^{\prime}$. Given a surface of infinite-genus $\Sigma$
with at least two nonplanar ends and a proper embedding of $S^{\prime}$ into
$\Sigma$ so that the two ends of the strip correspond to two distinct ends of
$\Sigma$, the handleshift on $S^{\prime}$ induces a handleshift on $\Sigma$,
where the homeomorphism acts as the identity on the complement of
$S^{\prime}$. In this paper, more flexibility is allowed, and we define the
following generalization.

######  Definition 2.5.

Let $S^{\prime}$ be the surface defined by taking the strip
$\mathbb{R}\times[-1,1]$, removing a closed disk of radius $\frac{1}{4}$ with
center $(n,0)$ for $n\in\mathbb{Z}$, and attaching any fixed topologically
non-trivial surface with exactly one boundary component to the boundary of
each such disk. A shift on $S^{\prime}$ is the homeomorphism that acts like a
translation, sending $(x,y)$ in $S^{\prime}$ to $(x+1,y)$ and which tapers to
the identity on $\partial S^{\prime}$.

Lanier and Loving use two particular cases of this generalization in [13].
Naming the full generalization a “shift” is in line with their paper. Note
that it is essential for the same surface to be glued to the boundary
component of each disk in order for the shift to be a homeomorphism of the
surface.

As above, given a surface $\Sigma$ with a proper embedding of $S^{\prime}$
into $\Sigma$ so that the two ends of the strip correspond to two different
ends of $\Sigma$, the shift on $S^{\prime}$ induces a shift on $\Sigma$, where
the homeomorphism acts as the identity on the complement of $S^{\prime}$.
Given a shift $h$ on $\Sigma$, the embedded copy of $S^{\prime}$ in $\Sigma$
is called the domain of $h$. In this paper, we produce special homeomorphisms
that can be obtained as a composition of three shift maps on such a surface
$\Sigma$ with an isolated puncture $p$ and that are loxodromic with respect to
the action of $\mathrm{Map}(\Sigma,p)$ on $\mathcal{A}(\Sigma,p)$. Instead of
working generally with surfaces that admit shift maps, we begin by letting $S$
be the biinfinite flute surface. Then, $S$ admits shift maps which shift a
countable collection of punctures on $S$. To prove Theorem 1.1 we first
construct mapping classes that are loxodromic with respect to the action of
$\mathrm{Map}(S,p)$ on $\mathcal{A}(S,p)$. We then use this surface as a
template for constructing the desired mapping classes for more general
surfaces $\Sigma$ by extending the shift maps on $S$ to shift maps on $\Sigma$
as follows.

######  Definition 2.6.

Let $S$ be the biinfinite flute surface. A surface $\Sigma$ with an isolated
puncture $p$ is of type $\mathcal{S}$ if there exists a proper embedding
$S\hookrightarrow\Sigma$ where $S$ contains $p$, the two non-isolated ends of
$S$ correspond to distinct ends of $\Sigma$, and such that a countably
infinite collection of connected components of $\Sigma\setminus S$ are of the
same (nontrivial) topological type. Note that when the components are once-
punctured disks, there are countably many isolated punctures of $S$ that
remain isolated punctures when embedded in $\Sigma$. Denote this special class
of connected components of $\Sigma\setminus S$ by $\mathcal{U}$, so that the
elements of $\mathcal{U}$ are all homeomorphic to a fixed surface $\Sigma_{0}$
with one boundary component. See Figure 1 for some examples of surfaces of
type $\mathcal{S}$.

Given a shift map $h$ on $S$, the support of $h$ is a strip
$\mathbb{R}\times[-1,1]$ with countably many punctures. When the set of
punctures in the support of $h$ only consists of those corresponding to
elements of $\mathcal{U}$, we can glue copies of $\Sigma_{0}$ onto the
punctures of this strip to produce a shift map on a surface $S^{\prime}$ as in
Definition 2.5. The embedding of $S$ in $\Sigma$ therefore gives an embedding
of $S^{\prime}$ in $\Sigma$ and the shift on $S^{\prime}$ in $\Sigma$ is
extended via the identity on $\Sigma\setminus S^{\prime}$ as usual. From this
construction, we immediately have one direction of the following lemma.

######  Lemma 2.7.

Given a surface $\Sigma$ with an isolated puncture, $\Sigma$ is of type
$\mathcal{S}$ if and only if $\Sigma$ admits shift maps.

###### Proof.

It is left to show that if $\Sigma$ has an isolated puncture $p$ and admits a
shift map, then $\Sigma$ is of type $\mathcal{S}$. To see this, we consider
the proper embedding of $S^{\prime}$ into $\Sigma$. Recall that $S^{\prime}$
is obtained from a punctured strip by gluing on countably many copies of any
surface $\Sigma_{0}$ with exactly one boundary component. Let
$\mathcal{T}=\\{T_{i}\\}$ denote the corresponding countable collection of
subsurfaces homeomorphic to $\Sigma_{0}$ in $\Sigma$, indexed by $\mathbb{Z}$.

Note that $E(S^{\prime})$ is a closed subset of $E(\Sigma)$, as is
$X=E(S^{\prime})\cup\\{p\\}$, and thus $X^{c}=E(\Sigma)\setminus X$ is open in
$E(\Sigma)$. The second countability of the topology on $E(\Sigma)$ implies
that $X^{c}$ is the union of countably many basis elements. If $X^{c}$ is in
fact clopen in $E(\Sigma)$, then $X^{c}$ is compact and is therefore a finite
union of basis elements. In this case, there exists a simple closed curve
$\gamma$ in $\Sigma$ with the following property: there exists a connected
component $K$ of $\Sigma\setminus\gamma$ such that the end space of
$\overline{K}=K\cup\gamma$ is exactly $X^{c}$. In this way, $\gamma$ cuts away
the ends of $\Sigma$ that are in $X^{c}$ (see Figure 4). We then have that

$\Sigma\setminus\left(\overline{K}\,\bigcup\,\left(\cup_{i}T_{i}\right)\right)$

is homeomorphic to the biinfinite flute surface and $\Sigma$ is of type
$\mathcal{S}$ with $\mathcal{T}$ playing the role of $\mathcal{U}$ in the
definition of a surface of type $\mathcal{S}$.

In general, we can only assume that $X^{c}$ is open, not clopen, so that it
can be expressed as the union of countably many basis elements for the
topology. Then, there exists a countable collection of simple closed curves
$\\{\gamma_{i}\\}$ and a countable collection of connected components $K_{i}$
of $\Sigma\setminus\gamma_{i}$ such that the end space of
$\bigcup_{i}\overline{K}_{i}=\bigcup_{i}K_{i}\cup\gamma_{i}$ is exactly
$X^{c}$ (see Figure 4). In this case,

$\Sigma\setminus\left(\left(\cup_{i}\overline{K}_{i}\right)\,\bigcup\,\left(\cup_{i}T_{i}\right)\right)$

is homeomorphic to the biinfinite flute surface and $\Sigma$ is of type
$\mathcal{S}$. ∎

\begin{overpic}[trim=130.08621pt 252.94499pt 83.11005pt
155.38005pt,clip={true},totalheight=278.24104pt]{typeSsurf1purple}
\put(39.0,78.0){$p$} \put(39.0,27.5){$p$} \put(50.0,78.0){\LARGE{$\gamma$}}
\put(31.0,21.0){$\gamma_{4}$} \put(25.0,25.0){\tiny{$\gamma_{3}$}}
\put(22.0,25.0){\tiny{$\gamma_{2}$}} \put(18.0,25.0){\tiny{$\gamma_{1}$}}
\put(14.0,25.0){\tiny{$\gamma_{0}$}} \put(9.0,25.0){\tiny{$\gamma_{-1}$}}
\put(60.0,22.5){\small{$\gamma_{5}$}} \put(71.0,22.0){\small{$\gamma_{6}$}}
\put(80.0,22.0){\small{$\gamma_{7}$}} \end{overpic} Figure 4. Examples showing
that surfaces with shift maps are always of type $\mathcal{S}$. For the first
surface, only one curve $\gamma$ is needed to cut away the extra topology of
$\Sigma$. In the second case, a countable collection of curves
$\\{\gamma_{i}\\}$ is needed.

Given this equivalent definition for a surface of type $\mathcal{S}$, we can
show that this class includes all infinite-type surfaces with an isolated
puncture outside of two sporadic classes that do not admit shift maps. We will
need the following definition.

######  Definition 2.8.

The Loch Ness Monster is the infinite-type surface with no planar ends and
exactly one non-planar end. An infinite-type surface is a fluted Loch Ness
Monster if it is obtained from the Loch Ness Monster in one of the two
following ways: 1) by deleting a finite, non-zero collection of isolated
points, or 2) deleting a countably infinite collection of isolated points
accumulating to exactly one point, which we also delete from the surface, or
accumulating onto the end of the Loch Ness Monster. See Figure 2 for examples
of fluted Loch Ness Monsters.

######  Lemma 2.9.

Let $\Sigma$ be an infinite-type surface with an isolated puncture. Then
$\Sigma$ is of type $\mathcal{S}$ unless $\Sigma$ is a flute surface with
finite (possibly zero) genus or is a fluted Loch Ness Monster surface.

###### Proof.

Let $\Sigma$ be an infinite-type surface with an isolated puncture $p$. If
$\Sigma$ has at least two non-planar ends, then $\Sigma$ admits a shift map
(in fact a handleshift). Similarly, if $\Sigma$ has at least two non-isolated
planar ends, then $\Sigma$ admits a shift map with these two ends
corresponding to the two ends of the strip $S^{\prime}$ in Definition 2.5.
Thus, if $\Sigma$ does not admit a shift map, $\Sigma$ has exactly one non-
isolated planar end and finite genus, i.e., a flute surface with finite genus,
or has exactly one non-planar end and up to one non-isolated planar end, i.e.,
a fluted Loch Ness Monster. ∎

Going back to the original definition of a surface of type $\mathcal{S}$,
there are a few more notable remarks regarding the relationship between $S$
and $\Sigma$. First, there is not necessarily an embedding of
$\mathrm{Map}(S,p)$ into $\mathrm{Map}(\Sigma,p)$ since if the support of a
shift $h$ on $S$ contains punctures that do not correspond to elements of
$\mathcal{U}$, then there may not be a way to extend that shift to $\Sigma$.
In particular, if $h$ shifts one puncture $x$ to another puncture $x^{\prime}$
but the topology of the surfaces glued to $x$ and $x^{\prime}$ are different,
there is no extension of $h$ to a shift of $\Sigma$. This will not affect our
arguments since there are countably many punctures of $S$ corresponding to the
elements of $\mathcal{U}$ which we move to the front of the cylinder for $S$
along with $p$, and we move all other punctures to the back of the cylinder.
Here we are choosing one non-isolated end of $S$ to correspond to the left
direction on the surface, and the other non-isolated end to correspond to
moving right on the surface so that there is a well-defined notion of the
front and back of $S$. In our constructions, we use shift maps on $S$ whose
support only contains the punctures on the front of $S$ so that all of these
shifts extend to $\Sigma$.

Second, and most importantly, we now show that proving Theorem 1.1 for
surfaces $\Sigma$ of type $\mathcal{S}$ can be reduced to the case of the
biinfinite flute $S$. In fact, this is the motivation for the original
definition of a surface of type $\mathcal{S}$. For simplicity, given a surface
$M$ with an isolated puncture $p$ and any points $a,b\in\mathcal{A}(M,p)$, we
write $d_{M}(a,b)$ for the distance between $a$ and $b$ in $\mathcal{A}(M,p)$

######  Lemma 2.10.

Let $\Sigma$ be an infinite-type surface with an isolated puncture $p$ of type
$\mathcal{S}$. Then the inclusion of $\mathcal{A}(S,p)$ into
$\mathcal{A}(\Sigma,p)$ is a $(2,0)$–quasi-isometric embedding.

###### Proof.

As $S\subset\Sigma$, it is clear that $d_{\Sigma}(a,b)\leq d_{S}(a,b)$ for any
$a,b\in\mathcal{A}(S,p)$.

To obtain the other inequality, let $S_{a,b}\subset S$ be a finite-type
subsurface of $S$ which contains $a,b$, the puncture $p$, and has complexity
at least $2$. Note that $S_{a,b}$ is then a finite-type subsurface of $\Sigma$
as well. Thus by [2, Corollary 4.3] applied to $S_{a,b}\subset\Sigma$ and to
$S_{a,b}\subset S$, we have

$d_{S}(a,b)\leq d_{S_{a,b}}(a,b)\leq 2d_{\Sigma}(a,b).$

Together, these imply that

$d_{\Sigma}(a,b)\leq d_{S}(a,b)\leq 2d_{\Sigma}(a,b),$

completing the proof. ∎

In particular, let $g\in\mathrm{Map}(S,p)$ be loxodromic with respect to the
action of $\mathrm{Map}(S,p)$ on $\mathcal{A}(S,p)$ with a $(K,C)$–quasi-
geodesic axis. If $g$ can be extended to an element of
$\mathrm{Map}(\Sigma,p)$, then this extension is loxodromic with respect to
the action of $\mathrm{Map}(\Sigma,p)$ on $\mathcal{A}(\Sigma,p)$, and the
extension will have a $(2K,C)$–quasi-geodesic axis.

##  3\. Coding arcs and standard position

Let $S$ be the biinfinite flute surface with a distinguished isolated puncture
$p$, and let $\\{p_{i}\\}_{i\in\mathbb{Z}}$ be any countably infinite discrete
collection of punctures on $S$ which exits both ends of the cylinder and does
not contain $p$. As described in Section 2.4, we choose one non-isolated end
of $S$ to correspond to the left direction and one to correspond to the right
direction, which gives a well-defined notion of a front and back of the
cylinder for $S$. We move all of the punctures in $\\{p_{i}\mid
i\in\mathbb{Z}\\}\cup\\{p\\}$ to the front of the cylinder for $S$ and all
other punctures to the back. We also move the distinguished puncture $p$ so
that it lies to the right of $p_{-1}$ and to the left of $p_{0}$. We will
consider the collection $\\{p_{i}\mid i\in\mathbb{Z}\\}\cup\\{p\\}$ of
punctures. We index this set with $\mathbb{Z}\cup\\{P\\}$, which we give the
ordering consisting of the usual ordering on $\mathbb{Z}$ with the additional
requirement that $-1<P<0$. The index $P$ corresponds to the distinguished
puncture $p$.

