# Finding and Combining Indicable Subgroups of
Big Mapping Class Groups

Carolyn R. Abbott  Department of Mathematics, Brandeis University, Waltham, MA
02543<EMAIL_ADDRESS>,  Hannah Hoganson  Department of
Mathematics, University of Maryland, College Park, MD 20742<EMAIL_ADDRESS>,  Marissa Loving  Department of Mathematics, University of Wisconsin-Madison,
Madison, WI 53706<EMAIL_ADDRESS>,  Priyam Patel  Department of
Mathematics, University of Utah, Salt Lake City, UT 84112
<EMAIL_ADDRESS>and  Rachel Skipper  Department of Mathematics,
University of Utah, Salt Lake City, UT 84112<EMAIL_ADDRESS>
###### Abstract.

We explicitly construct new subgroups of the mapping class groups of an
uncountable collection of infinite-type surfaces, including, but not limited
to, right-angled Artin groups, free groups, Baumslag-Solitar groups, mapping
class groups of other surfaces, and a large collection of wreath products. For
each such subgroup $H$ and surface $S$, we show that there are countably many
non-conjugate embeddings of $H$ into $\operatorname{Map}(S)$; in certain
cases, there are uncountably many such embeddings. The images of each of these
embeddings cannot lie in the isometry group of $S$ for any hyperbolic metric
and are not contained in the closure of the compactly supported subgroup of
$\operatorname{Map}(S)$. In this sense, our construction is new and does not
rely on previously known techniques for constructing subgroups of mapping
class groups. Notably, our embeddings of $\operatorname{Map}(S^{\prime})$ into
$\operatorname{Map}(S)$ are not induced by embeddings of $S^{\prime}$ into
$S$. Our main tool for all of these constructions is the utilization of
special homeomorphisms of $S$ called shift maps, and more generally, multipush
maps.

##  1\. Introduction

A fundamental question in low-dimensional topology asks which groups can arise
as subgroups of the diffeomorphism group, homeomorphism group, and mapping
class group of a surface $S$, denoted by
$\operatorname{Homeo}(S),\operatorname{Diffeo}(S),$ and
$\operatorname{Map}(S)$, respectively. One approach to producing such
subgroups is to consider embeddings of finite-type subsurfaces $S^{\prime}$
into an infinite-type surface $S$ that induce injections of
$\operatorname{Map}(S^{\prime})$ into $\operatorname{Map}(S)$. In this case,
every subgroup of $\operatorname{Map}(S^{\prime})$ is a subgroup of
$\operatorname{Map}(S)$. Another approach to this problem is to show that a
particular group $G$ acts by orientation-preserving isometries on a surface
$S$, which implies that $G$ can be realized as a subgroup of
$\operatorname{Homeo}(S),\operatorname{Diffeo}(S),$ and
$\operatorname{Map}(S)$. However, these two classical approaches have
limitations. For example, the strong Tits alternative holds for finite-type
mapping class groups, meaning every subgroup of
$\operatorname{Map}(S^{\prime})$ is either virtually abelian or contains a
free subgroup [Iva84, McC85]. In addition, Aougab, Patel, and Vlamis show that
only finite groups can arise as the isometry group of a hyperbolic metric on
$S$ whenever $S$ contains a _non-displaceable subsurface_ (see [APV, Lemma
4.2]). They also show that no uncountable group can be obtained as the
isometry group of a hyperbolic metric on any infinite-type surface. These
observations indicate that in order to fully understand big mapping class
groups, we need other constructions of subgroups in $\operatorname{Map}(S)$,
and we also need to understand the many different ways that a particular
subgroup can embed in $\operatorname{Map}(S)$. This is precisely the goal of
this paper.

To streamline the statements of our results below, we construct two
uncountable collections of surfaces for which particular results hold. The
precise definitions will appear in Section 3.2; we give a brief idea of the
types of surfaces contained in each collection here. The first collection,
which we call $\mathcal{B}_{\infty}$, contains the surface $S$ whose end space
is a Cantor set of nonplanar ends (the blooming Cantor tree surface) along
with the connect sum of $S$ and any surface $S^{\prime}$ with only planar
ends; see Definition 3.7. When $\Pi$ is a distinguished surface, we denote by
$\mathcal{C}(\Pi)$ certain surfaces that admit a map which acts as a shift
along a countable collection of copies of $\Pi$; see Definition 3.5. For
example, $\Pi$ could be a torus with one boundary component, in which case
$\mathcal{C}(\Pi)$ includes the ladder surface and the connect sum of a ladder
surface and any surface with only planar ends.

Our first construction produces right-angled Artin groups in
$\operatorname{Map}(S)$ that do not lie in
$\overline{\operatorname{Map}_{c}(S)}$. Although constructions of right-angled
Artin groups in the finite-type setting (e.g., the Clay–Leininger–Mangahas
Embedding Theorem [CLM12]) port immediately to the infinite-type setting
through subsurface inclusion, we emphasize that we construct subgroups of
$\operatorname{Map}(S)$ not arising from finite-type behavior.

######  Theorem 1.1.

For any surface $S\in\mathcal{B}_{\infty}$, there exists an infinite family of
non-isomorphic right-angled Artin subgroups of $\operatorname{Map}(S)$, each
of which embeds into $\operatorname{Map}(S)$ in countably many non-conjugate
ways. Moreover, the image of each of these embeddings is not completely
contained in $\overline{\operatorname{Map}}_{c}(S)$.

See Corollary 6.6 for a more precise statement outlining which right-angled
Artin subgroups we construct. As an example, each of the right-angled Artin
groups defined by the graphs in Figure 1 can be found as subgroups of the
mapping class group of the Loch Ness monster surface, the blooming Cantor tree
surface, and the plane minus a Cantor set.

Figure 1. Defining graphs for a few of the right-angled Artin groups found as
subgroups of big mapping class groups using Theorem 1.1.

Our second construction produces embeddings between big mapping class groups
that are not induced by embeddings of the underlying surfaces and that do not
preserve the notion of being compactly supported. Recall that a group is
called indicable if it admits a surjection onto $\mathbb{Z}$.

######  Theorem 1.2.

Let $\Pi$ be a distinguished surface. If $\operatorname{Map}(\Pi)$ is
indicable, then for any surface $S\in\mathcal{C}(\Pi)$, there exist countably
many non-conjugate embeddings of $\operatorname{Map}(\Pi)$ into
$\operatorname{Map}(S)$ that are not induced by an embedding of $\Pi$ into
$S$.

The above theorem is in line with a body of work dedicated to understanding
and constructing homomorphisms between mapping class groups; see, for example,
[ALS09, AS13, ALM21]. There are uncountably many distinguished surfaces $\Pi$
for which $\operatorname{Map}(\Pi)$ is indicable; see 5.4. When $\Pi$ has at
least two nonplanar ends, Theorem 1.2 holds with $\operatorname{PMap}(\Pi)$ in
place of $\operatorname{Map}(\Pi)$; see Corollary 5.3.

Theorem 1.2 also answers Question 4.75 from the AIM problem list on surfaces
of infinite type [AIM] which asks, “Given a homomorphism
$f\colon\operatorname{Map}(S)\to\operatorname{Map}(S^{\prime})$, does $f$
preserve the notion of being compactly supported?” Bavard, Dowdall and Rafi
[BDR20] show that the answer is yes for surjective homomorphisms, and
Aramayona, Leininger, and McLeay [ALM21] give an example of two surfaces and
self-maps for which the answer is no. In proving the theorem above, we show
that there is an uncountable family of surfaces and maps for which the answer
is also no.

These first two constructions are consequences of Theorems 6.3 and 5.2,
respectively. The latter is a general construction for embedding indicable
subgroups of mapping class groups into $\operatorname{Map}(S)$. Theorem 6.3 is
a combination theorem for indicable subgroups of $\operatorname{Map}(S)$, and
we summarize its statement as Theorem 1.3 below. The $\star$-product used in
the statement is a generalization of what is known in the literature as a
“free product with commuting subgroups,” a natural construction that has been
well-studied. Some basic group theoretic properties of such groups can be
found in [MKS04, Section 4.2, Problems 22–25]. The $\star$-product provides a
natural interpolation between free products and direct products and includes,
for example, graph products of groups. In particular, every right-angled Artin
group can be written as a $\star$-product.

######  Theorem 1.3.

Let $G_{i}$ be indicable groups that embed in $\operatorname{Map}(S_{i})$, for
$i=1,\ldots,n$, where $S_{i}$ is a surface with exactly one boundary
component. For each i, fix a surjective map $f_{i}\colon G_{i}\to\mathbb{Z}$,
and let $H_{i}$ be the kernel of $f_{i}$. The indicable group
$(G_{1},H_{1})\star\cdots\star(G_{n},H_{n})$ embeds in $\operatorname{Map}(S)$
for $S=S_{\Gamma}(\Pi)$, where $\Pi$ is obtained from $\\#_{n}S_{i}$ by
capping off $n-1$ boundary components.

We direct the reader to Section 3 for the definition of the surface
$S_{\Gamma}(\Pi)$, the construction of which was inspired by work of Allcock
[All06]. Importantly, the support of the homeomorphisms defined in our
construction is not all of $S_{\Gamma}(\Pi)$. Consequently, we may change the
topology outside the support of the homeomorphisms in any way we choose. In
this way, Theorem 1.3 actually shows that
$(G_{1},H_{1})\star\cdots\star(G_{n},H_{n})$ embeds in the mapping class group
of a wide class of infinite-type surfaces. For instance, we can arrange for
the edited surface to have a non-displaceable subsurface so that the subgroups
we construct cannot arise from a construction using isometries, as all such
surfaces have finite isometry groups.

A key aspect of the proof of Theorem 1.3 is a set of criteria for a collection
of _shift maps_ (or _multipush maps_) on an infinite-type surface to generate
a free group, given in Theorem 4.2. Shift maps are generalizations of
handleshifts, introduced by Patel and Vlamis in [PV18], that have become
integral to the theory of infinite-type surfaces. In particular, we augment
the generators of the groups $G_{i}$ in the statement of the theorem with
these shift maps (or multipush maps). The fact that shift maps do not lie in
$\overline{\operatorname{Map}_{c}(S)}$ implies that the subgroups we construct
are also not completely contained in $\overline{\operatorname{Map}_{c}(S)}$.
The only exceptions to this are when $S$ is finite type or the Loch Ness
Monster surface, for which
$\overline{\operatorname{Map}}_{c}(S)=\operatorname{Map}(S)$. We avoid the
technical statement of Theorem 4.2 here and direct the reader to Section 4.1.