Fix the simple closed curve $B_{0}$ bounding the puncture $p_{0}$ on $S$ shown
in Figure 5. More formally, to define $B_{0}$ we fix a complete hyperbolic
metric on $S$ and let $B_{0}$ be a horocycle at a height sufficiently far out
the cusp. Fix a shift map $H$ on $S$ whose domain contains exactly the
collection $\\{p_{i}\\}$ for $i\in\mathbb{Z}\cup P$ and which shifts $p_{i}$
to $p_{i+1}$ for all $i\in\mathbb{Z}\cup P$.

\begin{overpic}[trim=32.52127pt 343.28249pt 75.88371pt
61.42993pt,clip={true},width=289.07999pt]{Bzero2} \put(39.0,33.0){$P$}
\put(62.0,33.0){$0$} \put(14.0,33.0){$-1$} \put(85.0,33.0){$1$}
\put(42.5,-2.0){$P$} \put(55.0,-2.0){$0$} \put(69.0,-2.0){$1$}
\put(27.0,-2.0){$-1$} \end{overpic} Figure 5. The curves $B_{i}$ are in red.
The blue region is the domain of the shift map $H$.

######  Definition 3.1.

Define the simple closed curves $B_{i}:=H^{i}B_{0}$ for $i\in\mathbb{Z}\cup
P$. Then $B_{i}$ is a simple closed curve bounding the puncture $p_{i}$, where
$p_{P}=p$.

Our choice of left/right also gives a well-defined notion of an arc passing
over or under a puncture (or equivalently some $B_{i}$). In all pictures of
$S$ throughout the paper, we denote the special puncture $p$ by an “X,” and
rather than drawing the punctures $p_{i}$, we draw the simple closed curves
$B_{i}$ in $S$. We will use these simple closed curves to put arcs into
standard position as described later in this section.

###  3.1. Coding arcs

We use the simple closed curves $B_{i}$ to describe a way to code simple arcs
on $S$ starting and ending at $p$. We will use this code to quantify how long
two arcs fellow travel, which will be essential for proving the results of
this paper.

Suppose that $\gamma$ is an oriented arc on $S$ starting and ending at $p$
such that $\gamma$ can be homotoped to be completely contained on the front of
$S$. We code $\gamma$ as follows. First homotope $\gamma$ so that it is
disjoint from all $B_{i}$ with $i\in\mathbb{Z}\cup P$, with the exception that
$\gamma$ starts and ends at the puncture $p$ and therefore intersects $B_{P}$
exactly twice. The code for $\gamma$ always starts and ends with the character
$P_{s}$ (which stands for “puncture start”) and contains either the character
$k_{o}$ or the character $k_{u}$, where $k\in\mathbb{Z}\cup\\{P\\}$, whenever
$\gamma$ passes over or under the simple closed curve $B_{k}$ for
$k\in\mathbb{Z}\cup P$. These characters appear in the code for $\gamma$ in
the same order in which $\gamma$ passes over/under the curves $B_{k}$. For
example, since $-1<P<0$, the second character of the code for $\gamma$ must be
either $0_{o}$, $0_{u}$, $(-1)_{o}$, $(-1)_{u}$, $P_{o}$, or $P_{u}$, because
if $\gamma$ doesn’t immediately wrap around $p$ (which would lead to the
second character being $P_{o}$ or $P_{u}$), it must pass over or under either
$B_{0}$ or $B_{-1}$ before it can pass over or under $B_{k}$ for any $k\neq
0,-1$. Similarly, if the character $2_{o}$ or $2_{u}$ appears in the code,
each adjacent character must be one of $1_{o},1_{u},2_{o},2_{u},3_{o},$ or
$3_{u}$. To simplify notation, we write $k_{o/u}$ to mean that the character
could be $k_{o}$ or $k_{u}$. We will write $k_{o/u}k_{u/o}$ to mean that the
two adjacent characters are either $k_{o}k_{u}$ or $k_{u}k_{o}$; the $k_{u/o}$
is used to emphasize that the second character has the opposite subscript as
the first one.

######  Example 3.2.

Consider the arcs shown in Figure 6. The elements $k\in\mathbb{Z}\cup\\{P\\}$
shown under $S$ denote the subscript on the simple closed curves $B_{k}$. The
code for $\alpha$ is $P_{s}0_{o}1_{u}2_{o}2_{u}1_{u}0_{u}P_{s}$, the code for
$\beta$ is $P_{s}P_{u}P_{o}0_{o}1_{o}2_{o}2_{u}1_{o}0_{o}P_{s}$, and the code
for $\gamma$ is
$P_{s}(-1)_{o}(-2)_{o}(-2)_{u}(-1)_{u}P_{u}0_{u}1_{u}1_{o}0_{o}P_{s}$.

Now suppose $\gamma$ is an oriented arc on $S$ starting and ending at $p$ such
that no arc in its homotopy class is contained on the front of $S$. Since
$\gamma$ starts and ends at $p$, which is on the front of the surface, every
time $\gamma$ leaves the front of $S$ it must eventually re-enter the front.
We give the code $C$ to any subpath of $\gamma$ which is on the back of $S$.
Up to homotopy, we may assume that each time $\gamma$ exits then enters the
front of $S$, it does so “between” two simple closed curves $B_{k}$ and
$B_{k+1}$. In other words, there is an arc $\gamma^{\prime}$ in the homotopy
class of $\gamma$ whose code contains either $k_{o/u}C(k+1)_{o/u}$ or
$k_{o/u}Ck_{o/u}$ each time $\gamma^{\prime}$ leaves the front of $S$. We give
$\gamma$ the same code as $\gamma^{\prime}$. We emphasize that this implies
that the code of an arc does not distinguish the behavior of arcs $\gamma$ on
the back of $S$.

$P$
---
$0$
---
$1$
---
$2$
---
$3$
---
$-1$
---
$-2$
---
$-3$
---
$\alpha$
---
$P$
---
$0$
---
$1$
---
$2$
---
$3$
---
$-1$
---
$-2$
---
$-3$
---
$\beta$
---
$P$
---
$0$
---
$1$
---
$2$
---
$3$
---
$-1$
---
$-2$
---
$-3$
---
$\gamma$
---
Figure 6. Pictured are arcs on the front of the surface $S$. The X denotes the
puncture $p$ and the elements $k\in\mathbb{Z}\cup\\{P\\}$ shown under $S$
denote the subscript on the simple closed curves $B_{k}$. The codes for arcs
$\alpha,\beta,\gamma$ are given in Example 3.2.

By an abuse of notation, we typically blur the distinction between an arc and
its code, writing, for example,
$\alpha=P_{s}0_{o}1_{u}2_{o}2_{u}1_{u}0_{u}P_{s}$.

######  Definition 3.3.

Let $\gamma$ be an oriented arc on $S$ starting and ending at $p$. A code for
$\gamma$ is reduced if no two adjacent characters in the code are the same and
if the character immediately following the initial $P_{s}$ or preceding the
terminal $P_{s}$ is not $P_{o/u}$.

The appearance of repeated characters in the code of an arc indicates
backtracking in the arc. The following lemma is immediate.

######  Lemma 3.4.

If there are two arc $\gamma$ and $\gamma^{\prime}$, starting and ending at
$p$, whose codes differ only by the removal of two adjacent characters which
are equal, i.e., $k_{o}k_{o}$ or $k_{u}k_{u}$, then $\gamma$ and
$\gamma^{\prime}$ are homotopic.

$\gamma$
---
$\gamma^{\prime}$
---
$-3$
---
$-2$
---
$-1$
---
$P$
---
$0$
---
$1$
---
$2$
---
$3$
---
Figure 7. The arcs $\gamma$ and $\gamma^{\prime}$ are homotopic and have the
same reduced code: $\gamma^{\prime}$ is formed from $\gamma$ by the removal of
the pair $1_{o}1_{o}$.

######  Example 3.5.

The arcs $\gamma$ and $\gamma^{\prime}$ in Figure 7, with codes
$P_{s}0_{o}1_{o}1_{o}1_{u}2_{u}2_{o}1_{o}0_{o}P_{s}$ and
$P_{s}0_{o}1_{u}2_{u}2_{o}1_{o}0_{o}P_{s}$, respectively, are homotopic.

Note that if a triple appears in the code for an arc, it is reduced to a
single character according to our convention, as only _pairs_ of repeated
characters are removed. For example,
$P_{s}0_{o}1_{o}1_{o}1_{o}1_{u}0_{o}P_{s}$ is reduced to
$P_{s}0_{o}1_{o}1_{u}0_{o}P_{s}$.

Each homotopy class of curves on $S$ determines a reduced code, in the sense
that any two homotopic curves have the same reduced code. We write that two
codes are equal if they determine homotopic arcs. For example, we write

$P_{s}0_{o}1_{o}1_{o}1_{u}2_{u}1_{u}0_{u}P_{s}=P_{s}0_{o}1_{u}2_{u}1_{u}0_{u}P_{s}.$

The converse of this fact is not true, however, because the code does not
encode the behavior of arcs on the back of $S$; hence there can be non-
homotopic arcs with the same reduced code. This will not cause any problems in
this paper.

######  Definition 3.6.

The code length of an arc $\gamma$, denoted $\ell_{c}(\gamma)$, is the number
of characters in a reduced code for $\gamma$.

######  Convention 3.7.

When giving the code of an arc for which the numerical values of the
characters are unimportant (or unknown), we will use variables in the code.
Our convention is to use Roman letters to represent single characters and
Greek letters to represent strings of characters whose length is (possibly)
greater than one. For example, $\ell_{c}(a_{1}a_{2}a_{3})=3$ while
$\ell_{c}(a\gamma b)=\ell_{c}(\gamma)+2$.

Given a string of characters $\alpha=a_{1}a_{2}\ldots a_{n}$, we denote by
$\overline{\alpha}$ the reverse of $\alpha$, so that
$\overline{\alpha}=a_{n}a_{n-1}\ldots a_{2}a_{1}$. If $\alpha$ is an arc, then
$\overline{\alpha}$ is the same arc with the opposite orientation.

###  3.2. Standard position

In this section, we describe how to use the code for an arc to find its image
under a general class of shifts which we call “permissible”.

######  Definition 3.8.

We say a shift shifting to the right is a right shift, while a shift shifting
to the left is a left shift. A right shift is permissible if its domain $D$
stays on the front of our subsurface and contains a turbulent region
$(n_{1},n_{2})$, that is, there exist $n_{1},n_{2}\in\mathbb{Z}\cup\\{P\\}$
with $n_{1}<n_{2}$ such that $D$ contains $B_{k}$ for all
$k\in(-\infty,n_{1}]\cup[n_{2},\infty)$ but does not contain $B_{k}$ for any
$k\in(n_{1},n_{2})$. We call $(-\infty,n_{1})\cup[n_{2},\infty)$ the shift
region of $h$. Analogously, a left shift is permissible if its domain $D$
stays on the front of our subsurface and contains a turbulent region
$(n_{2},n_{1})$, that is, there exist $n_{1},n_{2}\in\mathbb{Z}\cup\\{P\\}$
with $n_{2}<n_{1}$ such that $D$ contains $B_{k}$ for all
$k\in(-\infty,n_{2}]\cup[n_{1},\infty)$ but does not contain $B_{k}$ for any
$k\in(n_{2},n_{1})$. The shift region for a left shift is
$(-\infty,n_{2}]\cup(n_{1},\infty)$.

######  Convention 3.9.

Throughout the paper, we will use both left and right shifts. For notational
simplicity, all general results about shifts will be stated for right shifts.
All statements of results, proofs, and figures will make this assumption as
well. However, all of our definitions and results (and their proofs) also hold
for left shifts, by modifying any proof for a right shift so that we
essentially replace all instances of $n_{1}$ with $n_{2}$ and vice versa and
replace all instances of the word “increasing” by the word “decreasing” and
vice versa. The only subtleties are that:

  * •

We retain the convention that $h(B_{n_{1}})=B_{n_{2}}$.

  * •

For the shift region intervals $(-\infty,n_{1})\cup[n_{2},\infty)$ that appear
for a right shift, we use $(-\infty,n_{2}]\cup(n_{1},\infty)$ for the left
shift. In particular, the $n_{2}$ is always contained in the shift region.

######  Remark 3.10.

It is worthwhile to mention that Convention 3.9 is equivalent to simply
redefining the order, given by the symbol $<_{rev}$, on
$\mathbb{Z}\cup\\{P\\}$ to be the opposite of the standard meaning of the
inequality sign $<$. For example, in this “reversed order” we would have
$5<_{rev}3$ and so on. Given this and using the standard meanings for
“increasing” and “decreasing” with respect to $<_{rev}$, all of the proofs for
shifts that shift to the left would go through identically as shifts that
shift to the right when one replaces each instance of $<$ with an $<_{rev}$.
Despite the simplicity of this reversed order, we found writing proofs with it
to be more confusing to the reader than applying the above convention.

In order to find the image of an arc using only its code, we will need to
consider paths whose endpoints are not on $p$.

######  Definition 3.11.

A segment is a simple path with at least one endpoint which is not a puncture,
and no endpoints on a puncture other than $p$. We code a segment in an
analogous way as we did arcs in Section 3.1. If a segment begins or ends on
$p$, then the initial or terminal character of the code is $P_{s}$,
respectively. Note that a segment can have at most one instance of $P_{s}$ in
its code. Given a segment $\gamma$, we denote the initial and terminal
character of its code by $\gamma^{i}$ and $\gamma^{t}$, respectively. A
segment is supported on an interval $(a,b)\subset\mathbb{Z}\cup\\{P\\}$ if the
numerical value of every character in its reduced code is contained in
$(a,b)$. A subsegment of $\gamma$ which is supported on an interval $(a,b)$ is
denoted $\gamma|_{(a,b)}$, with similar notation for half-open and closed
intervals. A segment is (strictly) monotone if the numerical value of the
characters in its reduced code are (strictly) monotone as a subset of
$\mathbb{Z}\cup\\{P\\}$. A segment with code $C$ is called a back loop.