There are a variety of indicable groups that can play the role of $G_{i}$ in
the statement of Theorem 1.3 (or the role of $G$ in the statement of Theorem
5.2). In particular, one can let $G_{i}$ be any indicable subgroup of the
mapping class group of a finite-type surface with exactly one boundary
component, for example, free groups (whose constructions come from pseudo-
Anosov elements), right-angled Artin groups, and braid groups. In the
constructions outlined below, we also produce new examples of indicable
subgroups of big mapping class groups that can be used as input for these
theorems, including solvable Baumslag-Solitar groups $BS(1,n)$ and a large
class of wreath products $G\wr H$. The following theorem is a particular case
of Proposition 4.3 and Theorem 4.5, which both hold for a more general class
of surfaces, including surfaces which contain a non-displaceable subsurface.
For instance, the theorem holds when $S$ is the connect sum of a Cantor tree
surface with a closed finite genus surface, whose isometry group must be
finite. We make the statement below to avoid technicalities.

######  Theorem 1.4.

If $S$ is a Cantor tree surface, then solvable Baumslag-Solitar groups
$BS(1,n)$ and wreath products $\mathbb{Z}^{n}\wr\mathbb{Z}$ for any $n\geq 1$
arise as subgroups of $\operatorname{Map}(S)$.

Note that solvable Baumslag-Solitar and $\mathbb{Z}^{n}\wr\mathbb{Z}$ cannot
embed in the mapping class group of any finite-type surfaces. Our theorem
gives the first construction of these groups (for $n>1$) in the mapping class
groups of many infinite-type surfaces. Lanier–Loving construct
$\mathbb{Z}\wr\mathbb{Z}$ as a subgroup of the mapping class group of any
infinite-type surface without boundary [LL20].

### Outline

Section 2 contains preliminaries on infinite-type surfaces, mapping class
groups, and shift and multipush maps. In Section 3, we give a construction of
surfaces from Schreier graphs and describe how to obtain non-conjugate
embeddings of subgroups generated by either shift or multipush maps. Our
constructions of specific subgroups of big mapping class groups begins in
Section 4, where we build embeddings of free groups, wreath products, and
solvable Baumslag–Solitar groups into big mapping class groups (Theorem 1.4).
In Section 5 we prove Theorem 1.2. Finally, in Section 6 we prove our
combination theorem (Theorem 1.3) for indicable subgroups before ending with
constructions of right-angled Artin subgroups, proving Theorem 1.1.

### Acknowledgements

The authors would like to thank Women in Groups, Geometry, and Dynamics
(WiGGD) for facilitating this collaboration, which was supported by NSF
DMS–1552234, DMS–1651963, and DMS–1848346. The authors also thank Mladen
Bestvina and Rylee Lyman for helpful conversations, as well as George Domat
for productive discussions about surfaces with indicable mapping class groups.

In addition, the authors acknowledge support from NSF grants DMS–1803368 and
DMS–2106906 (Abbott), DMS–1906095 and RTG DMS–1840190 (Hoganson), DMS–1902729
and DMS–2231286 (Loving), DMS–1937969 and DMS–2046889 (Patel), and DMS–
2005297 (Skipper). Skipper was also supported by the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation program
(grant agreement No.725773).

##  2\. Preliminaries

###  2.1. Ends of surfaces

Essential to the classification of infinite-type surfaces is the notion of an
end of a surface and the space of ends for an infinite-type surface $S$.

######  Definition 2.1.

An exiting sequence in $S$ is a sequence $\\{U_{n}\\}_{n\in\mathbb{N}}$ of
connected open subsets of $S$ 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 $S$ 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 $S$ 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 $S$ is an equivalence class of exiting sequences.

The space of ends of S, denoted by $E(S)$, is the set of ends of $S$ equipped
with a natural topology for which it is totally disconnected, Hausdorff,
second countable, and compact. In particular, $E(S)$ is homeomorphic to a
closed subset of a Cantor set. The definition of the topology on the space of
ends is not relevant to this paper and so is omitted.

Ends of $S$ 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 $S$ is a closed subspace of $E(S)$ and will be
denoted by $E^{g}(S)$; these are frequently called the _ends accumulated by
genus_. We have the following classification theorem of Kerékjártó [Ker23] and
Richards [Ric63]:

######  Theorem 2.2 (Classification of infinite-type surfaces).

The homeomorphism type of an orientable, infinite-type surface $S$ is
determined by the quadruple

$(g,b,E^{g}(S),E(S))$

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

There is a more complicated classification of infinite-type surfaces allowing
for non-compact boundary components due to Prishlyak–Mischenko [PM07]. We use
this classification once in Section 3.2, but in our setting, the surfaces we
are comparing have precisely the same boundary, so the classification reduces
to considering the triple $(g,E^{g}(S),E(S))$.

###  2.2. Mapping class group

The _mapping class group_ of $S$, denoted $\operatorname{Map}(S)$, is the set
of orientation-preserving homeomorphisms of $S$ up to isotopy that fix the
boundary pointwise. The natural topology on the set of homeomorphisms of $S$
is the compact-open topology, and $\operatorname{Map}(S)$ is endowed with the
induced quotient topology. Equipped with this topology,
$\operatorname{Map}(S)$ is a topological group. When $S$ is a finite-type
surface, this topology on $\operatorname{Map}(S)$ agrees with the discrete
topology, but when $S$ is of infinite type, the two topologies are distinct.
The _pure mapping class group_ , denoted $\operatorname{PMap}(S)$, is the
subgroup of $\operatorname{Map}(S)$ that fixes the set of ends of $S$
pointwise, and $\operatorname{Map}_{c}(S)$ is the subgroup of compactly
supported mapping classes.

######  Definition 2.3.

A mapping class $f\in\operatorname{Map}(S)$ is of _intrinsically infinite
type_ if $f\notin\overline{\operatorname{Map}}_{c}(S)$. A subgroup
$H\leq\operatorname{Map}(S)$ is of intrinsically infinite type if $H$ is not
completely contained in $\overline{\operatorname{Map}_{c}(S)}$.

Note that $\overline{\operatorname{Map}}_{c}(S)\leq\operatorname{PMap}(S)$.
When $S$ has at most one nonplanar end, $\overline{\operatorname{Map}}_{c}(S)$
is actually equal to $\operatorname{PMap}(S)$ [APV20]. In this paper, all of
the subgroups of $\operatorname{Map}(S)$ that we construct contain many
intrinsically infinite-type homeomorphisms and, therefore, cannot be
completely contained in $\overline{\operatorname{Map}}_{c}(S)$, except when
$S$ is finite-type or the Loch Ness Monster, in which case
$\overline{\operatorname{Map}}_{c}(S)=\operatorname{Map}(S)$. Recall that the
Loch Ness Monster surface is the unique infinite-genus surface with one end
(up to homeomorphism).

We are particularly interested in indicable groups and various ways of
embedding them in mapping class groups of infinite-type surfaces. A group $G$
is _indicable_ if there exists a surjective homomorphism $f\colon
G\to\mathbb{Z}$. We show in Lemma 5.1 that a group $G$ is indicable if and
only if there is a presentation for $G$ where the relators all have total
exponent sum zero in the generators of $G$. Importantly, many of our
constructions require an indicable subgroup $G$ of $\operatorname{Map}(S)$ as
an input, where $S$ is a surface with exactly one boundary component. There
are many examples of such groups that were mentioned in the introduction, but
there are also some restrictions on what groups $G$ can arise as subgroups of
mapping class groups, as is evidenced by the following lemma, which
generalizes the same result from the finite-type setting [FM12, Corollary
7.3].

######  Lemma 2.4 ([ACCL20, Corollary 3]).

If $S$ is an orientable surface with nonempty compact boundary, the mapping
class group fixing the boundary pointwise is torsion-free.

###  2.3. Push and shift maps

In this section, we define shift maps and push maps, which are central to all
of our constructions. A particular type of shift maps, called handle shifts,
were first studied by Patel-Vlamis in [PV18]. This inspired the following
definition of Abbott, Miller, and Patel [AMP]. A similar definition of shift
maps appears in [MR19] and [LL20].

######  Definition 2.5.

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

Given a surface $S$ with a proper embedding of $D_{\Pi}$ into $S$ so that the
two ends of the strip correspond to two different ends of $S$ (see Figure 2),
the shift on $D_{\Pi}$ induces a _shift_ on $S$, where the homeomorphism acts
as the identity on the complement of $D_{\Pi}$. If instead, we have a proper
embedding of $D_{\Pi}$ into $S$ where the two ends of the strip correspond to
the same end, we call the resulting homeomorphism on $S$ a _one-ended shift_.
Given a shift or one-ended shift $h$ on $S$, the embedded copy of $D_{\Pi}$ in
$S$ is called the _domain_ of $h$. By abuse of notation, we will sometimes say
that the domain of the shift or one-ended shift $h$ is $D_{\Pi}$ rather than
referring to it as an embedded copy of $D_{\Pi}$ in $S$ (when it is clear from
context to which embedded copy of $D_{\Pi}$ we are referring).

######  Remark 2.6.

If the surface $\Pi$ in Definition 2.5 has a nontrivial end space, then a
shift or one-ended shift $h$ on $S$ with domain $D_{\Pi}$ is not in
$\operatorname{PMap}(S)$ since there are ends of $S$ that are not fixed by
$h$. Thus, $h\notin\overline{\operatorname{Map}}_{c}(S)$ and is of
intrinsically infinite type. On the other hand, if $h$ is a shift map and if
$\Pi$ is a finite-genus surface with no planar ends, then $h$ is a power of a
handle shift on $S$, and the proof of [PV18, Proposition 6.3] again tells us
that $h\notin\overline{\operatorname{Map}}_{c}(S)$. However, the second
conclusion does not hold when $h$ is a power of a one-ended handle shift
since, in that case, it follows from work in [PV18] that
$h\in\overline{\operatorname{Map}_{c}(S)}$.

$\Pi$
---
$\Pi$
---
$\Pi$
---
$\Pi$
---
$D_{\Pi}$
---
Figure 2. A surface $S$ that admits a shift whose domain is an embedded copy
of $D_{\Pi}$.

We now use the construction of shift maps to introduce _finite shifts_ , which
will be used in Section 4.3 to construct certain wreath products. These are
constructed in a completely analogous way, starting with an annulus instead of
a biinfinite strip.

######  Definition 2.7.

Let $A_{\Pi}$ be a surface defined by taking the annulus

$([0,n]/0\sim n)\times[-1,1],$

removing an open disk of radius $\frac{1}{4}$ centered at the integer points,
and attaching any fixed topologically nontrivial surface $\Pi$ with exactly
one boundary component to the boundary of each disk. A _finite shift_ on
$A_{\Pi}$ is the homeomorphism that acts like a translation, sending $(x,y)$
to $(x+1,y)$ (modulo n) for $y\in[-1+\epsilon,1-\epsilon]$ and which tapers to
the identity on $\partial A_{\Pi}$. Given a surface $S$ with a proper
embedding of $A_{\Pi}$ into $S$, the finite shift on $A_{\Pi}$ induces a
_finite shift_ on $S$, where the homeomorphism acts as the identity on the
complement of $A_{\Pi}$. We call the embedded copy of $A_{\Pi}$ the _domain_
of the finite shift.

######  Definition 2.8.

A _push_ is any map that is a finite shift, a one-ended shift, or a shift map.

In Section 3, we will introduce the notion of a _multipush_ , which is roughly
a collection of push maps with disjoint supports, once we have developed some
further notation and language.

##  3\. Surfaces from graphs and non-conjugate embeddings

In this section, we begin by constructing a broad class of surfaces using an
underlying graph. We then introduce a specific type of homeomorphism called a
_multipush_ and show that these maps can be utilized to produce infinitely
many non-conjugate embeddings of certain groups into mapping class groups.

###  3.1. A construction of surfaces

The basic building block for this construction is a $d$–holed sphere. The
following definition of seams restricts to the normal notion of seams for a
3-holed sphere, i.e., a pair of pants.

######  Definition 3.1.

A set of _seams_ on a $d$–holed sphere is a collection of $d$ disjointly
embedded arcs such that each boundary component of the sphere intersects
exactly two components of the seams at two distinct points and such that the
collection of seams divides the sphere into two components. Call one component
the _front side_ and the other component the _back side_. These conditions
imply that each component is homeomorphic to a disk.

Starting from any graph $\Gamma$ with a countable vertex set and any surface
$\Pi$ with exactly one boundary component, we describe a procedure for
building a surface $S_{\Gamma}(\Pi)$. This mirrors a construction of Allcock
using the Cayley graph of a given group $G$ [All06].

For each vertex $v$ of valence $d+1$, start with a $(d+1)$–holed sphere.
Remove a disk on the interior of the front side, and attach the surface $\Pi$
along the boundary component. Call the resulting surface the _vertex surface
for $v$_, which we denote by $V_{v}$, and let $\Pi_{v}$ be the copy of $\Pi$
on $V_{v}$. For each edge of the graph, define the edge surface $E$ to be the
$2$–holed sphere with seams; topologically this is an annulus.

Whenever $u$ and $v$ are vertices of $\Gamma$ connected by an edge, connect
the vertex surfaces $V_{u}$ and $V_{v}$ with an edge surface $E(u,v)$ by
gluing one boundary component of the edge surface to a boundary component of
$V_{u}$ and the other boundary component of the edge surface to a boundary
component of $V_{v}$ so that the gluing is _compatible_ in the following
sense: the union of the seams separates $S_{\Gamma}$ into two disjoint
connected components, the _front_ and the _back_ , containing the front and,
respectively, the back of each vertex and edge surface. Call the resulting
surface $S_{\Gamma}(\Pi)$. See Figure 3 for an example. Notice that the
assumption that the vertex set $V(\Gamma)$ of $\Gamma$ is countable is
necessary for this construction to yield a surface. In particular, if
$V(\Gamma)$ is uncountable, then $S_{\Gamma}(\Pi)$ is not second countable,
and therefore cannot be a surface.

$\Pi$
---
$\Pi$
---
$\Pi$
---
$\Pi$
---
$\Pi$
---
$\Pi$
---
$\Pi$
---
$\Pi$
---
$\Pi$
---
Figure 3. An example of the surface $S_{\Gamma}(\Pi)$ where the graph $\Gamma$
is the Cayley graph of the group $\mathbb{Z}^{2}=\langle a,b:[a,b]\rangle$.

We also define a more general class of surfaces constructed by editing the
back of $S_{\Gamma}(\Pi)$ as follows. As above, fix a graph $\Gamma$ with a
countable vertex set and a surface with one boundary component $\Pi$, and let
$S=S_{\Gamma}(\Pi)$. Given any collection of surfaces $\\{\Omega_{v}\\}_{v\in
V(\Gamma)}$, only finitely many of which have boundary, we form the surface
$S\underset{v\in V(\Gamma)}{\\#}\Omega_{v}$ as follows. For each $v\in
V(\Gamma)$, take the connect sum of $V_{v}$ and the corresponding
$\Omega_{v}$. It is helpful to assume that the connect sum is done on the back
of $V_{v}$, since we will perform certain homeomorphisms on the front of $S$
later in the paper. We note that if every $\Omega_{v}$ is a sphere, then
$S\underset{v\in V(\Gamma)}{\\#}\Omega_{v}$ is homeomorphic to $S$. On the
other hand, by choosing the $\Omega_{v}$ to be more complicated, we can change
the homeomorphism type of $S$ by changing the genus or the space of ends.
Thus, even for a fixed surface $\Pi$, this construction will result in a large
family of surfaces, formed by varying the $\Omega_{v}$.

The underlying graph $\Gamma$ used to build $S_{\Gamma}(\Pi)$ throughout this
paper will often be a Schreier graph, which is defined as follows. Let $G$ be
a finitely generated group, $H$ a subgroup of $G$, and $T$ a finite generating
set for $G$. The Schreier graph $\Gamma(G,T,H)$ is the graph whose vertices
are the right cosets of $H$ and in which, for each coset $Hg$ and each $s\in
T$, there is an edge from $Hg$ to $Hgs$ labeled by $s$. If $Hg=Hgs$, there is
a loop labeled by $s$ at the vertex corresponding to $Hg$. Our assumption on
the finiteness of $T$ ensures that $\Gamma(G,T,H)$ has a countable vertex set.
When $\Gamma$ is a Schreier graph, let $\Pi_{Hg}$ be the copy of $\Pi$ on the
vertex surface corresponding to the coset $Hg$. In the special case when
$H=\\{1\\}$, the Schreier graph $\Gamma(G,T,\\{1\\})$ is simply the Cayley
graph of $G$ with respect to the generating set $T$, which we denote by
$\Gamma(G,T)$.

######  Definition 3.2.

Let $\Gamma$ be a Schreier graph for a triple $(G,T,H)$. A Schreier surface
associated to $(G,T,H)$ is a surface $S=S_{\Gamma}(\Pi)\underset{v\in
V(\Gamma)}{\\#}\Omega_{v}$ where $\Pi$ has exactly one boundary component and
is not a disk, and $\\{\Omega_{v}\\}$ is any collection of surfaces, only
finitely many of which have boundary.

We use Schreier graphs in our construction rather than simply Cayley graphs to
demonstrate the large class of surfaces our results apply to. When $\Pi$ is
compact, the surface $S_{\Gamma}(\Pi)$ will have the same end space as the
graph $\Gamma$. A Cayley graph for a finitely generated group will have $1,$
$2$ or a Cantor set of ends. On the other hand, there are many more
possibilities for a Schreier graph; any regular graph with even degree can be
realized as a Schreier graph [Gro77, Lub95]. For example, there are Schreier
graphs with any finite number of ends, or end spaces isomorphic to
$\mathbb{N}\cup\\{\infty\\}$ or $\\{-\infty\\}\cup\mathbb{Z}\cup\\{\infty\\}$,
and so our construction yields surfaces with these end spaces, as well. See
Figure 4 for two examples of Schreier graphs that cannot be realized as Cayley
graphs.

Figure 4. Each graph can be realized as a Schreier graph but not a Cayley
graph. The graph on the left has 3 ends, and the graph on the right has end
space homeomorphic to the $2$-point compactification of $\mathbb{Z}$.

We are now ready to define a _multipush_ on a Schreier surface.

######  Definition 3.3.

Let $\Gamma=\Gamma(G,T,H)$ be a Schreier graph. Fix a surface $\Pi$ with
exactly one boundary component. Let $S=S_{\Gamma}(\Pi)\underset{v\in
V(\Gamma)}{\\#}\Omega_{v}$ be a Schreier surface.

For each $s\in T$, we will construct a collection of push maps whose support
corresponds to connected components of the subgraph of $\Gamma$ which includes
only edges labeled by $s$ (see Figure 5). Fix a transversal $\mathcal{T}$ for
the set of double cosets $\\{Hg\langle s\rangle\mid g\in G\\}$, so that
$\mathcal{T}$ contains exactly one element from each double coset in the set.
In the case $H=\\{1\\}$, that is, when $\Gamma$ is a Cayley graph, the set
$\mathcal{T}$ is simply a transversal for (left cosets of) $\langle s\rangle$.
For each element $t$ in the transversal, we define a push $h_{t\langle
s\rangle}$ which maps $\Pi_{Hts^{i}}$ to $\Pi_{Hts^{i+1}}$. The support of
$h_{t\langle s\rangle}$ is contained in the front of

$\left(\bigcup_{i\in\mathbb{Z}}V_{Hts^{i}}\right)\,\bigcup\,\left(\bigcup_{i\in\mathbb{Z}}E(Hts^{i},Hts^{i+1})\right).$

Recall that $V_{Hts^{i}}$ is the vertex surface associated to the vertex
$Hts^{i}$ and $E(Hts^{i},Hts^{i+1})$ is the edge surface associated to the
edge $(Hts^{i},Hts^{i+1})$ for each $i\in\mathbb{Z}$. This support corresponds
to a connected component of $\Gamma$ with all edges labeled by $s$; see Figure
5. The _multipush_ $x_{s}$ associated to $s$ is the element of
$\operatorname{Map}(S)$ that acts simultaneously as the pushes $h_{t\langle
s\rangle}$ for each $t\in\mathcal{T}$. We let $D_{s}$ denote the domain of the
multipush $x_{s}$. If $h_{t\langle s\rangle}$ is not a finite shift for any
$t\in T$, we say $x_{s}$ is an infinite multipush.