The notion of left/right on the front of $S$ induces an orientation on
strictly monotone segments contained on the front of $S$ in the following way.
If the terminal endpoint of such a segment $\gamma$ is to the right of the
initial endpoint, then $\gamma$ is oriented to the right. Similarly, if the
terminal endpoint is to the left of the initial endpoint, then $\gamma$ is
oriented to the left. Since $\gamma$ is strictly monotone, one of the above
two possibilities must occur. We note that single characters of a code
represent strictly monotone segments and so can be oriented in this way.

We will use the code for an arc or segment to find the image of the arc or
segment under certain homeomorphisms of $S$. The process can be complicated.
We now introduce a new way of concatenating strings of characters which will
be more suited to finding the image of an arc or segment in certain
situations.

######  Definition 3.12.

Given two segments $\alpha$ and $\beta$ such that the terminal character of
$\alpha$ agrees with the initial character of $\beta$ and such that these two
characters have the same orientation, the efficient concatenation of $\alpha$
and $\beta$, denoted $\alpha+\beta$, is formed by removing the terminal
character of $\alpha$ to form a new string $\alpha^{\prime}$ and concatenating
this new string with $\beta$, resulting in $\alpha^{\prime}\beta$.

For example,

$P_{s}0_{o}+0_{o}1_{o}=P_{s}0_{o}1_{o},$

and

$P_{s}0_{o}1_{o}2_{o}2_{u}+2_{u}2_{u}2_{o}1_{o}1_{u}=P_{s}0_{o}1_{o}2_{o}2_{u}2_{u}2_{o}1_{o}1_{u}=P_{s}0_{o}1_{u},$

where the middle term is an unreduced code and the final term is a reduced
code. See Figure 8. We note that if $\alpha$ and $\beta$ can be efficiently
concatenated, then they cannot be concatenated, because $\alpha^{t}$ and
$\beta^{i}$ have the same orientation. By a similar reasoning, if two segments
can be concatenated, then they cannot be efficiently concatenated. Throughout
the paper, we only (efficiently) concatenate two segments when it is possible.

\begin{overpic}[unit=4pt,width=216.81pt]{effcat.pdf} \put(58.0,41.5){$0$}
\put(71.0,41.5){$1$} \put(84.0,41.5){$2$} \put(58.0,-1.0){$0$}
\put(71.0,-1.0){$1$} \put(84.0,-1.0){$2$} \end{overpic} Figure 8. The two
examples of efficient concatenation following Definition 3.12. In each, the
(reduced) code for the blue segment is the efficient concatenation of the
codes of the two black segments.

As written, the code of a segment is not well behaved under homotopy because
every segment is homotopically trivial or homotopic into a puncture. We will
introduce a standard position for segments on $S$ with the property that any
two segments that are homotopic rel endpoints will, in standard position, have
the same reduced code. Standard position will also allow us to find the image
of a segment under a permissible shift using only its code.

######  Definition 3.13.

Fix a simple closed curve $S_{j}$ as in Figure 9 for each
$j\in\mathbb{Z}\cup\\{P\\}$. Let $\mathcal{C}=\\{S_{j}\mid
j\in\mathbb{Z}\cup\\{P\\}\\}$.

$-3$
---
$-2$
---
$-1$
---
$P$
---
$0$
---
$1$
---
$2$
---
$3$
---
$S_{-2}$
---
$S_{-1}$
---
$S_{P}$
---
$S_{0}$
---
$S_{1}$
---
$S_{2}$
---
$S_{3}$
---
Figure 9. Some of the simple separating curves in $\mathcal{C}$ defined in
Definition 3.13.

For each $j\in\mathbb{Z}\cup\\{P\\}$, we orient the simple closed curve
$B_{j}$ clockwise and identify $B_{j}$ with the subset $\mathbb{S}^{1}$ of
$\mathbb{C}$ by a homeomorphism which preserves this orientation. Fix points
$b^{j}_{L},b^{j}_{R}\in B_{j}$ corresponding to $-1,1\in\mathbb{S}^{1}$,
respectively. Here $L$ and $R$ stand for left and right.

To describe standard position, we will sometimes move endpoints of segments
$\gamma$ to lie on various boundary components. When we do this, we will use
the following convention. Suppose $\gamma^{i}=k_{o/u}$ and we want to move the
initial endpoint of $\gamma$ onto the boundary component $B_{k}$. If $\gamma$
is oriented to the right, then we move the initial endpoint of $\gamma$ to
$b^{k}_{L}$, and if it is oriented to the left, we move the initial endpoint
to $b^{k}_{R}$. On the other hand, suppose $\gamma^{t}=(k^{\prime})_{o/u}$ and
we want to move the terminal endpoint of $\gamma$ onto the boundary component
$B_{k^{\prime}}$. If $\gamma$ is oriented to the right, we move the terminal
endpoint of $\gamma$ to $b^{k^{\prime}}_{R}$, and if it is oriented to the
left, we move the terminal endpoint to $b^{k^{\prime}}_{L}$. See Figure 10.
Moving endpoints in this way does not change the code for $\gamma$.

$k$
---
$k^{\prime}$
---
Figure 10. Adjusting the endpoints of $\gamma$ (black) results in the red
segment.

It will be useful to understand how a given segment interacts with the domain
of a permissible shift.

######  Definition 3.14.

Let $h$ be a permissible right shift with domain $D$ and turbulent region
$(n_{1},n_{2})$. Then both (the upper and lower) boundary components of $D$
have the same reduced code on $(n_{1},n_{2})$. For any
$k,k^{\prime}\in(n_{1},n_{2})$, we let $\partial D|_{[k,k^{\prime}]}$ be the
reduced code of the boundary components of $D$ on the interval
$[k,k^{\prime}]$ with similar notation for open and half-open intervals. If a
segment $\gamma$ has the same code as $\partial D$ on an interval
$(k_{1},k_{2})\subset(n_{1},n_{2})$, we say that $\gamma$ follows $\partial D$
or agrees with $\partial D$ on that interval. As noted in the convention
above, there is an analogous definition when $h$ is a left shift. When a
segment $\gamma$ supported on $[n_{1},n_{2}]$ intersects one component of
$\partial D$, we call this a half crossing. When such a $\gamma$ intersects
both components of $\partial D$ so that the code for the subsegment of
$\gamma$ between these two half crossings is empty, in the sense that this
subsegment does not pass over or under $B_{k}$ for any $k$, we call this a
full crossing. See Figure 11.

$n_{2}$
---
$\gamma$
---
$n_{1}$
---
Figure 11. The domain $D$ for a permissible shift is shown in red. The segment
$\gamma$ follows $\partial D$ on $[n_{1}+1,n_{1}+3]$, has a half crossing
between $B_{n_{2}-1}$ and $B_{n_{2}}$, and has a full crossing between
$B_{n_{1}+1}$ and $B_{n_{1}+2}$.

Even though $S$ is a straightforward surface, standard position for segments
on $S$ is necessarily complicated. Before introducing it formally in the next
two subsections, we briefly give an intuitive idea of standard position in the
following remark. On a first reading of this paper, we strongly suggest the
reader read this remark and study Figures 12–15 instead of reading the formal
definition of standard position given in Sections 3.2.1 and 3.2.2. The reader
may then safely skip to Section 3.3. If, later in the paper, the image of a
particular segment seems counterintuitive, likely this is because we put the
segment in standard position before taking its image. This would be a good
time to look back at Sections 3.2.1 and 3.2.2 with that example of a segment
in mind.

######  Remark 3.15.

Let $h$ be a permissible right shift with domain $D$ and turbulent region
$(n_{1},n_{2})$, and let $\gamma$ be a segment. To put $\gamma$ in standard
position with respect to $h$, we first homotope subsegments of $\gamma$ that
are contained in the region $(-\infty,n_{1})\cup(n_{2},\infty)$ to be
completely contained in $D$. For the subsegments of $\gamma$ contained in the
region $[n_{1},n_{2}]$, we homotope $\gamma$ so that crossings are full
crossings whenever possible. We will always be able to make crossings full
except near $n_{1}$ or $n_{2}$, because $n_{1}$ and $n_{2}$ are where $\gamma$
leaves the turbulent region and enters the shift region, where we have already
ensured that it is contained in $D$. We also homotope $\gamma$ to minimize the
number of full crossings. If $\gamma$ contains a loop
$(n_{1})_{o/u}(n_{1})_{u/o}$ or $(n_{2})_{o/u}(n_{2})_{u/o}$, we homotope
$\gamma$ so that it crosses one connected component of $\partial D$
immediately before the loop and the other connected component immediately
after. In other words, $\gamma$ has to enter one side of $D$ before the loop
and exit the other side of $D$ after the loop. If $\gamma$ contains the
character $C$, we homotope it so that the subsegment of $\gamma$ represented
by $C$ exits and reenters the front of $S$ at the same place, in the sense
that the endpoints of this subsegment lie on the same simple closed curve
$S_{i}$. Finally, we also homotope each endpoint of $\gamma$ to lie either on
the nearest $B_{k}$ (if it is in the shift region) or the nearest $S_{k}$ (if
it is in the turbulent region).

Standard position will be slightly different for those segments which contain
back loops. We first discuss standard position for segments that do not
contain back loops.

####  3.2.1. Segments without back loops

Given a segment $\gamma$ whose reduced code does not contain $C$ and a
permissible right shift $h$ with domain $D$ and turbulent region
$(n_{1},n_{2})$, we put $\gamma$ into standard position with respect to $h$ as
follows.

If the endpoints of $\gamma$ are not contained in $(n_{1},n_{2})$ and it is
possible to homotope $\gamma$ completely inside of the domain of $h$, we do
so, and we move the endpoints of $\gamma$ to lie on the $B_{k}$ curve numbered
by the first and last characters of the code as described above.

Otherwise, $\gamma$ can be written as the concatenation
$\gamma_{1}\cdots\gamma_{k}$ of disjoint connected maximal subsegments such
that each $\gamma_{i}$ is either:

  1. (a)

supported on either $(-\infty,n_{1}]$ or $[n_{2},\infty)$; or

  2. (b)

supported on $(n_{1},n_{2})$.

We now homotope each $\gamma_{i}$ individually. If $\gamma_{i}$ satisfies (a),
then we move the endpoints of $\gamma_{i}$ onto the $B_{k}$ curve numbered by
the first and last character of the code as above and homotope the interior of
$\gamma_{i}$ to lie completely inside the domain of $h$. We homotope segments
$\gamma_{i}$ satisfying (b) using the following procedure:

$\scriptstyle{n_{1}}$
---
$\scriptstyle{n_{2}}$
---
$\scriptstyle{n_{1}}$
---
$\scriptstyle{n_{2}}$
---
$\scriptstyle{n_{1}}$
---
$\scriptstyle{n_{2}}$
---
$\scriptstyle{n_{1}}$
---
$\scriptstyle{n_{2}}$
---
(a) Putting $\gamma$ in standard position.

$\scriptstyle{n_{1}}$
---
$\scriptstyle{n_{2}}$
---
(b) A simple segment with the same code as $\gamma$ which we say is in
standard position.

Figure 12. The domain $D$ for a permissible shift is shown in red. In black is
a segment $\gamma$ that has no $(n_{i})_{o/u}(n_{i})_{u/o}$ for $i=1,2$.

  1. Step (i):

If the initial character of the segment is $k_{o/u}$, move the initial
endpoint of the segment onto the separating curve $S_{k}$ if $k_{o/u}$ is
oriented to the right and onto $S_{k+1}$ if it is oriented to the left. If the
terminal character of the segment is $k^{\prime}_{o/u}$, move the terminal
endpoint of the segment onto $S_{k^{\prime}}$ if $k^{\prime}_{o/u}$ is
oriented to the left and onto $S_{k^{\prime}+1}$ if it is oriented to the
right. Now move the endpoints along the $S_{j}$ containing them to reduce the
number of full and half crossings, if possible, without creating any self-
intersections. In particular, the endpoints should not lie in the domain of
$h$. See Figure 12.

There is one caveat to the rule above. Note that when $\gamma_{i}$ is type
(b), then $\gamma_{i+1}$ and $\gamma_{i-1}$ are type (a), otherwise
$\gamma_{i}$ would not be maximal. In the case that either of these
neighboring segments is exactly a loop around $n_{1}$ or $n_{2}$ we must
adjust the position of the endpoints of $\gamma_{i}$ along $S_{j}$. If
$\gamma_{i+1}=(n_{k})_{o}(n_{k})_{u}$, for $k=1$ or $2$, we require that the
terminal endpoint of $\gamma_{i}$ is above $D$ and require that the terminal
endpoint is below $D$ when $\gamma_{i+1}=(n_{i})_{u}(n_{i})_{o}$. Similarly,
if $\gamma_{i-1}=(n_{i})_{o}(n_{i})_{u}$, for $i=1$ or $2$, we require that
the initial endpoint of $\gamma_{i}$ is below $D$ and require that the initial
endpoint of $\gamma_{i}$ is above $D$ when
$\gamma_{i-1}=(n_{i})_{u}(n_{i})_{o}$. See Figure 13. Note that this
repositioning of endpoints can cause additional crossings of $D$ as in Figure
14, but this is the appropriate configuration for our calculations.

  2. Step (ii):

Homotope the segment rel endpoints to make all crossings full. Since Step (i)
ensures that the endpoints of $\gamma_{i}$ are always outside the domain $D$,
this is always possible.

  3. Step (iii):

Homotope the segment rel endpoints to reduce the number of crossing by
removing all bigons that bound disks and have one side on the segment
$\gamma_{i}$ and the other side on $\partial D$.