$\Pi$
---
$\Pi$
---
$\Pi$
---
$\Pi$
---
$\Pi$
---
$\Pi$
---
$\Pi$
---
$\Pi$
---
$\Pi$
---
$\Pi$
---
$\Pi$
---
$\Pi$
---
$\Pi$
---
Figure 5. A portion of the domain $D_{a}$ (in blue) of the multipush $x_{a}$
on the surface $S_{\Gamma}(\Pi)$, where $\Gamma$ is the Cayley graph
$\Gamma=\Gamma(\mathbb{F}_{2},\\{a,b\\})$ for $\mathbb{F}_{2}=\langle
a,b\rangle$.

Since supports of the pushes $h_{t\langle s\rangle}$ are disjoint, the
multipush $x_{s}$ is a well-defined homeomorphism of the surface. Note that if
$Hg=Hgs$, then the edge $E(Hgs^{i},Hgs^{i+1})$ in the support of $h_{g\langle
s\rangle}$ is a loop. In particular, finite pushes can occur as part of a
multipush. In Figure 5, all pushes are infinite, but the multipush with the
orange domain shown in Figure 6 has both a finite and infinite push.

######  Remark 3.4.

We emphasize that multipushes are not induced by an action on the graph
$\Gamma$, even when $\Gamma$ is a Cayley graph. The construction simply uses
the labeling of the vertices of $\Gamma$ to define the homeomorphism $x_{s}$.

$\Pi$
---
$\Pi$
---
$\Pi$
---
$\Pi$
---
$\Pi$
---
Figure 6. A multipush on a surface corresponding to the Schreier graph on the
left in Figure 4 that contains both a finite and infinite push. The domain of
the multipush is highlighted in orange.

###  3.2. Non-conjugate embeddings

Given a shift map $h$ corresponding to an embedding of $D_{\Pi}$ into a
surface $S$, one can define a new and distinct shift map $h^{\prime}$ on $S$
by omitting some of the surfaces $\Pi_{i}$ from the domain, as long as
infinitely many remain. This gives another embedding of $D_{\Pi}$ into $S$.
See Figure 7. Since there are uncountably many infinite subsets of
$\mathbb{Z}$, we can construct uncountably many distinct embeddings of
$D_{\Pi}$ into $S$, and thus uncountably many distinct shift maps on $S$, in
this way. The same argument goes through for one-ended shifts as well.
Similarly, one infinite multipush $x_{s}$ associated to a generator $s\in T$
on a surface $S=S_{\Gamma}(\Pi)$ can be used to produce uncountably many
distinct domains for multipushes associated to $s$ by simply omitting some of
the copies of $\Pi$ from the domains of $x_{s}$ for all $s\in T$.

$\Pi$
---
$\Pi$
---
$\Pi$
---
$\Pi$
---
$D_{\Pi}$
---
$D_{\Pi}$
---
$\Pi$
---
$\Pi$
---
$\Pi$
---
$\Pi$
---
Figure 7. Two different embeddings of $D_{\Pi}$. The figure at the top
corresponds to a shift $h$ and the embedding shown at the bottom corresponds
to a new and distinct shift $h^{\prime}$ obtained by leaving two copies of
$\Pi$ out of the domain of $h$.

In many cases, these distinct domains give rise to isomorphic but non-
conjugate subgroups of $\operatorname{Map}(S)$. First, consider the case of
shift maps. Given a shift map $h$ on $S$, we define a new shift map
$h^{\prime}$ on $S$ by removing some copies of $\Pi$ from the domain of the
shift. The groups $\langle h\rangle$ and $\langle h^{\prime}\rangle$ are
isomorphic subgroups of $\operatorname{Map}(S)$. If they were conjugate, not
only would $\operatorname{supp}(h)$ and $\operatorname{supp}(h^{\prime})$ be
homeomorphic, but their complements $S\setminus\operatorname{supp}(h)$ and
$S\setminus\operatorname{supp}(h^{\prime})$ would also be homeomorphic. There
are many surfaces $\Pi$ and $S$ for which this latter condition fails. For
example, $\Pi$ could be a handle, that is, a torus with one boundary
component, and $S=S_{{\Gamma}}(\Pi)$ could be obtained from
${\Gamma}(\mathbb{Z},\\{1\\})$ and connect sum with a surface with only planar
ends. In this case, $h=x_{1}$ is a shift map and
$S\setminus\operatorname{supp}(h)$ has genus zero, while
$S\setminus\operatorname{supp}(h^{\prime})$ has nonzero genus coming from the
copies of $\Pi$ that were removed from the domain of $h$. Therefore, these
complements are not homeomorphic and the embeddings of $\mathbb{Z}$ as
$\langle h\rangle$ and $\langle h^{\prime}\rangle$ are non-conjugate. In this
example, $\Pi$ is a handle, but we can generalize to allow $\Pi$ to be any
surface with a countable end space and one boundary component, so long as
removing copies of $\Pi$ from the domain of the shift map still produces non-
homeomorphic subsurfaces $S\setminus\operatorname{supp}(h)$ and
$S\setminus\operatorname{supp}(h^{\prime})$. This motivates the following
definition.

######  Definition 3.5.

A distinguished surface is a surface $\Pi$ with exactly one boundary
component, satisfying at least one of the following:

  1. (1)

$\Pi$ has finite genus,

  2. (2)

$E(\Pi)$ consists of finitely many planar ends, or

  3. (3)

$E(\Pi)$ consists of finitely many nonplanar ends.

For each distinguished surface $\Pi$, let $\mathcal{C}(\Pi)$ be the collection
of surfaces $S$ that admit an embedding of $D_{\Pi}$ such that the following
holds. If $\Pi$ satisfies (1), then $S\setminus D_{\Pi}$ has finite (possibly
zero) genus. If $\Pi$ satisfies (2) or (3), then $S\setminus D_{\Pi}$ has
finitely many planar or nonplanar ends, respectively. If $\Pi$ falls into more
than one of the above categories, then $\mathcal{C}(\Pi)$ should consist of
surfaces that satisfy either of the conditions on $S\setminus D_{\Pi}.$

If $\Pi$ is a distinguished surface and $S\in\mathcal{C}(\Pi)$, then $S$
admits a shift $h$ with domain $D_{\Pi}$. If $h^{\prime}$ is another shift on
$S$ whose domain is embedded by omitting finitely many copies of $\Pi$ from
$D_{\Pi}$, then each of the three conditions on $\Pi$ ensures that
$S\setminus\operatorname{supp}(h)$ and
$S\setminus\operatorname{supp}(h^{\prime})$ are not homeomorphic. In
particular, $S\setminus\operatorname{supp}(h)$ and
$S\setminus\operatorname{supp}(h^{\prime})$ will have different genus or will
contain a different number of planar or nonplanar ends. Similarly, if
$h^{\prime}$ and $h^{\prime\prime}$ are obtained from $h$ by omitting
different (finite) numbers of copies of $\Pi$ from $D_{\Pi}$, then the
complements of their supports are not homeomorphic.

The collection $\mathcal{C}(\Pi)$ for a distinguished surface $\Pi$ is
uncountable. To see this, suppose $\Pi$ has finite genus or $E(\Pi)$ consists
of finitely many nonplanar ends. Then $S$ can be any surface such that
$S\setminus D_{\Pi}$ has only planar ends. On the other hand, if $E(\Pi)$
consists of finitely many planar ends, then $S$ can be any surface so that
$S\setminus D_{\Pi}$ has no planar ends. In either case, there are uncountably
many such $S$.

The definition of a distinguished surface $\Pi$ and the collection of surfaces
$\mathcal{C}(\Pi)$ ensure that we can “count” the number of copies of $\Pi$
that have been removed the domain of a shift, thus producing non-conjugate
embeddings. We could expand the definition of a distinguished surface and the
collection $\mathcal{C}(\Pi)$ to encompass a larger family of surfaces for
which this is possible, but we choose the streamlined definition above for
simplicity, while still demonstrating that our results hold for a broad class
of surfaces.

We have shown that there are countably many non-conjugate infinite cyclic
subgroups in $\operatorname{Map}(S)$. In Section 5, we will use these
different embeddings of $\mathbb{Z}$ to construct non-conjugate embeddings of
indicable subgroups into $\operatorname{Map}(S)$ (Theorem 1.2). The following
lemma summarizes the discussion above.

######  Lemma 3.6.

Let $S$ be any surface in the uncountable collection $\mathcal{C}(\Pi)$ for a
distinguished surface $\Pi$. There exist countably many non-conjugate
embeddings of the subgroup generated by the shift map on $S$ with domain
$D_{\Pi}$ into $\operatorname{Map}(S)$.

We now turn our attention to constructing non-conjugate embeddings of
subgroups generated by multipushes. Let $S=S_{\Gamma}(\Pi)$ be infinite-type,
and let $x_{s}$ be the multipush defined by $s\in T$. In the same way as for a
shift map, by omitting copies of $\Pi$ from the domain of $x_{s}$ so that the
complements of the supports are not homeomorphic, we obtain a non-conjugate
embedding of $\langle x_{s}\rangle$ in $\operatorname{Map}(S)$.

If several multipushes $x_{s}$ for $s\in T$ have common copies of $\Pi$ in
their supports, such as in Figure 14, more care needs to be taken. It is
possible to remove copies of $\Pi$ from the domains of all the multipushes to
obtain new multipushes $x_{s}^{\prime}$ in such a way that $\langle x_{s}\mid
s\in T\rangle\cong\langle x_{s}^{\prime}\mid s\in T\rangle$ and so that the
complements of the supports of the subgroups are not homeomorphic. One way to
formalize this is to consider the surface $S_{m}=S\underset{v\in
V(\Gamma)}{\\#}\Omega_{v}$ where exactly $m$ of the $\Omega_{v}$ are
homeomorphic to $\Pi$ with the boundary component capped off, and the
remainder of the $\Omega_{v}$ are spheres. By the classification of surfaces,
the surfaces $S$ and $S_{m}$ are homeomorphic, and this homeomorphism induces
an isomorphism of mapping class groups
$\operatorname{Map}(S)\cong\operatorname{Map}(S_{m})$. Let $x_{s}^{(m)}$ be
the multipush on the surface $S_{m}$ defined by $s\in T$. Notice that
$G=\langle x_{s}\mid s\in T\rangle$ is isomorphic to $\langle x_{s}^{(m)}\mid
s\in T\rangle$ because they are generated by multipushes with the same
supports $\pi_{1}$-embedded into different surfaces. Let
$G_{m}\leq\operatorname{Map}(S)$ be the image of $\langle x_{s}^{(m)}\mid s\in
T\rangle\leq\operatorname{Map}(S_{m})$ under the isomorphism of mapping class
groups, so that $G\cong G_{m}$. By construction, there are $m$ copies of $\Pi$
that are not in the support of $G_{m}$, while all copies of $\Pi$ are in the
support of $G$, and so $G$ and $G_{m}$ are not conjugate. Similarly, whenever
$m\neq n$, the groups $G_{m}$ and $G_{n}$ are isomorphic and non-conjugate.

For the remainder of the paper, when we say that we remove copies of $\Pi$
from the supports of multipushes, we will mean that we do so in the above
manner, so that the resulting groups are isomorphic.

If the ends of $\Gamma$ contain a Cantor set, then there are uncountably many
non-conjugate copies of $G$ in $\operatorname{Map}(S)$. To see this, use the
procedure above to edit the domains of the multipush maps by removing a
collection of copies of $\Pi$ that accumulate onto a closed subset of the
Cantor set of ends of $S=S_{\Gamma}(\Pi)$. By removing copies of $\Pi$ that
accumulate onto non-homeomorphic closed subsets of the Cantor set, we obtain a
non-conjugate embedding of $G$ into $\operatorname{Map}(S)$.

Above, we assumed that $S=S_{\Gamma}(\Pi)$. However, the argument applies more
broadly. For example, if $S=S_{\Gamma}(\Pi)\underset{v\in
V(\Gamma)}{\\#}\Omega_{v}$ and each $\Omega_{v}$ has only planar ends and
$\Pi$ has nonzero finite genus, then adding two finite collections of handles
of differing sizes to some $\Omega_{v}$ still results in the complements of
the domains being non-homeomorphic subspaces. More generally, we could let
$\Pi$ be any surface with a countable end space and one boundary component (of
which there are uncountably many), so long as removing two finite collections
of $\Pi$ of differing cardinalities still results in the complement
subsurfaces being non-homeomorphic. This observation leads to the definition
of the following family of surfaces.

######  Definition 3.7.

Let $\mathcal{B}$ be the collection of Schreier surfaces
$S=S_{\Gamma}(\Pi)\underset{v\in V(\Gamma)}{\\#}\Omega_{v}$ such that
$S_{\Gamma}(\Pi)$ is infinite-type, $\Pi$ has a countable end space, and the
surfaces $\Omega_{v}$ are compatible with $\Pi$ in following sense: if
$Y=\bigcup_{s\in T}\operatorname{supp}(x_{s})$, with $Y^{\prime}=\bigcup_{s\in
T}\operatorname{supp}(x_{s}^{\prime})$ and $Y^{\prime\prime}=\bigcup_{s\in
T}\operatorname{supp}(x_{s}^{\prime\prime})$, respectively, where
$x_{s}^{\prime}$ is obtained by moving $m$ copies of $\Pi$ out of the domain
of $x_{s}$ and $x_{s}^{\prime\prime}$ is obtained by moving $n$ copies of
$\Pi$ out of the domain of $x_{s}$ with $m\neq n$, then $S\setminus
Y^{\prime}$ and $S\setminus Y^{\prime\prime}$ are non-homeomorphic in $S$. Let
$\mathcal{B}_{\infty}$ be the subset of $\mathcal{B}$ consisting of those
Schreier surfaces built from infinite-type surfaces $\Pi$.

The above discussion demonstrates that $\mathcal{B}$ is uncountable and proves
the following lemma.

######  Lemma 3.8.

Let $S$ be any surface in the uncountable collection $\mathcal{B}$. Letting
$G$ be the subgroup of $\operatorname{Map}(S)$ generated by the multipush maps
$x_{s}$ on $S$ for $s\in T$, there exist countably many non-conjugate copies
of $G$ in $\operatorname{Map}(S)$. If $\Gamma$ has a Cantor set of ends, then
there are uncountably many non-conjugate copies of $G$ in
$\operatorname{Map}(S)$.

###  3.3. Non-isometric embeddings

Throughout the paper, all constructions of subgroups will utilize push and
multipush maps. If the complement of the domain of a (multi)push is not simply
connected, then the map cannot act as an isometry for any hyperbolic metric on
$S$. We can use the collection of subsurfaces $\\{\Omega_{v}\\}$ from the
construction of a Schreier surface $S$ to ensure this condition holds, and so
all of our constructions can produce subgroups that are not contained in the
isometry group of $S$ for any hyperbolic metric on $S$.

For many surfaces, this is not simply an artifact of our particular
construction. By choosing the collection $\\{\Omega_{v}\\}$ carefully, we can
often ensure that the resulting surface $S$ has a non-displaceable subsurface,
and hence its isometry group (with respect to any hyperbolic metric) contains
only finite groups [APV, Lemma 4.2]. In particular, the groups we construct
could not arise from a construction using isometries for any such surface.

##  4\. Free groups, wreath products, and Baumslag-Solitar groups

In this section, we use shift maps and multipushes to construct free groups,
certain wreath products, and solvable Baumslag-Solitar groups as subgroups of
big mapping class groups.

###  4.1. Free groups

The construction of Schreier surfaces from Section 3.1 was motivated by the
following construction of a free subgroup of intrinsically infinite type.

######  Example 4.1.

Let $\Gamma$ be the Cayley graph of the free group $\mathbb{F}_{2}=\langle
a,b\rangle$, which is the Schreier graph
$\Gamma(\mathbb{F}_{2},\\{a,b\\},\\{\text{id}\\})$, and build the Schreier
surface $S=S_{\Gamma}(\Pi)$ with $\Pi$ a torus with one boundary component.
See Figure 5. This Schreier surface is homeomorphic to the blooming Cantor
tree, that is, the surface with no boundary components, no planar ends, and a
Cantor set of nonplanar ends. The multipushes $x_{a}$ and $x_{b}$ generate a
copy of $\mathbb{F}_{2}$ in $\operatorname{PMap}(S)$. To see this, observe
that for any $g\in\langle a,b\rangle$, the multipush $x_{a}$ maps $\Pi_{g}$ to
$\Pi_{ga}$, and similarly for $x_{b}$. Thus, the only way for a word
$w\in\langle x_{a},x_{b}\rangle$ to act trivially on the surface is if the
corresponding word in $\langle a,b\rangle$ is trivial. Moreover, Remark 2.6
shows that this copy of $\mathbb{F}_{2}$ in $\operatorname{PMap}(S)$ is not
contained in $\overline{\operatorname{Map}}_{c}(S)$.

In this example, it is straightforward to prove that a non-trivial word
$w\in\langle x_{a},x_{b}\rangle$ acts non-trivially on the surface because
$\mathbb{F}_{2}$ has no relations and $\Gamma$ is a tree, so we only need to
track where $w$ sends $\Pi_{\text{id}}$. With a more nuanced analysis of the
action of $w$, however, we can show that multipushes generate a free group in
a much more general setting. Recall that the collection $\mathcal{B}$ of
Schreier surfaces was defined in Definition 3.7.

######  Theorem 4.2.

Let $\Gamma$ be a Schreier graph for a triple $(G,T,H)$ and $S$ any associated
Schreier surface. The set $\\{x_{\alpha}\mid\alpha\in T\\}$ generates a free
group of rank $|T|$ in $\operatorname{Map}(S)$. If $\lvert T\rvert=1$ and
$\Gamma$ is finite, then we require that at least one $\Omega_{v}$ is not a
sphere.

Moreover, when $S\in\mathcal{B}$, there exist countably many non-conjugate
embeddings of such a free group in $\operatorname{Map}(S)$, none of which can
lie entirely in the isometry group for any hyperbolic metric on $S$. If $S$ is
not finite type and not the Loch Ness monster surface, these free groups
cannot be completely contained in $\overline{\operatorname{Map}}_{c}(S)$.

###### Proof.

Let $w=t_{1}\ldots t_{k}$ be a nontrivial, freely reduced word in the free
group generated by the set $T$, and let $x_{w}:=x_{t_{k}}\cdots x_{t_{1}}$ be
the product of multipushes. We aim to show that $x_{w}$ is nontrivial in
$\operatorname{Map}(S)$. We first observe that if
$x_{w}(\Pi_{Hg})=\Pi_{Hgw}\neq\Pi_{Hg}$ for any coset $Hg$, then $x_{w}$ is
nontrivial in $\operatorname{Map}(S)$. We may therefore assume that $x_{w}$
returns each $\Pi_{Hg}$ to itself. In particular, this implies that $Hgw=Hg$
for all $g\in G$, and so the edge path given by labels $(t_{1},\dots,t_{k})$
in $\Gamma(G,T,H)$ based at any vertex describes a cycle.

First consider a one-generated group $G$. If $\Gamma$ is infinite, then we
must have $H=\\{\text{id}\\}$, in which case
$\Gamma(G,T,H)=\Gamma(\mathbb{Z},\\{1\\},\\{\operatorname{id}\\})$ is the
Cayley graph of $\mathbb{Z}$ with its standard generator. Since this graph has
no cycles, each element $x_{w}$ with $w\in G$ is non-trivial in
$\operatorname{Map}(S)$. On the other hand, suppose $\Gamma$ is a finite cycle
of order $k$, and consider the multipush $x_{t}$, where $t$ is the generator
of $G$. Then, $x_{t}^{k}$ represents a cycle in $\Gamma$, but the requirement
that some $\Omega_{i}$ is not a sphere guarantees that the curve $\gamma$ and
$x_{t}^{k}(\gamma)$ cobound a surface with non-trivial topology. See Figure 8
for the case $k=3$. Thus $\gamma$ and $x_{t}^{k}(\gamma)$ are not homotopic,
so $x_{t}^{k}$ is non-trivial and $\langle x_{t}\rangle\cong\mathbb{Z}$.