  4. Step (iv):

Finally, there may be a choice of where a full crossing occurs. If there is
such a choice, then it will always be possible to homotope rel endpoints so
that the crossing occurs between two adjacent characters $k_{o/u}$,
$(k^{\prime})_{o/u}$ with $k,k^{\prime}\in(n_{1},n_{2})$ such that the $o/u$
pattern of $k$ and/or $k^{\prime}$ does not match that of $\partial D$, and
our convention is to make this choice for the largest possible $k,k^{\prime}$.
For example, in Figure 12, the bottom strand could fully cross the domain
between $(n_{1}+1)_{o}$ and $(n_{1}+1)_{u}$ or between $(n_{1}+1)_{u}$ and
$(n_{1}+2)_{u}$; we make the latter choice.

At this point, we have a collection of disjoint subsegments, each in standard
position. The final step is to connect the endpoints of these segments, in the
order they were originally connected, by segments called connectors.
Connectors always occur between characters with numerical value $n_{1}$ and
$n_{1}+1$ or $n_{2}$ and $n_{2}-1$. For the purposes of the code, we picture
these segments extended slightly in either direction to overlap with the
segments on either side so that the code of a connector will always be a pair
$j_{o/u}(j+1)_{o/u}$ or $(j+1)_{o/u}j_{o/u}$. Thus, if a connector $\alpha$
connects disjoint subsegments $\delta_{1}$ and $\delta_{2}$ (in that order),
then $\alpha=(\delta_{1})^{t}(\delta_{2})^{i}$ and we can write
$\delta_{1}+\alpha+\delta_{2}$. Note that this code is equivalent to the
concatenation $\delta_{1}\delta_{2}$. By construction, connectors always have
one endpoint inside and one endpoint outside the the domain $D$ of the shift.
After applying this procedure, the resulting segment in standard position may
no longer be simple. However, it will have the same code as our original
$\gamma$.

$\scriptstyle{n_{1}}$
---
$\scriptstyle{n_{1}}$
---
$\scriptstyle{n_{1}}$
---
$\scriptstyle{n_{1}}$
---
(a) Putting $\gamma$ in standard position.

$\scriptstyle{n_{1}}$
---
(b) A simple segment with the same code as $\gamma$ which we say is in
standard position.

Figure 13. A segment $\gamma$ that contains multiple copies occurrences of
$(n_{1})_{o}(n_{1})_{u}$.

The following lemma summarizes the above procedure.

######  Lemma 3.16.

A segment $\gamma$ in standard position with respect to a permissible right
shift $h$ which is not completely contained in the domain of $h$ can be
written as the efficient concatenation of (possibly empty) segments in the
following way.

(1)
$\gamma=\gamma_{1}^{tu}+\gamma_{1}^{c1}+\gamma_{1}^{sh}+\gamma_{1}^{c2}+\gamma_{2}^{tu}+\gamma_{2}^{c1}+\cdots+\gamma_{n}^{sh}+\gamma_{n}^{c2},$

where for each $i$,

  * •

$\gamma_{i}^{tu}$ is supported on the turbulent region $(n_{1},n_{2})$, has
been put into standard position following steps (i)–(iv) above, and has both
endpoints outside of the domain $D$;

  * •

$\gamma_{i}^{sh}$ is supported on the shift region
$(-\infty,n_{1}]\cup[n_{2},\infty)$, is completely contained in the domain
$D$, and has both endpoints on the $B_{k}$ curves; and

  * •

$\gamma_{i}^{c1},\gamma_{i}^{c2}$ are connectors, each of which has code
length 2, is supported on either $[n_{1},n_{1}+1]$ or $[n_{2}-1,n_{2}]$, and
has one endpoint on $B_{n_{1}}$ or $B_{n_{2}}$ and one endpoint outside the
domain $D$ on $S_{n_{1}+1}$ or $S_{n_{2}}$.

We will sometimes use the notation $\gamma_{i}^{conn}$ for a connector if it
is not important if this subsegment is the first connector $\gamma_{i}^{c1}$
or the second connector $\gamma_{i}^{c2}$.

It is often convenient to abuse notation and say that a simple segment is in
“standard position” even if it is not the result of the above procedure
because these segments are easier to draw and think about. What we mean by
this is that the segment intersects the boundary of $D$ in the same place as
it would in standard position. In Figures 12–15, we give a segment, the steps
to put it in standard position, and an example of a simple segment with the
same code which we also say is in standard position.

$\scriptstyle{n_{1}}$
---
$\scriptstyle{n_{1}}$
---
$\scriptstyle{n_{1}}$
---
$\scriptstyle{n_{1}}$
---
(a) Putting $\gamma$ in standard position.

$\scriptstyle{n_{1}}$
---
(b) A simple segment with the same code as $\gamma$ which we say is in
standard position.

Figure 14. A segment $\gamma$ that contains $(n_{1})_{o}(n_{1})_{u}$. Note
that the requirements on the endpoints of $\gamma_{i}$ in step (i) of the
procedure for standard position require that we have one connector cross the
top of $\partial D$ and one connector cross the bottom of $\partial D$.
$\scriptstyle{n_{1}}$
---
$\scriptstyle{n_{2}}$
---
$\scriptstyle{n_{1}}$
---
$\scriptstyle{n_{2}}$
---
Figure 15. A segment $\gamma$ that has endpoints in
$(-\infty,n_{1}]\cup[n_{2},\infty)$ and can be homotoped to be completely
inside $D$.

Because each segment in (1) has fixed endpoints, its image under $h$ is well-
defined up to homotopy rel endpoints. Thus we may use the decomposition of
$\gamma$ to find its image under $h$ as follows

$h(\gamma)=h(\gamma_{1}^{tu})+h(\gamma_{1}^{c1})+h(\gamma_{1}^{sh})+h(\gamma_{1}^{c2})+h(\gamma_{2}^{tu})+h(\gamma_{2}^{c1})+\cdots+h(\gamma_{n}^{sh})+h(\gamma_{n}^{c2}).$

Since in standard position $\gamma$ may not be simple, its image under $h$ may
not be simple. However, this code corresponds to a unique (homotopy class of)
simple segment with the same endpoints. It is important that we use efficient
concatenation when calculating $h(\gamma)$. Using regular concatenation, the
code for $\gamma$ is simply
$\gamma_{1}^{tu}\gamma_{1}^{sh}\ldots\gamma_{n}^{tu}\gamma_{n}^{sh}$. However,
it is not true that $h(\gamma_{1}^{tu})h(\gamma_{1}^{sh})\ldots
h(\gamma_{n}^{tu})h(\gamma_{n}^{sh})$ is a code for $\gamma$; in fact, much of
the time this code does not define a segment. See Example 3.17. Most of the
interesting behavior in the image of an arc or segment under a permissible
shift actually comes from the full and half crossings. Since each connector
contributes a half crossing, they are essential for determining the image of
$\gamma$.

######  Example 3.17.

Consider the permissible shift $h$ shown in Figure 11 along with the segment
$\gamma$. In Figure 16, we put $\gamma$ in standard position and find its
image under the shift. Using code, we have
$\gamma=(n_{1}+1)_{u}(n_{1}+2)_{o}(n_{1}+3)_{o}(n_{2}-1)_{o}(n_{2})_{u}$. Note
that every character in this code except the terminal $(n_{2})_{u}$ is fixed
by $h$, and $h((n_{2})_{u})=(n_{2}+1)_{u}$. If we simply compute $h(\gamma)$
character by character, we get
$h(\gamma)=(n_{1}+1)_{u}(n_{1}+2)_{o}(n_{1}+3)_{o}(n_{2}-1)_{o}(n_{2}+1)_{u}$.
However, this is not a well-defined code since $n_{2}-1$ and $n_{2}+1$ are not
adjacent. On the other hand, using efficient concatenation, we see that

$\displaystyle h(\gamma)$
$\displaystyle=h(\gamma^{tu})+h(\gamma^{conn})+h(\gamma^{sh})$
$\displaystyle=h((n_{1}+1)_{u}(n_{1}+2)_{o}(n_{1}+3)_{o}(n_{2}-1)_{o})+h((n_{2}-1)_{o}(n_{2})_{u})+h((n_{2})_{u})$
$\displaystyle=(n_{1}+1)_{u}(n_{1}+2)_{o}(n_{1}+3)_{o}(n_{2}-1)_{o}+(n_{2}-1)_{o}(n_{2})_{o}(n_{2}+1)_{u}+(n_{2}+1)_{u}$
$\displaystyle=(n_{1}+1)_{u}(n_{1}+2)_{o}(n_{1}+3)_{o}(n_{2}-1)_{o}(n_{2})_{o}(n_{2}+1)_{u}.$

Note that all but the final pair $(n_{2}-1)_{o}(n_{2})_{u}$ are fixed by $h$.
This final pair is a half crossing and in standard position it is a connector.

$n_{1}$
---
$n_{2}$
---
$n_{1}$
---
$n_{2}$
---
Figure 16. The segment $\gamma$ from Example 3.17 in standard position (left)
and its image under the shift whose domain is shown (right).

In fact, given a segment $\gamma=\gamma^{tu}$ supported on $(n_{1},n_{2})$, we
can further decompose $\gamma$ into subsegments which are disjoint from $D$
and pairs which fully cross $D$ in such a way that makes it straightforward to
find its image under $h$. Write

(2)
$\gamma=\gamma^{d}_{1}+\gamma^{e}_{1}+\cdots+\gamma^{d}_{s}+\gamma^{e}_{s},$

where each $\gamma^{d}$ is a maximal subsegment disjoint from $D$, each
$\gamma^{e}$ fully crosses $D$, and $\ell_{c}(\gamma^{d})\geq 2$,
$\ell_{c}(\gamma^{e})=2$, when non-empty.

Using the above decomposition, in an unreduced code we have

$h(\gamma)=\gamma^{d}_{1}+h(\gamma^{e}_{1})+\dots+\gamma^{d}_{s}+h(\gamma^{e}_{s}).$

Every non-empty $\gamma^{e}_{j}$ will have image which follows $\partial D$,
loops around $n_{2}$, and follows $\partial D$ back, so that

$h(\gamma^{e}_{j})=\partial
D|_{[(\gamma^{e}_{j})^{i},n_{2})}(n_{2})_{o/u}(n_{2})_{u/o}\overline{\partial
D|_{[(\gamma^{e}_{j})^{t},n_{2})}}.$

As in Example 3.17, if we don’t use efficient concatenation then we can write
$\gamma=\gamma^{d}_{1}\gamma^{d}_{2}\ldots\gamma^{d}_{s}$. Applying $h$ to
each of these subsegments individually would yield $h(\gamma)=\gamma$, since
each of these subsegments is fixed by $h$, which is not the correct image.

####  3.2.2. Segments with back loops.

If $\gamma$ is a segment with code equal to $C$, we require that $\gamma$ has
both endpoints on some separating curve $S_{i}$ in our collection
$\mathcal{C}$. We also assume that the endpoints of $C$ lie outside the
domain, and, moreover, that $\gamma$ does not intersect $D$. There are two
possibilities for $\gamma$, either $\gamma$ both enters and exits the front of
$S$ at the top or bottom or (up to taking inverses) $\gamma$ enters at the top
and exits at the bottom of the front of the surface. Recall that we define the
top/bottom of the front of $S$ with respect to the notion of right/left on the
front of $S$. In the first case, this implies that the endpoints of $\gamma$
are both above or both below $D$, respectively, while in the second case one
will be above and one will be below. In either case, this convention implies
that $\gamma\cap D=\emptyset$ and $h(\gamma)=\gamma$.

Suppose next that $\gamma$ is a segment whose code contains $C$ but also
contains other characters. Suppose for simplicity that the code for $\gamma$
contains a single instance of $C$, so that $\gamma=\tau_{1}C\tau_{2}$, where
$\tau_{i}$ does not contain $C$ for $i=1,2$. We put $\gamma$ in standard
position as follows. First note that by definition of the code, we must either
have $(\tau_{1})^{t}=(\tau_{2})^{i}$ (if $\tau_{1}$ and $\tau_{2}$ have
opposite orientations) or $(\tau_{1})^{t}=(\tau_{2})^{i}\pm 1$ (if
$(\tau_{1})^{t}$ and $(\tau_{2})^{i}$ have the same orientation). See Figure
17. Without loss of generality, suppose $\tau_{1}^{t}$ is oriented to the
right. Consider the (disjoint) segments $\tau_{1}$ and $\tau_{2}$, put them in
standard position as in the previous section, and homotope the endpoints of
$C$ as in the previous paragraph. Note the endpoints of $C$ will lie on the
curve $S_{(\tau_{1})^{t}+1}$. We now have three disjoint segments with codes
$\tau_{1}$, $\tau_{2}$, and $C$. We will add (possibly empty) segments called
back loop connectors from the terminal point of $\tau_{1}$ to the initial
point of $C$ and from the terminal point of $C$ to the initial point of
$\tau_{2}$, respectively, to form a connected segment. See Figure 17. We code
these back loop connectors with the characters $C^{-},C^{+}$, respectively, so
that we can easily discuss their image. In particular, by a slight abuse of
notation, we replace $C$ in the code for $\gamma$ with $C^{-}CC^{+}$.

Figure 17. Various segments $\gamma$ containing back loops. The segments on
the right are in standard form. Back loops are blue and back loop connectors
are green. Note that in the second, third, and fifth examples, at least one of
$C^{\pm}$ is empty.

Recall that without loss of generality, $(\tau_{1})^{t}$ is oriented to the
right. If the terminal endpoint of $\tau_{1}$ lies outside $D$ and on the same
side of $D$ as the initial point of the back loop $C$, then the back loop
connector $C^{-}$ is empty. An analogous statement holds for $C^{+}$, where we
use the initial endpoint of $\tau_{2}$ in place of the terminal endpoint of
$\tau_{1}$. Each non-empty back loop connector $C^{\pm}$ either:

  1. (i)

has one endpoint on $B_{(\tau_{1})^{t}}$ or $B_{(\tau_{2})^{i}}$ (depending on
whether it is $C^{-}$ or $C^{+}$), the other endpoint on
$S_{(\tau_{1})^{t}+1}$, and half-crosses $D$; or

  2. (ii)

has both endpoints on $S_{(\tau_{1})^{t}+1}$ which are outside of $D$ and
fully crosses $D$.

###  3.3. Gaps in segments

When we use the code of a segment to find its image under a permissible shift,
we first break it into smaller subsegments using standard position. When we do
this, we always use efficient concatenation, so that the codes of the
individual pieces overlap in a single character. The goal of efficient
concatenation is to ensure that we do not lose any information about the
segment by breaking it into pieces. However, we need to be careful when we do
this. If we first break a segment into subsegments and then put each
subsegment into standard position, it is possible that we will lose some
information. In particular, we may cause there to be a “gap” in the segment.
Based on standard position, these gaps can only occur when breaking a segment
in the interior of the turbulent region $(n_{1},n_{2})$.

To see this, suppose we break a segment $\gamma=\gamma_{1}+\gamma_{2}$ into
two subsegments such that the numerical value of
$\gamma_{1}^{t}=\gamma_{2}^{i}$ is $j\in(n_{1},n_{2})$. If we put each
$\gamma_{i}$ into standard position individually, it is possible that
$\gamma_{1}^{t}$ and $\gamma_{2}^{i}$ lie on opposite sides of $D$ (see Figure
18). In this case, we have lost the full crossing between them. Recall that a
shift fixes the surface outside of its domain. In the region $(n_{1},n_{2})$,
a segment in standard position is disjoint from the domain of the shift except
where there is a full crossing (see the decomposition in equation (2)), so the
full crossings are essential for determining the image of a segment. Thus we
cannot use $\gamma_{1}$ and $\gamma_{2}$ to find the correct image of
$\gamma$.

Figure 18. A segment $\gamma$ (left) is split into subsegments
$\gamma=\gamma_{1}+\gamma_{2}$ and each $\gamma_{i}$ is individually put into
standard position with respect to the shift $h$ whose domain is shown (right).
This causes a loss of information when we take the images of $\gamma_{i}$
individually under $h$: since $h(\gamma_{i})=\gamma_{i}$ for $i=1,2$, we have
$h(\gamma_{1})+h(\gamma_{2})=\gamma_{1}+\gamma_{2}=\gamma$, but $\gamma$ is
clearly not fixed by $h$.

In order to ensure that we do not lose any information when working with a
segment and subsegments in the turbulent region, we always first homotope the
whole segment $\gamma$ to be a simple segment in standard position. We then
fix this particular representative of the homotopy class of $\gamma$ for the
remainder of the time we work with it. Thus, when we break $\gamma$ into
subsegments $\gamma_{1}$ and $\gamma_{2}$, we do not put these into standard
position individually. To be precise, we choose the endpoints $\gamma_{1}^{t}$
and $\gamma_{2}^{i}$ to be the intersection of $\gamma$ and the appropriate
separating curves in our collection $\mathcal{C}$ and we do not allow any
further homotopies of $\gamma_{1}$ or $\gamma_{2}$. This will always ensure
that there are no gaps between $\gamma_{1}$ and $\gamma_{2}$.

In certain cases, it may be simpler to break $\gamma$ into subsegments in a
different way, and we do this whenever it will avoid technicalities. For
example, in Figure 18, we may simply choose not to divide $\gamma$ into
subsegments at all. On the other hand, we could also choose to make
$\gamma_{1}$ or $\gamma_{2}$ longer than is strictly necessary in a particular
calculation in order to avoid a potential loss of information.

###  3.4. Taking inverses of segments

In general, if we have a segment $\gamma$ and a subsegment $\zeta$ of
$h(\gamma)$, there is not necessarily a subsegment of $\gamma$ which we may
call $h^{-1}(\zeta)$. In other words, not every subsegment of $h(\gamma)$ is
the image of a subsegment of $\gamma$. It is important here that when we think
of a subsegment of $\gamma$, we are fixing $\gamma$ in standard position. That
is, we are thinking of a reduced code for $\gamma$, rather than any
(unreduced) code representing $\gamma$.

For example, consider the shift shown in Figure 19(a). Here, the pink
subsegment $\zeta_{1}$ of $h(\gamma)$ is not the image of any subsegment of
$\gamma$. Rather, it is properly contained in the image of the purple
subsegment of $\gamma$, specifically because it is in the image of the full
crossing of the purple segment with $D$. Explicitly,

$\gamma=0_{o}1_{u}2_{o}2_{u}1_{u},\quad
h(\gamma)=0_{o}1_{u}2_{o}2_{u}1_{u}0_{o}0_{u}1_{u}2_{u}2_{o}1_{u}1_{o}2_{o}2_{u}1_{u}0_{u},$

and one can see that if the subsegment $\zeta_{1}=1_{u}2_{o}2_{u}1_{u}$ of
$h(\gamma)$ then there is no subsegment $\gamma^{\prime}$ of $\gamma$ for
which $h(\gamma^{\prime})=\zeta_{1}$.

$\gamma$
---
$h(\gamma)$
---
$B_{0}$
---
$B_{0}$
---
(a) A subsegment which does not have a preimage.

$\gamma$
---
$h(\gamma)$
---
(b) A subsegment which has a preimage.

Figure 19. A segment $\gamma$ and its image $h(\gamma)$ are shown. In (A), the
pink subsegment of $h(\gamma)$ does not have an inverse; it is a proper
subpath of the image of the purple subsegment of $\gamma$. In (B), the pink
subsegment of $h(\gamma)$ has an inverse; it is the image of the purple
subsegment of $\gamma$.

However, we may take an inverse image of a subsegment $\zeta$ of $h(\gamma)$
whenever we know that $\zeta$ is the image of a subsegment of $\gamma$. This
is the case, for example, when $\zeta=h(\gamma)$; in other words, it is true
that $\gamma=h^{-1}(h(\gamma))$. This can also happen when the initial and
terminal characters of a subsegment of $h(\gamma)$ are the images of the
initial and terminal characters of a subsegment of $\gamma$. For example, in
Figure 19(b), a direct computation will show that the initial and terminal
characters of $\zeta_{2}$ (in pink) are the images of the initial and terminal
characters of the purple subsegment of $\gamma$. Therefore $\zeta_{2}$ is the
image of the purple subsegment of $\gamma$, which is precisely the result of
calculating $h^{-1}(\zeta_{2})$.

##  4\. Loop Theorem

As in Section 3, we let $S$ be the biinfinite flute surface with a
distinguished puncture $p$ and fix the collection of simple closed curves
$\\{B_{i}\mid i\in\mathbb{Z}\cup\\{P\\}\\}$ as in Definition 3.1. Let $h$ be a
permissible shift (see Definition 3.8) and $k\in\mathbb{Z}\cup\\{P\\}$. By an
abuse of notation, we may occasionally write $h(k)$, by which we mean that
$h(k)$ is the label of $h(B_{k})$. Thus, given any segment whose code is
$k_{o/u}$, we have $h(k_{o/u})=h(k)_{o/u}$.

Recall that a segment is a path which does not have both endpoints on $p$.

######  Definition 4.1.

A segment in standard position is trivial if it can be homotoped rel endpoints
to one of the following:

  1. (a)

a segment contained in one of the separating curves $S_{i}\in\mathcal{C}$; or

  2. (b)

a point.

We will use the notation $\emptyset$ to denote the reduced code for a trivial
segment. For example, we write $k_{o}k_{o}=\emptyset$.

######  Definition 4.2.

A loop is a segment that has one of the following forms:

  1. (1)

$\delta_{1}a_{o/u}a_{u/o}\delta_{2}$ for some $a$ where
$\delta_{1}^{i}=\delta_{2}^{t}$,

  2. (2)

$a_{o/u}a_{u/o}$ for some $a$, or

  3. (3)

$C$. See Figure 20 for examples of loops.

\begin{overpic}[width=252.94499pt,trim=28.90755pt 650.43pt 79.49744pt
3.61371pt,clip]{loopex} \end{overpic} Figure 20. Some examples of loops from
Definition 4.2. The loop in red fits Case (1), the loop in blue fits Case (2),
and the loop in purple fits Case (3).

A loop satisfying (1) is called a regular loop, a loop satisfying (2) is
called an over-under loop, and, as in Definition 3.11, a loop satisfying (3)
is called a back loop. Note that regular loops always contain an over-under
loop but not all over-under loops can be extended to a regular loop. A single
loop is a back loop, an over-under loop, or a regular loop such that
$\delta_{i}$ does not contain a loop for $i=1,2$.

With the above definitions, the rest of this section is devoted to analyzing
the following question.

######  Question 4.3.

Let $h$ be a permissible shift. When does $h$ send a loop to a trivial
segment?

The reason we introduced standard position is to ensure that this question is
well defined. The issue is that homotopies of a loop can change whether or not
its image is trivial, even if those homotopies keep the endpoints on a fixed
separating closed curve (see Figure 21). Thus it is important that, given a
loop, we first put it in standard position before applying the permissible
shift $h$. This will remove any possible ambiguity in the image of the loop.

Figure 21. The permissible shift $h$ translates to the right. The first two
loops on the left are homotopic via homotopies which keep the endpoints on the
same fixed separating curve. However, one image is trivial while the other is
not. Figure 22.  The second loop from Figure 21 is in standard position. We
show why its image under $h$ is trivial.

In this section, we first introduce a kind of cancellation in the image of a
segment and its code which we call cascading cancellation. This kind of
cancellation will cause technical problems throughout the paper, and much of
Section 6 is devoted to understanding how to control it. We then prove the
Loop Theorem (Theorem 4.8), which answers Question 4.3. We end the section
with a discussion of several technical consequences of the Loop Theorem which
will be useful later.

###  4.1. Cascading cancellation

######  Definition 4.4.

We call an arc $\gamma$ symmetric if $\gamma=\delta
q_{1}q_{2}\overline{\delta}$ for any characters $q_{1}$, $q_{2}$. In other
words, a reduced code for $\gamma$ is palindromic with the exception of the
middle two characters. Note in particular that this implies that $q_{1}$,
$q_{2}$ have the same numerical value.

Recall that we find the image of a path with code $\alpha\beta$ under a
permissible shift $f$ as follows. Let $q$ be the last character of $\alpha$
and $q^{\prime}$ be the first character of $\beta$. Then

$f(\alpha\beta)=f(\alpha)+f(qq^{\prime})+f(\beta).$

While $f(\alpha),f(qq^{\prime}),$ and $f(\beta)$ are all reduced codes, it is
possible that the efficient concatenation will cause there to be cancellation.
For example, if $f(\alpha)=1_{u}1_{o}2_{o}3_{o}$ and
$f(qq^{\prime})=3_{o}3_{o}2_{o}1_{o}0_{o}P_{o}$, then
$f(\alpha)+f(qq^{\prime})=1_{u}0_{o}P_{o}$. When this type of cancellation
occurs, that is, when a character of $f(qq^{\prime})$ does not appear in a
reduced code of the image, we say there is cancellation involving
$f(qq^{\prime})$. In our example, there is cancellation involving
$f(qq^{\prime})$ and $f(\alpha)$. When it is necessary to be more precise, we
may also say there is (respectively, is not) cancellation involving a
character $s$, if $s$ appears in the unreduced code but not the reduced code
(respectively, appears in both the unreduced code and the reduced code) of the
image. Thus in our example, there is no cancellation involving $1_{u}$ but
there is cancellation involving $2_{o}$. Our goal is to understand, in
general, when there is cancellation with a particular character in a path
under a permissible shift.

Suppose $\alpha=\gamma q_{1}$ and $h$ is a permissible shift. Let $q$ be the
terminal character of $\gamma$. It is tempting to believe that if we can show
that there is no cancellation involving $h(q_{1})$ within $h(qq_{1})$, then
there is no cancellation involving $h(q_{1})$ at all. However, this is not
sufficient, for it is possible that there is “cascading cancellation.” Before
giving a formal definition, we illustrate this phenomenon with an example.

######  Example 4.5.

Consider the segment $\delta=3_{u}2_{u}1_{o}0_{u}$ and the permissible left
shift $h$ whose domain is shown in Figure 23. Then

$h(\delta)=h(3_{u}2_{u})+h(2_{u}1_{o})+h(1_{0}0_{u}).$

We have

$h(1_{o}0_{u})=1_{o}0_{u},$ $h(2_{u}1_{o})=2_{u}1_{o},$

and

$h(3_{u}2_{u})=3_{u}2_{u}1_{o}0_{u}(-1)_{u}(-1)_{o}0_{u}1_{o}2_{u}2_{u}.$

Putting this together, we obtain
$h(\delta)=3_{u}2_{u}1_{o}0_{u}(-1)_{u}(-1)_{o}.$

\begin{overpic}[width=180.67499pt]{red_green.pdf}
\put(35.0,71.0){${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\delta}$}
\put(35.0,25.5){${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}h(\delta)}$}
\put(4.0,-5.0){$-1$} \put(25.0,-5.0){$0$} \put(41.0,-5.0){$1$}
\put(57.0,-5.0){$2$} \put(73.0,-5.0){$3$} \put(89.0,-5.0){$4$}
\put(4.0,39.0){$-1$} \put(25.0,39.0){$0$} \put(41.0,39.0){$1$}
\put(57.0,39.0){$2$} \put(73.0,39.0){$3$} \put(89.0,39.0){$4$} \end{overpic}
Figure 23. Above: the segment $\delta$ in standard position. The domain of the
left shift $h$ is shown in green. Below: the image $h(\delta)$.

There is no cancellation involving either of the terms when computing
$h(2_{u}1_{0})+h(1_{o}0_{u})$. However, when computing
$h(3_{u}2_{u})+h(2_{u}1_{0})$, we see that $h(2_{u}1_{o})$ completely cancels
with an initial segment of $h(3_{u}2_{u}3)$, and $h(1_{o}0_{u})$ completely
cancels with the next subsegment of $h(3_{u}2_{u})$. Therefore, there is in
fact cancellation involving $h(0_{u})$ in $h(\delta)$.

######  Definition 4.6.

Formally, given an arc $\delta_{1}\dots\delta_{n}$ and a permissible shift
$f$, we say there is cascading cancellation involving $f(\delta_{n})$ if there
is cancellation involving $f(\delta_{n})$ in
$f(\delta_{1}\delta_{2}\dots\delta_{n})$ but there is no cancellation
involving $f(\delta_{n})$ in $f(\delta_{n-1}\delta_{n})$.