$\Pi$
---
$\Pi$
---
$\Pi$
---
$\gamma$
---
$x_{t}^{3}(\gamma)$
---
Figure 8. A surface $S$ built from the Cayley graph of
$\mathbb{Z}/3\mathbb{Z}=\langle t\rangle$. The curve $\gamma$ is not homotopic
to its image under $x_{t}^{3}$ due to the handle on the back of $S$.

Now assume $|T|=n\geq 2$, so that every vertex of $\Gamma(G,T,H)$ has degree
$2n\geq 4$. Let $p\colon\tilde{\Gamma}\rightarrow{\Gamma}$ be the universal
cover of the labelled graph $\Gamma$, which is a tree of valency $2n$ with
edge labels in the set $T$. Construct the Schreier surface
$\tilde{S}=S_{\tilde{\Gamma}}(\Pi)\,\underset{\tilde{v}\in
V(\tilde{{\Gamma}})}{\\#}\,\Omega_{\tilde{v}}$, where
$\Omega_{\tilde{v}}=\Omega_{v}$ whenever $p(\tilde{v})=v$. By construction,
$\tilde{S}$ is a cover of $S=S_{\Gamma}(\Pi)\underset{v\in
V(\Gamma)}{\\#}\Omega_{v}$. See Figure 9 for an example.

For each $t\in T$, let $\tilde{x}_{t}$ be the multipush on $\tilde{S}$
obtained by identifying $\tilde{\Gamma}$ with the Cayley graph of the free
group with basis $T$. The covering map $P\colon\tilde{S}\rightarrow S$ induces
a homomorphism from the group generated by the multipushes on $\tilde{S}$ to
the group generated by the multipushes on $S$ by mapping $\tilde{x}_{t}\mapsto
x_{t}$. Recall that, by assumption, $w=t_{1}\ldots t_{k}$ is a non-trivial
reduced word in the free generating set $T$ and $x_{w}=x_{t_{k}}\dots
x_{t_{1}}$. Let $\tilde{x}_{w}=\tilde{x}_{t_{k}}\dots\tilde{x}_{t_{1}}$.

Suppose towards a contradiction that $x_{w}$ is trivial in
$\operatorname{Map}(S)$. Then the following commutative diagram of
homeomorphisms shows that $\tilde{x}_{w}$ is a deck transformation.

${S_{\tilde{\Gamma}}(\Pi)}$${S_{\tilde{\Gamma}}(\Pi)}$${S_{\Gamma}(\Pi)}$${S_{\Gamma}(\Pi)}$$\scriptstyle{\tilde{x}_{w}}$$\scriptstyle{P}$$\scriptstyle{P}$$\scriptstyle{x_{w}=\text{id}}$

$\Pi$
---
$\Pi$
---
$\Pi$
---
$\Pi$
---
$\Pi$
---
$\Pi$
---
$\Pi$
---
$\Pi$
---
$x_{a}$
---
$x_{b}$
---
$x_{b}$
---
$x_{b}$
---
$\widetilde{x}_{a}$
---
$\widetilde{x}_{a}$
---
$\widetilde{x}_{a}$
---
$\widetilde{x}_{b}$
---
$\widetilde{x}_{b}$
---
$\widetilde{x}_{b}$
---
$P$
---
$p$
---
$a$
---
$b$
---
$\widetilde{a}$
---
$\widetilde{b}$
---
$a$
---
$b$
---
$b$
---
Figure 9. An example of the surface $\tilde{S}=S_{\tilde{\Gamma}}(\Pi)$ and
lifts of multipushes $x_{a},x_{b}$.

On the other hand, since $\tilde{x}_{w}$ is a multipush, it moves every vertex
surface of $\tilde{S}$ at most $k$ steps away from itself, a bounded distance.
We claim this is a contradiction. Indeed, as the covering map sends vertex
surfaces to vertex surfaces and edge surfaces to edge surfaces, respecting the
edge labels in $T$, we see that any deck transformation of
$P\colon\tilde{S}\rightarrow S$ is determined by a deck transformation of the
covering $p\colon\tilde{\Gamma}\rightarrow\Gamma$. One readily checks that for
any such nontrivial deck transformation and for all $j\geq 1$, there exists a
vertex $v$ in the tree $\tilde{\Gamma}$ such that the distance from $v$ to its
image is larger than $j$, and we have obtained our contradiction.

When $S\in\mathcal{B}$, it follows from Lemma 3.8 that there are countably
many non-conjugate embeddings of the free group $\mathbb{F}_{|T|}$ in
$\operatorname{Map}(S)$. By the argument in Section 3.3, none of these
embeddings lie in the isometry group for any hyperbolic metric on $S$.
Finally, when $S$ is not finite-type or the Loch Ness Monster (in which case
$\overline{\operatorname{Map}_{c}(S)}=\operatorname{Map}(S)$), each multipush
in the argument above is a collection of shift maps, so Remark 2.6 completes
the proof. ∎

It follows from the proof of Theorem 4.2 that the support of every non-trivial
element of $\mathbb{F}_{|T|}$ is not contained in the union of the vertex
surfaces. This is clear if $w$ does not fix every $\Pi_{Hg}$, because the
shift domains are contained in the support of $x_{w}$. On the other hand,
suppose $w$ fixes each $\Pi_{Hg}$. Since $x_{w}$ is a collection of pushes, it
therefore restricts to the identity on each vertex surface. However, the proof
of the theorem shows that $x_{w}$ is a non-trivial homeomorphism, and so the
support of $x_{w}$ cannot be contained in the union of the vertex surfaces.
See Figure 10 for an example of what the image of a loop $\gamma$ might look
like after the application of $x_{w}$ when $w$ is trivial in $G$. This will be
a crucial ingredient in the proof of Theorem 6.3.

$\gamma$
---
$x_{w}(\gamma)$
---
Figure 10. A portion of the Schreier surface for
$(\mathbb{Z}^{2},\\{a,b\\},\\{1\\})$ and the image of the curve $\gamma$ under
the element $x_{bab^{-1}a^{-1}}$.

###  4.2. Shift Maps that do not generate a free group

The construction above uses a countable collection of intersecting push maps
to ensure the resulting group is free. The following example demonstrates why
this is necessary by showing that the group generated by two shift maps with
minimal intersection is not free. We use the convention that
$[x,y]=xyx^{-1}y^{-1}$ and choose a right action.

$h_{a}$
---
$h_{b}$
---
$(0,0)$
---
Figure 11. The shifts $h_{a}$ and $h_{b}$ do not generate a free group.

Let $\Gamma$ be the four-ended tree with a single vertex of valence four and
all other vertices of valence two. Identify ${\Gamma}$ with the coordinate
axes in $\mathbb{R}^{2}$ to get a labeling of the vertices as integer
coordinates. Let $\Pi$ be any surface with one boundary component that is not
a disk, and construct the surface $S=S_{{\Gamma}}(\Pi)$. There is a horizontal
shift $h_{a}$ corresponding to the $+(1,0)$ map on the $x$–axis and a vertical
shift $h_{b}$ corresponding to the $+(0,1)$ map on the $y$–axis, as shown in
Figure 11. The intersection of the supports of these shifts is contained in
the front of $V_{(0,0)}$, the vertex surface at $(0,0)$. It can be checked
that the support of $[h_{a},h_{b}]$ is contained in the fronts of
$V_{(-1,0)}$, $V_{(0,0)}$, and $V_{(0,-1)}$ and the adjoining edge surfaces.
The word $w=h_{a}h_{b}h_{a}^{2}$ maps
$\\{\Pi_{(0,-1)},\Pi_{(-1,0)},\Pi_{(0,0)}\\}$ to the collection
$\\{\Pi_{(1,0)},\Pi_{(2,0)},\Pi_{(3,0)}\\}$. Thus, the elements
$[h_{a},h_{b}]$ and $w[h_{a},h_{b}]w^{-1}$ have disjoint supports and so
commute. More generally, the words $w_{n}=h_{a}^{3n+1}h_{b}h_{a}^{2}$ map
$\\{\Pi_{(0,-1)},\Pi_{(-1,0)},\Pi_{(0,0)}\\}$ to
$\\{\Pi_{(1+3n,0)},\Pi_{(2+3n,0)},\Pi_{(3+3n,0)}\\}$. From this, we see that
$H:=\langle h_{a},h_{b}\rangle$ is not a free group and actually contains
copies of $\mathbb{Z}^{n}$ for all $n$.

In fact, $H$ is isomorphic to a 2–generated subgroup of an infinite strand
braid group. To see this, note that the group structure of $H$ is not
dependent on the surface $\Pi$ that we attach, so we may assume $\Pi$ is a
punctured disk. We can also realize each shift domains as a disk with
countably many punctures with two distinct accumulation points on the
boundary. Because braid groups are mapping class groups of punctured disks,
this viewpoint allows us to realize $H$ as a subgroup of the infinite strand
braid group in which braids are allowed to have non-compact support. In
particular, $H$ is isomorphic to the subgroup of this braid group generated by
the elements $h_{a}$ and $h_{b}$, viewed as braids with non-compact support.

###  4.3. Wreath products

Recall that if $H$ acts on a set $\Lambda$, then the (restricted) wreath
product $G\wr_{\Lambda}H$ is defined as

$G\wr_{\Lambda}H=G^{\Lambda}\rtimes_{\gamma}H,$

that is, the semidirect product of $H$ with the direct sum of copies of $G$
indexed by $\Lambda$. Here, $G^{\Lambda}=\oplus_{\Lambda}G$ and is the set of
$(g_{\lambda})_{\lambda\in\Lambda}$. The automorphism $\gamma\colon
H\to\operatorname{Aut}(G^{\Lambda})$ is defined by
$\gamma(h)(G_{\lambda})=hG_{\lambda}h^{-1}=G_{h\lambda}$, so that $H$ acts on
$G^{\Lambda}$ by permuting the coordinates according to the action on the
indices. When it is clear from context, or when $\Lambda=H$, we may simply
write $G\wr H$.