Understanding and controlling cascading cancellation is the difficult part of
many of the proofs in this paper. The remainder of this subsection is devoted
to theorems that will allow us to control cascading cancellation for
permissible shifts. We will often be in the following situation: there is a
segment $\gamma=\gamma_{1}\gamma_{2}$ whose image under a shift $h$ we would
like to understand and we know that $h(\gamma_{1})$ has some desired quality
(such as containing a loop, for example). In order to show that the desirable
behavior of $h(\gamma_{1})$ persists in $h(\gamma)$, we need to ensure that
there is no cancellation involving $h(\gamma_{1})$ and $h(\gamma_{2})$ by
controlling cascading cancellation. In Section 6, we will revisit this topic
and prove some additional results that allow us to control cascading
cancellation for the particular homeomorphisms we construct, which are
compositions of shifts.

######  Lemma 4.7.

Let $h$ be a permissible right shift with domain $D$ and turbulent region
$(n_{1},n_{2})$. Let $\alpha$ be a strictly monotone segment supported on
$(n_{1},n_{2})$. Then in a reduced code $h(\alpha)$ has $n$ loops around
$n_{2}$, where $n$ is the number of times $\alpha$ fully crosses $D$.

###### Proof.

Without loss of generality, we will assume that $\alpha$ is strictly monotone
increasing, that is, the numerical value $\alpha^{i}$ is strictly less than
that of $\alpha^{t}$, as the conclusion is invariant under replacing $\alpha$
by $\overline{\alpha}$. As in Section 3.2.1, put $\alpha$ in standard
position. Since no segment in standard position which is supported on
$(n_{1},n_{2})$ will be completely contained in the domain of the shift, we
may write

$\alpha=\alpha^{d}_{1}+\alpha^{e}_{1}+\dots+\alpha^{d}_{s}+\alpha^{e}_{s},$

where each $\alpha_{j}^{d}$ is a maximal subsegment disjoint from $D$, each
$\alpha_{j}^{e}$ fully crosses $D$, and $\ell_{c}(\alpha_{j}^{d})\geq 2$,
$\ell_{c}(\alpha_{j}^{e})=2$ when non-empty. Notice that if
$\alpha^{d}_{j}\neq\emptyset$ and $\alpha^{d}_{j+1}\neq\emptyset$, then also
$\alpha^{e}_{j}\neq\emptyset$ by maximality.

Under the above decomposition, in an unreduced code we have

$h(\alpha)=\alpha^{d}_{1}+h(\alpha^{e}_{1})+\dots+\alpha^{d}_{s}+h(\alpha^{e}_{s}),$

where every non-empty $\alpha^{e}_{j}$ will have image

$h(\alpha^{e}_{j})=\partial
D|_{[(\alpha^{e}_{j})^{i},n_{2})}(n_{2})_{o/u}(n_{2})_{u/o}\overline{\partial
D|_{[(\alpha^{e}_{j})^{t},n_{2})}}.$

Thus each full crossing $\alpha_{j}^{e}$ contributes a loop around $n_{2}$ in
an unreduced code for $h(\alpha)$. We must show that such loops persist in a
reduced code for $h(\alpha)$.

If $\alpha^{e}_{j}$, $\alpha^{e}_{j+1}\neq\emptyset$ and
$\alpha^{d}_{j+1}=\emptyset$ then a calculation shows that in a reduced code

$h(\alpha^{e}_{j}+\alpha^{d}_{j+1}+\alpha^{e}_{j+1})=h(\alpha^{e}_{j}+\alpha^{e}_{j+1})=h(\alpha^{e}_{j})+h(\alpha^{e}_{j+1}).$

In particular, there is no cancellation between $h(\alpha^{e}_{j})$ and
$h(\alpha^{e}_{j+1})$ and the loops around $n_{2}$ persist. See Figure 24.

\begin{overpic}[width=252.94499pt]{monotonesegment} \end{overpic} Figure 24.
Above: The case where
$\alpha_{j}^{e}+\alpha_{j}^{d}+\alpha_{j+1}^{e}=\alpha_{j}^{e}+\alpha_{j+1}^{e}$
in the proof of Lemma 4.7, where $\alpha_{j}^{e}$ is in purple and
$\alpha_{j+1}^{e}$ is in red. Below: the non-trivial image of this segment
under $h$.

Now assume that $\alpha_{j+1}^{d}$ is nonempty. The maximality of
$\alpha_{j+1}^{d}$ implies that $\alpha^{e}_{j}$ and $\alpha^{e}_{j+1}$ are
nonempty as well. We must show that there is no cancellation between the loops
around $n_{2}$ in $h(\alpha^{e}_{j})$ and $h(\alpha^{e}_{j+1})$ so that both
loops persist in a reduced code for
$h(\alpha^{e}_{j})+\alpha^{d}_{j+1}+h(\alpha^{e}_{j+1})$. Recall that in
standard position, a full crossing occurs between two adjacent characters
$k_{o/u}$, $(k^{\prime})_{o/u}$ with $k,k^{\prime}\in(n_{1},n_{2})$ such that
the $o/u$ pattern of $k$ and/or $k^{\prime}$ does not match that of $\partial
D$, and our convention is to make this choice for the largest possible
$k,k^{\prime}$.

The subtlety arises because the code for $h(\alpha^{e}_{j})+\alpha^{d}_{j+1}$
may not be reduced if an initial subsegment of
$\alpha^{d}_{j+1}=h(\alpha_{j+1}^{d})$ agrees with $\partial D$, in which case
$(\alpha_{j}^{e})^{t}$ agrees with $\partial D$. If $(\alpha_{j}^{e})^{t}$ is
the character of the full crossing $\alpha^{e}_{j}$ that does _not_ agree with
$\partial D$, then this character will block cancellation between
$h(\alpha^{e}_{j})$ and $\alpha^{d}_{j+1}$, so that the loops around $n_{2}$
in $h(\alpha^{e}_{j})$ and $h(\alpha^{e}_{j+1})$ cannot cancel. On the other
hand, if $(\alpha_{j+1}^{e})^{i}$ is the character of the full crossing
$\alpha^{e}_{j+1}$ that does _not_ agree with $\partial D$, then this
character will block cancellation between the two $n_{2}$ loops, even if
$\alpha_{j+1}^{d}$ fully cancels in a reduced code for
$h(\alpha^{e}_{j})+\alpha^{d}_{j+1}+h(\alpha^{e}_{j+1})$.

In the last case, where $(\alpha_{j}^{e})^{t}$ and $(\alpha_{j+1}^{e})^{i}$
both agree with $\partial D$, the largest possible choice of $k,k^{\prime}$
for the full crossing $\alpha_{j}^{e}$ would in fact result in a segment
$\alpha^{e}_{j}+\alpha^{d}_{j+1}+\alpha^{e}_{j+1}$ that is fully disjoint from
$D$, contradicting the maximality of $\alpha_{j+1}^{d}$, so that this case
cannot occur. See Figure 25.

Thus, there is no cancellation between the loops around $n_{2}$ in
$h(\alpha^{e}_{j})$ and $h(\alpha^{e}_{j+1})$, which proves the lemma. ∎

$\alpha_{j}^{e}$
---
$\alpha_{j+1}^{e}$
---
$\alpha_{j+1}^{d}$
---
Figure 25. Above: A picture of
$\alpha^{e}_{j}+\alpha^{d}_{j+1}+\alpha^{e}_{j+1}$, not in standard position,
where $(\alpha_{j}^{e})^{t}$ and $(\alpha_{j+1}^{e})^{i}$ both agree with
$\partial D$. Below: The same subsegment of $\alpha$ in standard position.

###  4.2. The Loop Theorem

In this subsection, we prove the Loop Theorem, which describes the form a loop
must have if its image under a shift is trivial. In particular, the image of
any loop which does not have the form as stated in the theorem is non-trivial.
In addition, any segment that is not a loop cannot have trivial image under a
shift since the numerical values the initial and terminal characters, and thus
their images, differ.

######  Theorem 4.8 (Loop Theorem).

Let $h$ be a permissible right shift with turbulent region $(n_{1},n_{2})$.
Suppose $\beta$ is a non-trivial loop such that $h(\beta)=\emptyset$. Then
either $\beta=k_{o}\delta k_{o}$ or $\beta=k_{u}\delta k_{u}$ where

  1. (i)

$k\in(n_{1},n_{2})$,

  2. (ii)

$\delta=\gamma(n_{1})_{o/u}(n_{1})_{u/o}\bar{\gamma}$, and

  3. (iii)

$\gamma$ follows $\partial D$ between $k$ and $n_{1}$.

###### Proof.

Put $\beta$ in standard position. We will consider the image $h(\beta)$. By
the discussion in Section 3.4, we have $\beta=h^{-1}(h(\beta))$. By
assumption, $h(\beta)$ is trivial, and therefore it is homotopic rel endpoints
to either a segment contained in the separating curve $S_{k}$ for some
$k\in[n_{1},n_{2})$ or a point. If $h(\beta)$ is homotopic rel endpoints to a
point, then $\beta=h^{-1}(h(\beta))$ is also homotopic rel endpoints to a
point, in which case $\beta$ is trivial, which is a contradiction. So suppose
that $h(\beta)$ is homotopic rel endpoints to an embedded subsegment $\sigma$
of the separating curve $S_{k}$ for some $k\in(n_{1},n_{2}]$. Then $\beta$ is
homotopic rel endpoints to $h^{-1}(\sigma)$.

Let $D$ denote the domain of $h$. Since $\beta$ is in standard position, the
endpoints of $\sigma$ (which are the endpoints of $\beta$) are not contained
in $D$. There are two possibilities. If $\sigma\cap D=\emptyset$, then $\beta$
is homotopic rel endpoints to $h^{-1}(\sigma)=\sigma$ and so $\beta$ is
trivial, which is a contradiction. On the other hand, if $\sigma\cap
D\neq\emptyset$, then $\sigma$ must fully cross $D$. A direct computation
shows that, up to reversing orientation,

$h^{-1}(\sigma)=\overline{\partial
D}|_{[k,n_{1})}(n_{1})_{o/u}(n_{1})_{u/o}\partial D|_{(n_{1},k]},$

for $k\in(n_{1},n_{2})$, as desired. ∎

We note one immediate consequence of Theorem 4.8.

######  Corollary 4.9.

Suppose $\beta$ is any of the following:

  1. (1)

a loop containing more than one single loop;

  2. (2)

a loop containing a back loop; or

  3. (3)

a loop with endpoints outside of the turbulent region.

Then $h(\beta)$ is non-trivial.

Some of these consequences may be surprising, so we now give a more intuitive
(and informal) discussion of why the image of loops as in the corollary are
non-trivial.

We first consider the case of segments which contain more than one single
loop. At first glance, it may seem that if a segment $\beta$ is composed of
loops which each satisfy the conclusion of Theorem 4.8, then $h(\beta)$ will
be trivial. The problem arises in how these loops fit together to form
$\beta$. Suppose we have two loops, $\beta_{1}$ and $\beta_{2}$ which each
satisfy the conclusion of Theorem 4.8. If the numerical value of
$\beta_{1}^{t}$ is not the same as that of $\beta_{2}^{i}$, then in order to
“connect” $\beta_{1}$ to $\beta_{2}$, we must add a segment between the
terminal point of $\beta_{1}$ and the initial point of $\beta_{2}$. This
segment is either disjoint from $D$, in which case it is fixed by $h$, or it
fully crosses $D$. In the former case, this segment will appear in the image
of $\beta$, while in the latter case, the image of this full crossing will be
non-trivial by Lemma 4.7 applied to the full crossing. If the numerical values
are the same and $\beta=\beta_{1}\beta_{2}$, then $\beta$ will be trivial. On
the other hand, if there is some segment connecting the terminal point of
$\beta_{1}$ to the initial point of $\beta_{2}$, then as before, the image of
this segment will not be trivial and will force $h(\beta)$ to be non-trivial.
See Figure 26 for an example of this.

Suppose next that $\beta$ contains a back loop $C$. Intuitively, it seems
reasonable that if $\beta=C\beta^{\prime}\bar{C}$ and $\beta^{\prime}$ is a
loop as in Theorem 4.8, then $h(\beta)$ is trivial. However, this is not the
case. To see this, notice that if we put $\beta^{\prime}$ in standard position
then it will have one endpoint above $D$ and one below $D$. However, $C$ and
$\bar{C}$ are the same loop. If $C$ is either above or below $D$, then this
will cause $\beta$ to have a full crossing between $\beta^{\prime}$ and one of
$C$ or $\bar{C}$. The image of this full crossing will be non-trivial by Lemma
4.7 applied to the full crossing, which will prevent any cancellation between
$h(C)=C$ and $h(\bar{C})=\bar{C}$. On the other hand, if $C$ exits at the top
and enters at the bottom, then $\bar{C}$ exits at the bottom and enters at the
top. This will again force $\beta$ to have a full crossing between
$\beta^{\prime}$ and one of $C$ or $\bar{C}$, preventing cancellation between
the images of the back loops.

Finally, if $\beta$ is a loop whose endpoints are outside the turbulent
region, say both in $[n_{2},\infty)$, and $\beta|_{(n_{1},n_{2})}$ has the
form in the conclusion of Theorem 4.8, it seems possible that $h(\beta)$ is
trivial. For example, suppose that
$\beta=(n_{2})_{o}\beta^{\prime}(n_{2})_{o}$, where $\beta^{\prime}$ is as in
Theorem 4.8. Then since $h(\beta^{\prime})$ is trivial, $h(\beta^{\prime})$
will be a segment contained in the separating curve $S_{n_{2}}$. On the other
hand, the images of the connectors $(n_{2})_{o}(\beta^{\prime})^{i}$ and
$(\beta^{\prime})^{t}(n_{2})_{o}$ will each be of the form
$(n_{2}+1)_{o}(n_{2})_{o/u}(n_{2}-1)_{o/u}$ or
$(n_{2}-1)_{o/u}(n_{2})_{o/u}(n_{2}+1)_{o}$. In particular, we have
$h(\beta)=(n_{2}+1)_{o}(n_{2})_{o/u}(n_{2})_{u/o}(n_{2}+1)_{o}$, which is non-
trivial.

\begin{overpic}[width=216.81pt]{compoundloopcor.pdf}
\put(15.0,90.0){\tiny${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\beta_{1}}$}
\put(28.0,90.0){\tiny${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\beta_{2}}$}
\put(24.0,96.0){\tiny$\beta$} \put(54.0,65.0){\tiny$h(\beta)$}
\put(22.5,58.5){\tiny$C$} \put(25.75,58.5){\tiny$\overline{C}$}
\put(49.0,24.0){\tiny$\beta$} \put(65.0,13.0){\tiny$h(\beta)$} \end{overpic}
Figure 26. Examples of loops whose images are non-trivial. From top to bottom,
these correspond to Corollary 4.9 (1), (2), and (3).

###  4.3. Consequences of the Loop Theorem

In this subsection, we record several (technical) consequences of Theorem 4.8
which will be useful in later sections. The first shows that loops in the
shift region persist under the image of shifts.

######  Lemma 4.10.

Let $h$ be a permissible right shift and $\gamma$ any non-trivial, simple arc
with no back loops. If $k\in(-\infty,n_{1})\cup[n_{2},\infty)$ and
$k_{o/u}k_{u/o}$ appears in a reduced code for $\gamma$, then there is no
cancellation involving $h(k_{o/u}k_{u/o})$ in an unreduced code for
$h(\gamma)$. In particular, such a pair $k_{o/u}k_{u/o}$ will yield a pair
$(k+1)_{o/u}(k+1)_{u/o}$ in a reduced code for $h(\gamma)$.

###### Proof.

We argue for $k_{o}k_{u}$ without loss of generality. Fix $k$ as above and put
$\gamma$ into standard position. If in standard position $\gamma$ is
completely contained in the domain of $h$, then the result is clear, so
suppose this is not the case. We assume for contradiction that there is
cancellation with $h(k_{o}k_{u})$ and, without loss of generality, that it is
with $h(k_{u})=(k+1)_{u}$, as otherwise we replace $\gamma$ with
$\overline{\gamma}$ and argue identically for $k_{o}$ instead. Let
$\gamma^{\prime}$ be the minimal subsegment of $\gamma$ beginning with
$k_{o}k_{u}$ so that $h(\gamma^{\prime})$ had cancellation with $h(k_{u})$.
The only way cancellation in an unreduced code can involve $h(k_{u})$ is if a
subsegment of the form $(k+1)_{o}(k+1)_{u}\delta(k+1)_{u}$ appears in an
unreduced $h(\gamma^{\prime})$ where a reduced code for $\delta$ is trivial.
Since $k\in(-\infty,n_{1})\cup[n_{2},\infty)$, any $(k+1)_{u}$ in
$h(\gamma^{\prime})$ must be the image under $h$ of $k_{u}$. Therefore,
$\gamma^{\prime}=k_{o}k_{u}\eta k_{u}$ where $h(k_{u}\eta
k_{u})=(k+1)_{u}\delta(k+1)_{u}$, which has trivial reduced code. This
contradicts Theorem 4.8 as $k\notin(n_{1},n_{2})$. ∎

The second consequence of our loop theorems shows that characters of a segment
that lie in the turbulent region which disagree with the domain of a shift
persist in a reduced code of the image of the segment under that shift.

######  Lemma 4.11.

Let $h$ be a permissible right shift with domain $D$. Let $\delta$ be a simple
segment whose support intersects $(n_{1},n_{2})$ non-trivially, and suppose
$\delta$ contains a character $b$ with numerical value in $(n_{1},n_{2})$
which disagrees with $\partial D$. Then $b$ persists in a reduced code for
$h(\delta)$.

In other words, if $\delta=\delta_{1}b\delta_{2}$, then
$h(\delta)=\sigma_{1}b\sigma_{2}$, where $\sigma_{1}b=h(\delta_{1}b)$ and
$b\sigma_{2}=h(b\delta_{2})$.

###### Proof.

Write $b=k_{o/u}$, where by assumption $k\in(n_{1},n_{2})$.

######  Claim 4.12.

Since $b$ disagrees with $\partial D$, any occurrence of $b$ in $h(\delta)$
must also appear in $\delta$.

###### Proof.

Since $b$ disagrees with $\partial D$, the segment $\delta$ cannot be
homotoped rel endpoints to be completely contained in $D$. Thus in standard
position, $\delta$ can be written as the efficient concatenation of
subsegments which are disjoint from $D$, subsegments of length two which are
either full or half crossings, and subsegments supported on
$(-\infty,n_{1}]\cup[n_{2},\infty)$ (see Lemma 3.16). We consider the images
of each type of subsegment in turn. The subsegments which are disjoint from
$D$ are fixed by $h$. The image of subsegments supported on
$(-\infty,n_{1})\cup[n_{2},\infty)$ have empty intersection with
$(n_{1},n_{2})$, and so cannot contain $b$. The remaining subsegments are half
crossings, full crossings, or segments that include a character whose
numerical value is $n_{1}$. Any subsegment of the image of any of these
subsegments supported on $(n_{1},n_{2})$ agrees with $\partial D$ and so
cannot contain $b$. This proves the claim. ∎

Now suppose towards a contradiction that $b$ does not persist in $h(\delta)$.
Then there must be another instance of $b$ in an unreduced code for
$h(\delta)$ which cancels with $b$. By the claim, $\delta$ must contain a
subsegment of the form $b\sigma b$ whose image is trivial. Theorem 4.8 then
implies that $b\sigma b=\overline{\partial
D|_{(n_{1},k]}}(n_{1})_{o/u}(n_{1})_{u/o}\partial D|_{(n_{1},k]}$. However,
$b=k_{o/u}$ disagrees with $\partial D$ by assumption, and therefore $b\sigma
b$ cannot have this form, which is a contradiction. ∎

##  5\. Constructing the homeomorphisms

As in Section 3, let $S$ be the biinfinite flute with an isolated puncture
$p$, and let $\\{B_{i}\mid i\in\mathbb{Z}\cup P\\}$ be the simple closed
curves from Definition 3.1 bounding a collection of punctures $\\{p_{i}\mid
i\in\mathbb{Z}\cup P\\}$, where $p_{P}=p$. As in that section, we move the
punctures $\\{p_{i}\mid i\in\mathbb{Z}\cup P\\}$ to the front of $S$ and all
other punctures to the back of $S$.

In this section, we define a countable collection of elements
$\\{g_{n}\\}_{n\in\mathbb{N}}$ in $\mathrm{Map}(S,p)$. In the following
sections we will show that these elements are of intrinsically infinite type
and are loxodromic with respect to the action of $\mathrm{Map}(S,p)$ on
$\mathcal{A}(S,p)$.

######  Definition 5.1.

For each $n\in\mathbb{N}$, define

(3) $g_{n}:=h^{(n)}_{3}\circ h^{(n)}_{2}\circ h_{1},$

where $h_{1}$ and $h_{i}^{(n)}$ for $i=2,3$ are permissible shifts defined as
follows. Let $h_{1}$ be the right shift whose domain includes all simple
closed curves $B_{j}$ for $j\in\mathbb{Z}\setminus\\{0\\}$ and passes under
$p$ and $B_{0}$. Let $h_{2}^{(n)}$ be the left shift whose domain includes
only $\\{B_{j}\mid j\in(-\infty,-n-1]\cup[n+1,\infty)\\}$, passes over $p$ and
$B_{0}$, and, when $j\in[-n,-1]\cup[1,n]$, passes under $B_{j}$ for $j$ odd
and over $B_{j}$ for $j$ even. Finally, let $h_{3}^{(n)}$ be the right shift
whose domain includes only $\\{B_{j}\mid
j\in(-\infty,-n-1]\cup[n+1,\infty)\\}$ and passes under $p$ and over $B_{j}$
for all $j\in[-n,P)\cup(P,n]$. See Figure 27.

\begin{overpic}[width=180.67499pt]{theshiftsh1} \put(3.0,5.0){\tiny{$-3$}}
\put(15.5,5.0){\tiny{$-2$}} \put(29.0,5.0){\tiny{$-1$}}
\put(42.5,5.0){\tiny$P$} \put(54.5,5.0){\tiny{$0$}} \put(67.5,5.0){\tiny{$1$}}
\put(79.5,3.0){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tiny{$h_{1}$}}
\end{overpic} (a) The shift map $h_{1}$.

\begin{overpic}[width=607.06534pt]{theshiftsh2} \put(53.0,60.0){\tiny{$0$}}
\put(61.5,60.0){\tiny{$1$}} \put(70.0,60.0){\tiny{$2$}}
\put(78.5,60.0){\tiny{$3$}} \put(45.0,60.0){\tiny{$P$}}
\put(36.0,60.0){\tiny{$-1$}}
\put(28.5,58.75){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tiny{$h_{2}^{(1)}$}}
\put(53.0,41.0){\tiny{$0$}} \put(61.5,41.0){\tiny{$1$}}
\put(70.0,41.0){\tiny{$2$}} \put(78.5,41.0){\tiny{$3$}}
\put(45.0,41.0){\tiny{$P$}} \put(36.0,41.0){\tiny{$-1$}}
\put(28.5,40.0){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tiny{$h_{2}^{(2)}$}}
\put(53.0,22.0){\tiny{$0$}} \put(61.5,22.0){\tiny{$1$}}
\put(70.0,22.0){\tiny{$2$}} \put(86.25,22.0){\tiny{$n+1$}}
\put(80.25,22.0){\tiny{$n$}} \put(45.0,22.0){\tiny{$P$}}
\put(36.0,22.0){\tiny{$-1$}} \put(27.5,22.0){\tiny{$-2$}}
\put(16.5,22.0){\tiny{$-n$}}
\put(9.5,20.75){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tiny{$h_{2}^{(n)}$}}
\put(53.0,3.0){\tiny{$0$}} \put(61.5,3.0){\tiny{$1$}}
\put(70.0,3.0){\tiny{$2$}} \put(86.25,3.0){\tiny{$n+1$}}
\put(80.25,3.0){\tiny{$n$}} \put(45.0,3.0){\tiny{$P$}}
\put(36.0,3.0){\tiny{$-1$}} \put(27.5,3.0){\tiny{$-2$}}
\put(16.5,3.0){\tiny{$-n$}}
\put(9.5,1.75){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tiny{$h_{2}^{(n)}$}}
\end{overpic} (b) The shift map $h_{2}^{(n)}$ for various $n$. In order from
top to bottom, the first two pictures are the case of $n=1,2$ (respectively)
and the bottom two are for general $n$ odd and general $n$ even
(respectively).

\begin{overpic}[width=607.06534pt]{theshiftsh3} \put(12.0,51.0){\tiny{$-3$}}
\put(23.0,51.0){\tiny{$-2$}} \put(33.5,51.0){\tiny{$-1$}}
\put(53.5,51.0){\tiny{$0$}} \put(64.0,51.0){\tiny{$1$}}
\put(44.0,51.0){\tiny{$P$}}
\put(71.0,49.0){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tiny{$h_{3}^{(1)}$}}
\put(1.5,27.5){\tiny{$-4$}} \put(12.0,27.5){\tiny{$-3$}}
\put(23.0,27.5){\tiny{$-2$}} \put(33.5,27.5){\tiny{$-1$}}
\put(53.5,27.5){\tiny{$0$}} \put(64.0,27.5){\tiny{$1$}}
\put(75.0,27.5){\tiny{$2$}} \put(44.0,27.5){\tiny{$P$}}
\put(80.0,25.5){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tiny{$h_{3}^{(2)}$}}
\put(0.0,4.0){\tiny{$-n-2$}} \put(10.0,4.0){\tiny{$-n-1$}}
\put(22.5,4.0){\tiny{$-n$}} \put(33.5,4.0){\tiny{$-1$}}
\put(53.5,4.0){\tiny{$0$}} \put(64.0,4.0){\tiny{$1$}}
\put(75.0,4.0){\tiny{$n$}} \put(44.0,4.0){\tiny{$P$}}
\put(79.0,2.0){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tiny{$h_{3}^{(n)}$}}
\end{overpic} (c) The shift map $h_{3}^{(n)}$ for various $n$. In order from
top to bottom, the first two pictures are the case of $n=1,2$ and the bottom
picture is the case of general $n$.

Figure 27. The various shift maps used in the construction of
$g_{n}=h_{3}^{(n)}\circ h_{2}^{(n)}\circ h_{1}$.

In the language of the previous sections, we have that $n_{1}=-1$ and
$n_{2}=1$ for $h_{1}$. For $h_{2}^{(n)}$, $n_{1}=n+1$ and $n_{2}=-n-1$. For
$h_{3}^{(n)}$, $n_{1}=-n-1$ and $n_{2}=n+1$.

In order to prove that each $g_{n}$ is loxodromic with respect to the action
of $\mathrm{Map}(S,p)$ on $\mathcal{A}(S,p)$, we introduce a collection of
arcs $\\{\alpha_{i}^{(n)}\mid i\in\mathbb{Z}\\}$ which are invariant under
$g_{n}$. We will show in Proposition 8.2 that for $i\geq 0$, these arcs form a
quasi-geodesic half-axis for $g_{n}$ in $\mathcal{A}(S,p)$.

######  Definition 5.2.

For each fixed $n$, we define a sequence of arcs
$\\{\alpha_{i}^{(n)}\\}_{i=-\infty}^{\infty}$ in $\mathcal{A}(S,p)$ as the
images under successive applications of $g_{n}$ of the fixed initial arc
$\alpha_{0}=P_{s}0_{o}0_{u}P_{s}$. That is to say that
$\alpha^{(n)}_{i}=g^{i}_{n}(\alpha_{0})$ for any $i\in\mathbb{Z}$. See Figure
28 for the case $n=1$ and Figure 29 for general $n\geq 1$.