We now construct a collection of wreath products in big mapping class groups.
The most straightforward example of this construction is when $S$ is a surface
which admits a shift whose domain is an embedded copy of $D_{\Pi}$ for some
surface $\Pi$ with one boundary component. For any
$G\leq\operatorname{Map}(\Pi)$, we generalize a construction of Lanier and
Loving [LL20] to construct $G\wr\mathbb{Z}$ as a subgroup of
$\operatorname{Map}(S)$. When $G$ is chosen to be the infinite cyclic group
generated by a single Dehn twist, we recover [LL20, Theorem 4].

######  Proposition 4.3.

Let $G\leq\operatorname{Map}(\Pi)$, where $\Pi$ is a surface with a single
boundary component. Let $S$ be a surface and $H\leq\operatorname{Map}(S)$ be
generated by a collection of pushes and multipushes, all of whose domains are
(unions of) embedded copies of $A_{\Pi}$ or $D_{\Pi}$. Index the copies of
$\Pi$ in these domains by $\Lambda$. The wreath product $G\wr_{\Lambda}H$ is a
subgroup of $\operatorname{Map}(S)$.

###### Proof.

Let $h_{1},\dots,h_{n}$ be the generators of $H$, so $\Lambda$ is a set
indexing the copies of $\Pi$ contained in the union of the domains of the
$h_{i}$. Each $h_{i}$ permutes the copies of $\Pi$ in its domain and so acts
on $\Lambda$: if $\lambda\in\Lambda$, then $h_{i}(\lambda)$ is defined to be
the index of $h_{i}\left(\Pi_{\lambda}\right)$. This induces an action of $H$
on $\Lambda$.

Let $G\leq\operatorname{Map}(\Pi)$, and let $G_{\lambda}\cong G$ be the
corresponding subgroup of $\operatorname{Map}(S)$ supported on
$\Pi_{\lambda}$. Whenever $\lambda\neq\lambda^{\prime}$, the subgroups
$G_{\lambda}$ and $G_{\lambda^{\prime}}$ have disjoint supports and commute,
so $\langle G_{\lambda}\mid\lambda\in\Lambda\rangle=G^{\Lambda}$. For any
$h\in H$ and $\lambda\in\Lambda$, we have $hG_{\lambda}h^{-1}=G_{h(\lambda)}$
and $H\cap G_{\lambda}=\\{1\\}$. Therefore, the subgroup of
$\operatorname{Map}(S)$ generated by $\langle
H,G_{\lambda}\mid\lambda\in\Lambda\rangle$ is isomorphic to
$G\wr_{\Lambda}H$.∎

We illustrate this proposition with several examples.

######  Example 4.4.

Proposition 4.3 applies whenever $S$ and $H$ are one of the following.

  1. (1)

Let $S$ be a surface with an embedded copy of $D_{\Pi}$, and let $H$ be
generated by a (possibly one-ended) shift $h$, so that $H\cong\mathbb{Z}$. The
index set $\Lambda$ is simply $\mathbb{Z}$, and $h$ acts on $\Lambda$ as
addition by 1.

  2. (2)

Let $S$ be a Schreier surface for a triple $(A,T,B)$ such that
$t_{1},\dots,t_{n}\in T$ correspond to biinfinite geodesics in
$\Gamma(A,T,B)$. Let $H$ be the subgroup of $\operatorname{Map}(S)$ generated
by the multipushes $x_{t_{1}},\dots,x_{t_{n}}$. By Theorem 4.2,
$H\cong\mathbb{F}_{n}$. In this case, the index set $\Lambda$ is the
collection of right cosets $\\{Ba\mid a\in A\\}$. Each generator $x_{t_{i}}$
acts on $\Lambda$ as follows: if $Ba\in\Lambda$, then $x_{t_{i}}\cdot
Ba=Bat_{i}$.

  3. (3)

Let $S=S_{\Gamma}(\Pi)$ be the surface described in Section 4.2, and let
$H=\langle h_{a},h_{b}\rangle$ be the subgroup of $\operatorname{Map}(S)$
constructed in that section. In this case, $H$ is not free. The index set
$\Lambda$ is the set $\\{(0,n),(n,0)\mid n\in\mathbb{Z}\\}$, and the
generators $h_{a}$ and $h_{b}$ act on $\Lambda$ as addition by $(1,0)$ and
$(0,1)$, respectively.

When $S\in\mathcal{B}$, it follows from Lemma 3.8 that there are countably
many non-conjugate embeddings of $G\wr H$ in $\operatorname{Map}(S)$ for $G,H$
as in the statements of Proposition 4.3. Moreover, none of these embeddings
lie in the isometry group for any hyperbolic metric on $S$ by the discussion
in Section 3.3 or in $\overline{\operatorname{Map}_{c}(S)}$ by Remark 2.6.

###  4.4. Solvable Baumslag-Solitar groups

For our third and final construction in this section, we focus on solvable
Baumslag-Solitar groups. Fixing a positive integer $n$, recall that the
Baumslag-Solitar group $BS(1,n)$ is the group with presentation

$BS(1,n)=\langle a,t\mid tat^{-1}a^{-n}\rangle.$

The condition on the surface $S$ in the following theorem involves a partial
order on $E(S)$ and the notion of self-similarity of a set of ends. The
precise definitions are not important for this paper; we only use that this
condition implies that $S$ admits a shift map of a Cantor set of ends [FPR22].
We refer the reader to [MR19] for precise definitions. The surfaces satisfying
this condition include, for example, the Cantor tree, the blooming Cantor
tree, and a Cantor tree with finite genus and finitely many punctures, but it
is a much more general class of surfaces.

######  Theorem 4.5.

Let $S$ be a surface such that $E(S)$ contains a self-similar subset that
contains a Cantor set of maximal ends. Then $BS(1,n)\leq\operatorname{Map}(S)$
for all $n>0$.

$D_{\Pi}$
---
Figure 12. The most straightforward example of a surface $S$ for which Theorem
4.5 shows that $BS(1,n)\leq\operatorname{Map}(S)$ for all $n>0$.

###### Proof.

Let $S$ be as in the statement of the theorem. By [FPR22, Lemma 3.4], the
surface $S$ admits a shift map $h$ with domain $D_{\Pi}$, where $\Pi$ is a
surface with one boundary component that contains a Cantor set of maximal
ends, called $\mathcal{C}$. Index the copies of $\Pi$ in $D_{\Pi}$ by
$\mathbb{Z}$.

\begin{overpic}[width=361.34999pt,trim=36.135pt 419.16621pt 75.88371pt
180.67499pt]{images/BS12.pdf} \put(20.0,2.75){\small{$\alpha_{0}^{-1}$}}
\put(49.0,2.75){\small{$\alpha_{1}^{-1}$}}
\put(78.0,2.75){\small{$\alpha_{2}^{-1}$}}
\par\put(13.0,17.0){\footnotesize{$\alpha_{0}^{0}$}}
\put(27.0,17.0){\footnotesize{$\alpha_{1}^{0}$}}
\put(42.0,17.0){\footnotesize{$\alpha_{2}^{0}$}}
\put(55.0,17.0){\footnotesize{$\alpha_{3}^{0}$}}
\put(71.0,17.0){\footnotesize{$\alpha_{4}^{0}$}}
\put(84.0,17.0){\footnotesize{$\alpha_{5}^{0}$}}
\par\par\put(9.0,31.5){\tiny{$\alpha_{0,1}^{1}$}}
\put(15.5,31.5){\tiny{$\alpha_{0,2}^{1}$}}
\put(23.0,31.5){\tiny{$\alpha_{1,1}^{1}$}}
\put(30.0,31.5){\tiny{$\alpha_{1,2}^{1}$}}
\put(38.0,31.5){\tiny{$\alpha_{2,1}^{1}$}}
\put(45.0,31.5){\tiny{$\alpha_{2,2}^{1}$}}
\put(52.0,31.5){\tiny{$\alpha_{3,1}^{1}$}}
\put(59.5,31.5){\tiny{$\alpha_{3,2}^{1}$}}
\put(66.5,31.5){\tiny{$\alpha_{4,1}^{1}$}}
\put(73.5,31.5){\tiny{$\alpha_{4,2}^{1}$}}
\put(81.0,31.5){\tiny{$\alpha_{5,1}^{1}$}}
\put(87.5,31.5){\tiny{$\alpha_{5,2}^{1}$}} \par\end{overpic} Figure 13. The
curves for $k=-1,0,1$ when $n=2$. The green, blue, and red arrows indicate
$\phi_{1},\phi_{0}$, and $\phi_{-1}$, respectively.

We will first construct a collection of homeomorphisms of $\Pi$. For each
$k\in\mathbb{Z}$, we will define a collection of simple closed curves which
divide $\mathcal{C}$ into clopen sets. When $k=0$, define an arbitrary
countable collection of disjoint clopen sets of $\mathcal{C}$ enclosed by a
collection of simple closed curves $\\{\alpha^{0}_{i}\\}_{i\in\mathbb{Z}}$.
When $k=1$, for each $i$, divide the maximal ends contained in
$\alpha^{0}_{i}$ into $n$ clopen sets using simple closed curves
$\alpha^{1}_{i,1},\ldots,\alpha^{1}_{i,n}$. Continue in this manner for all
$k\geq 2$. When $k=-1$, for each $i\equiv 0\mod n$, let $l=i/n$ and let
$\alpha^{-1}_{l}$ be a simple closed curve such that
$\alpha^{-1}_{l},\alpha^{0}_{i},\alpha^{0}_{i+1},\ldots,\alpha^{0}_{i+n-1}$
cobound an $(n+1)$–holed sphere. Thus, $\alpha_{l}^{-1}$ groups together the
ends that are cut away by
$\alpha^{0}_{i},\alpha^{0}_{i+1},\ldots,\alpha^{0}_{i+n-1}$. Continue in this
manner for all $k\leq-2$. See Figure 13.

We now define an element of $\operatorname{Map}(\Pi)$ for each
$k\in\mathbb{Z}$. The mapping class $\phi_{0}$ is the shift that sends
$\alpha^{0}_{i}$ to $\alpha^{0}_{i+1}$ for all $i\in\mathbb{Z}$. The mapping
class $\phi_{1}$ is the shift that sends $\alpha^{1}_{i,j}$ to
$\alpha^{1}_{i,j+1}$ when $1\leq j<n$ and $\alpha^{1}_{i,n}$ to
$\alpha^{1}_{i+1,1}$. Define $\phi_{k}$ when $k\geq 2$ analogously. The
mapping class $\phi_{-1}$ is the shift that sends $\alpha^{-1}_{l}$ to
$\alpha^{-1}_{l+1}$. Define $\phi_{k}$ for $k\leq-2$ analogously.