$-2$
---
$-1$
---
$0$
---
$1$
---
$\scriptstyle{4}$
---
$\scriptstyle{8}$
---
$\scriptstyle{2}$
---
$\scriptstyle{1}$
---
$\scriptstyle{2}$
---
$\scriptstyle{12}$
---
$\scriptstyle{6}$
---
$\alpha_{-1}^{(1)}$
---
$\alpha_{0}$
---
$\scriptstyle{1}$
---
$2$
---
$-1$
---
$1$
---
$3$
---
$0$
---
$2$
---
$-1$
---
$1$
---
$3$
---
$0$
---
$2$
---
$-1$
---
$1$
---
$3$
---
$0$
---
$\alpha_{1}^{(1)}$
---
$\scriptstyle{1}$
---
$\alpha_{2}^{(1)}$
---
$\scriptstyle{8}$
---
$\scriptstyle{4}$
---
$\scriptstyle{2}$
---
$\scriptstyle{1}$
---
$\scriptstyle{10}$
---
$\scriptstyle{4}$
---
$\scriptstyle{6}$
---
$\scriptstyle{2}$
---
$\scriptstyle{4}$
---
$\scriptstyle{10}$
---
Figure 28. The arcs $\alpha_{i}^{(1)}$ for $i=-1,0,1,2$ drawn as train tracks
for simplicity.

When $n=1$, the arcs $\alpha_{i}^{(1)}$ are the most straightforward and thus
most useful for building intuition. We suggest that the reader keep these arcs
in mind while reading the remainder of the paper. We now give the code for
$\alpha_{1}^{(1)}$ and $\alpha_{2}^{(1)}$, which the reader can compare to
Figure 28:

$\displaystyle\alpha_{1}^{(1)}=$ $\displaystyle
P_{s}0_{o}1_{o}2_{o}2_{u}1_{o}0_{o}0_{u}1_{o}2_{u}2_{o}1_{o}0_{o}P_{s}$
$\displaystyle\alpha_{2}^{(1)}=$ $\displaystyle
P_{s}0_{o}1_{o}2_{o}2_{u}1_{o}0_{o}P_{u}(-1)_{u}(-1)_{o}P_{u}0_{o}1_{o}2_{o}2_{u}1_{o}0_{o}P_{u}(-1)_{o}(-1)_{u}P_{u}0_{o}1_{o}2_{u}2_{o}1_{o}$
$\displaystyle
0_{o}P_{u}P_{o}0_{o}1_{o}2_{o}3_{o}3_{u}2_{o}1_{o}0_{o}P_{o}P_{u}0_{o}1_{o}2_{o}2_{u}1_{o}0_{o}P_{u}(-1)_{u}(-1)_{o}P_{u}0_{o}1_{o}2_{u}2_{o}1_{o}0_{o}$
$\displaystyle
P_{u}(-1)_{o}(-1)_{u}P_{u}0_{o}1_{o}2_{u}2_{o}1_{o}0_{o}P_{u}P_{o}0_{o}1_{o}2_{o}2_{u}1_{o}0_{o}0_{u}1_{o}2_{u}2_{o}1_{o}0_{o}P_{o}P_{u}0_{o}1_{o}2_{o}$
$\displaystyle
2_{u}1_{o}0_{o}P_{u}(-1)_{u}(-1)_{o}P_{u}0_{o}1_{o}2_{o}2_{u}1_{o}0_{o}P_{u}(-1)_{o}(-1)_{u}P_{u}0_{o}1_{o}2_{u}2_{o}1_{o}0_{o}P_{u}P_{o}0_{o}$
$\displaystyle
1_{o}2_{o}3_{u}3_{o}2_{o}1_{o}0_{o}P_{o}P_{u}0_{o}1_{o}2_{o}2_{u}1_{o}0_{o}P_{u}(-1)_{u}(-1)_{o}P_{u}0_{o}1_{o}2_{u}2_{o}1_{o}0_{o}P_{u}(-1)_{o}(-1)_{u}$
$\displaystyle P_{u}0_{o}1_{o}2_{u}2_{o}1_{o}0_{o}P_{s}$

While it is possible to compute the images of arcs under
$g_{n}=h_{3}^{(n)}\circ h_{2}^{(n)}\circ h_{1}$ by hand, we have also written
a computer program to implement this. The interested reader should contact the
authors for more details.

$n+1$
---
$-1$
---
$n+2$
---
$0$
---
$-2$
---
$-n+1$
---
$-n$
---
$\alpha_{2}^{(n)}$
---
$\scriptstyle{8}$
---
$\scriptstyle{4}$
---
$\scriptstyle{2}$
---
$\scriptstyle{1}$
---
$\scriptstyle{10}$
---
$\scriptstyle{4}$
---
$\scriptstyle{6}$
---
$\scriptstyle{2}$
---
$\scriptstyle{4}$
---
$\scriptstyle{10}$
---
$n+1$
---
$-1$
---
$n+2$
---
$0$
---
$\scriptstyle{1}$
---
$\alpha_{1}^{(n)}$
---
$\alpha_{0}$
---
$\scriptstyle{1}$
---
$2$
---
$-1$
---
$1$
---
$3$
---
$0$
---
$n-1$
---
$-1$
---
$n$
---
$0$
---
$-2$
---
$-n$
---
$-3$
---
$-n-1$
---
$1$
---
$2$
---
$\scriptstyle{4}$
---
$\scriptstyle{8}$
---
$\scriptstyle{10}$
---
$\scriptstyle{1}$
---
$\scriptstyle{12}$
---
$\scriptstyle{6}$
---
$\scriptstyle{6}$
---
$\alpha_{-1}^{(n)}$
---
Figure 29. The arcs $\alpha_{i}^{(n)}$ for $i=-1,0,1,2$ and $n\geq 1$. In
$\alpha_{-1}^{(n)}$, the blue appears when $n$ is odd, while the green appears
when $n$ is even.

######  Remark 5.3.

In light of how complicated the arcs $\alpha_{i}^{(n)}$ are when $i\neq 0,1$,
it may be surprising that only single loops can have trivial image under each
of the shifts $h_{1},h_{2}^{(n)},$ and $h_{3}^{(n)}$ (see Theorem 4.8 and
Corollary 4.9). Figure 30 gives the intermediate steps so that one can see how
the image under $g_{1}$ of a complicated arc such as $\alpha^{(1)}_{-1}$
becomes the straightforward arc $\alpha_{0}$. The figure shows the collection
of single loops with trivial image at each stage of computing
$g_{1}(\alpha^{(1)}_{-1})=\alpha_{0}$. A similar phenomenon happens for $n>1$.

$\alpha_{-1}^{(1)}$
---
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}h_{1}}$
---
$h_{1}(\alpha_{-1}^{(1)})$
---
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}h_{2}^{(1)}}$
---
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}h_{3}^{(1)}}$
---
$h_{2}^{(1)}\circ h_{1}(\alpha_{-1}^{(1)})$
---
$\alpha_{0}$
---
Figure 30. Each of the strands in blue are single loops whose image is trivial
under the shift whose domain is shown in red. Each blue strand fits the form
of Theorem 4.8.

Our method for using the arcs $\alpha_{i}^{(n)}$ to prove that $g_{n}$ is a
loxodromic isometry of $\mathcal{A}(S,p)$ is inspired by the foundational work
of Bavard in [5]. Bavard constructs an element $f$ of the mapping class group
of the plane minus a Cantor set which is loxodromic with respect to the
associated relative arc graph. We give a brief outline of Bavard’s methods
here.

Bavard constructs a collection of simple paths $\beta_{i}$ which start at an
isolated puncture and end at some point of the Cantor set. These paths are
invariant under the action of a chosen homeomorphism $f$ and have the property
that $\beta_{i+1}=f(\beta_{i})$. They are constructed so that for all $i$, the
path $\beta_{i+1}$ begins by following the same path as $\beta_{i}$. Roughly,
if a path $\gamma$ begins by following the same path as $\beta_{i}$ then
Bavard says the path $\gamma$ ‘begins like’ $\beta_{i}$ (see [5, Section 2]).
Bavard uses this definition to define a function from the vertex set of a
certain graph of paths (defined similarly to the relative arc graph but with
paths instead of arcs) to $\mathbb{Z}_{\geq 0}$ by sending a path $\gamma$ to
the maximal $i\in\mathbb{Z}_{\geq 0}$ so that $\gamma$ begins like
$\beta_{i}$. Aramayona, Fossas, and Parlier [2] show that the relative arc
graph is quasi-isometric to the graph whose vertex set is isotopy class of
paths with at least one endpoint on the distinguished isolated puncture.
Therefore this function can then be used to estimate distances in the relative
arc graph and ultimately to show that the collection of paths
$\\{\beta_{i}\mid i\in\mathbb{Z}_{\geq 0}\\}$ forms a geodesic half-axis for
the element $f$. The key fact that Bavard uses is that if $\delta$ is a path
which begins like $\beta_{i}$ and $\gamma$ is any path disjoint from $\delta$,
then $\gamma$ begins like $\beta_{i-1}$ [5, Lemma 2.4].

Our arcs $\\{\alpha_{i}^{(n)}\\}_{i\in\mathbb{Z}}$ do not satisfy the same
property as Bavard’s paths. One notable difference is that our arcs start and
end at the puncture $p$. Because of this, for a fixed $n$,
$\alpha_{i+1}^{(n)}$ does not begin by following the entirety of
$\alpha_{i}^{(n)}$. However, we will show that it does begin by following the
first half of $\alpha_{i}^{(n)}$. In light of this, we modify Bavard’s notion
of ‘begins like’ and make the following definition.

######  Definition 5.4.

Fix an $n\in\mathbb{N}$. An arc $\delta$ starts like $\alpha^{(n)}_{i}$ if the
maximal initial or the maximal terminal segment of $\delta$ which agrees with
an initial or terminal segment of $\alpha^{(n)}_{i}$ (not necessarily
respectively) has code length at least
$\lfloor\frac{1}{2}\ell_{c}(\alpha^{(n)}_{i})\rfloor$. Recall that code length
was defined in Definition 3.6.

In Section 6, we investigate the properties of arcs that start like
$\alpha_{i}^{(n)}$ for some $i$, and in Section 7 we use these properties to
prove Theorem 7.10, which is an analog of [5, Lemma 2.4]. In our more general
context, proving this result is quite a bit more involved than the proof of
[5, Lemma 2.4]. One reason for this is that the behavior of our arcs
$\alpha_{i}^{(n)}$ is much more complicated than the paths from [5].

##  6\. Arcs that start like $\alpha_{i}^{(n)}$

The main goal of this section is to prove that the images of arcs which start
like $\alpha^{(n)}_{i}$ under $g_{n}$ must start like $\alpha^{(n)}_{i+1}$
(see Corollary 6.9). This will follow from Proposition 6.2 and Theorem 6.4.
Proposition 6.2 states that the image of the first half of $\alpha_{i}^{(n)}$
is the first half of $\alpha_{i+1}^{(n)}$. We need to be careful about what we
mean by this because of the possibility that we cause there to be a gap when
we break $\alpha_{i}^{(n)}$ into its first and second half (see Section 3.3).
Given a segment $\gamma$, let $\mathring{\gamma}$ be the initial subsegment of
$\gamma$ with code length $\lfloor\frac{1}{2}\ell_{c}(\gamma)\rfloor$. For a
fixed $i$ and $n$, we have
$\alpha_{i}^{(n)}=\mathring{\alpha}_{i}^{(n)}+\alpha^{\prime}$ for some
$\alpha^{\prime}$. As in Section 3.3, it is possible that when we put
$\mathring{\alpha}_{i}^{(n)}$ and $\alpha^{\prime}$ (or their images) into
standard position individually with respect to one of the shifts
$h_{1},h_{2}^{(n)}$, or $h_{3}^{(n)}$, we will lose information when we take
its image. In fact, this does happen when we put
$h_{2}^{(n)}(h_{1}(\mathring{\alpha}_{i}^{(n)}))$ and
$h_{2}^{(n)}(h_{1}(\alpha^{\prime}))$ into standard position with respect to
$h_{3}^{(n)}$ (see Figure 31). We avoid this issue by extending
$\mathring{\alpha}_{i}^{(n)}$ by one character and considering
$\mathring{\alpha}_{i}^{(n)}0_{u}$. Note that $0_{u}$ is fixed by $g_{n}$.
Thus when we say that the image of the first half of $\alpha_{i}^{(n)}$ is the
first half of $\alpha_{i+1}^{(n)}$ we precisely mean that
$g_{n}(\mathring{\alpha}_{i}^{(n)}0_{u})=\mathring{\alpha}_{i+1}^{(n)}0_{u}$
as a subsegment of $g_{n}(\alpha_{i}^{(n)})=\alpha_{i+1}^{(n)}$, in a reduced
code. In other words, $g_{n}$ sends everything before the central $0_{u}$ in
$\alpha_{i}^{(n)}$ to $\mathring{\alpha}_{i+1}^{(n)}$.

######  Remark 6.1.

When dealing with the arcs $\alpha_{i}^{(n)}$, the floor function in
$\mathring{\alpha}_{i}^{(n)}$ is actually unnecessary, as
$\ell_{c}(\alpha_{i}^{(n)})$ will always be even with the central two
characters being $0_{o}0_{u}$. However, at this point in the paper we have not
proven this fact, which is a consequence of Proposition 6.2, so we will not
use it in what follows.

$h_{2}^{(1)}$
---
$h_{1}(\mathring{\alpha}_{1}^{(1)})$
---
$h_{1}(0_{o}0_{u})$
---
$h_{2}^{(1)}\circ h_{1}(0_{o}0_{u})$
---
$h_{2}^{(1)}\circ h_{1}(\mathring{\alpha}_{1}^{(1)})$
---
$h_{3}^{(1)}$
---
Figure 31. Here we consider
$h_{1}(\mathring{\alpha}_{1}0_{u})=h_{1}(\mathring{\alpha}_{1})+h_{1}(0_{o}0_{u})$
and take the images of each subsegment separately. The images are drawn as
train tracks for clarity, but we omit the weights as they are immaterial. When
we put the images $h_{2}^{(n)}(h_{1}(\mathring{\alpha}_{1}))$ and
$h_{2}^{(n)}(h_{1}(0_{o}0_{u}))=0_{o}0_{u}$ in standard position individually
with respect to $h_{3}^{(n)}$ (the second picture), there is a gap in the
segment between $p$ and $B_{1}$, which would have been a full crossing. This