Let $\phi\in\operatorname{Map}(S)$ be the element which simultaneously acts as
$\phi_{k}$ on $\Pi_{k}$, the $k-$th copy of $\Pi$, for each $k\in\mathbb{Z}$
and as the identity elsewhere. Let $h\in\operatorname{Map}(S)$ be the shift
whose domain is $D_{\Pi}$. See Figure 12.

Let $f\colon BS(1,n)\to\operatorname{Map}(S)$ be the map defined by
$f(a)=\phi$ and $f(t)=h$. We will show that $f$ is an isomorphism onto its
image, i.e., $\operatorname{Map}(S)$ contains an isomorphic copy of $BS(1,n)$.

For each $k\in\mathbb{Z}$, the mapping class $f(tat^{-1})=h\phi h^{-1}$ first
shifts $\Pi_{k}$ to the left, applies $\phi$, which now acts as $\phi_{k-1}$
on $\Pi_{k}$, and then shifts $\Pi_{k}$ back to the right. By construction,
$\phi_{k-1}$ applied to $\Pi_{i}$ is equivalent to $\phi_{k}^{n}$ applied to
$\Pi_{i}$. It follows that

$f(tat^{-1})=h\phi h^{-1}=\phi^{n}=f(a^{n}).$

Therefore, $f$ is a well-defined homomorphism.

Suppose there exists $g\in BS(1,n)$ such that $f(g)$ is the identity of
$\operatorname{Map}(S)$. Using the relation in $BS(1,n)$, the element $g$ can
be written as $g=t^{i}a^{k}t^{-j}$ for some $k\in\mathbb{Z}$ and
$i,j\in\mathbb{Z}_{\geq 0}$. Since $f(g)=h^{i}\phi^{k}h^{-j}$ is the identity,
it must fix each $\Pi_{i}$, and so we must have $i=j$. Consider the surface
$\Pi_{0}$. Then, $f(g)$ first shifts $\Pi_{0}$ to the left $j$ times, applies
$\phi^{k}$ (which acts as $\phi_{-j}^{k}$ on $\Pi_{-j})$, and then shifts back
to $\Pi_{0}$. The result is that $f(g)$ acts as the shift $\phi_{-j}^{k}$ on
$\Pi_{0}$. The only way that $f(g)$ can act as the identity on $\Pi_{0}$ is if
$k=0$. Thus, $g=t^{i}a^{0}t^{-i}=1$, and $f$ is injective, as desired. ∎

The construction above embeds solvable Baumslag-Solitar groups into mapping
class groups of certain _infinite-type_ surfaces. This is in contrast to the
finite-type case, where $BS(1,n)$ is never a subgroup of the mapping class
group due to the Tits alternative: every subgroup of such a mapping class
group either contains a free subgroup or is virtually abelian [Iva84, McC85].
Since $BS(1,n)$ is solvable, it does not contain any free subgroups, but it is
also not virtually abelian.

Note that the techniques in Section 3.2 show that given one embedding of
$BS(1,n)$ into $\operatorname{Map}(S)$, we can produce countably many non-
conjugate copies of $BS(1,n)$ in $\operatorname{Map}(S)$, but we must edit the
surface $\Pi$ to contain, for example, a single handle in order to satisfy the
conditions in Definition 3.5. Once again, none of these embeddings lie in the
isometry group for any hyperbolic metric on $S$ by the discussion in Section
3.3 or in $\overline{\operatorname{Map}_{c}(S)}$ by Remark 2.6.

We expect that a similar construction can be used to embed $BS(m,n)$ into
$\operatorname{Map}(S)$ when $m\neq 1$. However, the lack of a normal form for
elements in $BS(m,n)$ significantly increases the complexity of the proof.

##  5\. Indicable groups

In this section, we give a general construction for embedding any indicable
group which arises as a subgroup of a mapping class group of a surface with
one boundary component into a big mapping class group in countably many non-
conjugate and intrinsically infinite-type ways. We will need the following
lemma in our construction.

######  Lemma 5.1.

A group $G$ is indicable if and only if there exists a presentation $G=\langle
T\mid R\rangle$ such that for each $r\in R$, the total exponent sum of $r$
with respect to the generators $T$ is zero.

Before presenting the proof of the lemma, we give an example that motivates
the argument. Consider the Baumslag-Solitar group $BS(1,n)$ with its standard
presentation $BS(1,n)=\langle a,t\mid tat^{-1}a^{-n}\rangle$. This
presentation does not have the desired property since the total exponent sum
of the relator in the generators $a$ and $t$ is $1-n$. However, there exists a
homomorphism $f\colon BS(1,n)\to\mathbb{Z}$ defined by letting $f(a)=0$ and
$f(t)=1$, so the lemma tells us that there must be a presentation of $BS(1,n)$
with the desired property. If we augment the generator $a$ to be $at$ instead,
then

$BS(1,n)=\left\langle at,t\,\middle|\,(t\cdot at\cdot t^{-1}\cdot
t^{-1})\cdot\underbrace{t(at)^{-1}\cdots t(at)^{-1}}_{n\text{
times}}\right\rangle,$

and the relator has zero total exponent sum in the generators $at$ and $t$. In
this presentation, the generators of $BS(1,n)$ both map to $1$ under the
homomorphism $f$, and we will use this property in the proof of the lemma.

###### Proof of Lemma 5.1.

Given a group $G=\langle T\,|\,R\rangle$ with all relators having total
exponent sum zero, there is a well-defined homomorphism $f\colon
G\to\mathbb{Z}$ defined by sending each generator to $1\in\mathbb{Z}$.

For the other direction, assume there exists a homomorphism $f\colon
G\to\mathbb{Z}$, and let $N=\ker(f)$. Let $N=\langle V\mid W\rangle$ be a
presentation for $N$, and let $a\in G$ be such that $f(a)=1$. Then since
$G/N\cong\mathbb{Z}$, the group $G$ is generated by $T^{\prime}=\\{a\\}\cup
V$. If we augment the generators in $V\subseteq T^{\prime}$ by $a$, then
$T=\\{a\\}\cup\\{av:v\in V\\}$ is also a generating set for $G$. Importantly,
the image of every one of these generators under $f$ is $1\in\mathbb{Z}$.

Let $G=\langle T\mid R\rangle$ be the presentation of $G$ for the generating
set $T$. If $r\in R$ is a relator, then $r$ is a word in $\langle T\rangle$
that is the identity in $G$. Thus, $f(r)=0$, and given that every element of
$T$ maps to $1\in\mathbb{Z}$, the total exponent sum of $r$ with respect to
$T$ must be zero. Therefore, $\langle T\mid R\rangle$ is one such desired
presentation for $G$. ∎

We can now begin our construction. Take any indicable group $G$ that arises as
a subgroup of $\operatorname{Map}(\Pi)$, where $\Pi$ is a surface with exactly
one boundary component. Let $h$ be a shift map on an infinite-type surface $S$
whose domain is an embedded copy of $D_{\Pi}$ in $S$. As discussed in Section
3, this includes a wide range of surfaces, including surfaces
$S_{\Gamma}(\Pi)$ built from any graph with countable vertex set that contains
a biinfinite path.

The most trivial way to embed $G$ into $\operatorname{Map}(S)$ is to let $G$
act on one copy of $\Pi$ in $S$. Indexing the copies of $\Pi$ in $D_{\Pi}$ by
$\mathbb{Z}$ and taking any subset of $I$ of $\mathbb{Z}$, $G$ can also act
simultaneously on the subsurfaces $\Pi_{i}$ of $S$ for $i\in I$. Varying over
all subsets of $\mathbb{Z}$ gives an uncountable collection of copies of $G$
in $\operatorname{Map}(S)$. Unlike these embeddings, the construction in the
next theorem produces an uncountable collection of copies of $G$ which do not
lie in the isometry group of $S$, even if $G$ lies in the isometry group of
$\Pi$, and do not lie in $\overline{\operatorname{Map}}_{c}(S)$, even if $\Pi$
is compact. See Definition 3.5 for the definition of a distinguished surface
and the family $\mathcal{C}(\Pi)$.

######  Theorem 5.2.

Let $\Pi$ be a distinguished surface and $G\leq\operatorname{Map}(\Pi)$ an
indicable group. Given a surface $S\in\mathcal{C}(\Pi)$, there are countably
many non-conjugate embeddings of $G$ in $\operatorname{Map}(S)$ such that no
embedded copy is contained in $\overline{\operatorname{Map}}_{c}(S)$ and no
embedded copy is contained in the isometry group for any hyperbolic metric on
$S$.

###### Proof.

Let $h\in\operatorname{Map}(S)$ be a shift with domain $D_{\Pi}$, and let $G$
be an indicable group. Fix a presentation $\langle T\mid R\rangle$ of $G$ such
that each $r\in R$ has total exponent sum zero with respect to $T$, which
exists by Lemma 5.1. Since $G$ is a subgroup of $\operatorname{Map}(\Pi)$, $G$
acts by homeomorphisms on each $\Pi_{i}$ in $D_{\Pi}$. For each $g\in G$, let
$\bar{g}\in\operatorname{Map}(S)$ be the element that acts as $g$
simultaneously on each $\Pi_{i}$ in $D_{\Pi}$. We claim that the group
generated by $\overline{T}=\\{\bar{t}h:t\in T\\}$ in $\operatorname{Map}(S)$
is isomorphic to $G$. Let $\phi\colon
F_{T}\to\langle\overline{T}\rangle\leq\operatorname{Map}(S)$ be the surjective
map defined by $t\mapsto\bar{t}h$ for all $t\in T$, where $F_{T}$ is the free
group on the generators $T$. We claim a word is in the kernel of this map if
and only if it represents a trivial element in $G$.

Notice that $h$ and $\bar{t}$ commute as elements of $\operatorname{Map}(S)$
so that for any word $w\in F_{T}$ with total exponent sum $k\in\mathbb{Z}$,
the image $\phi(w)$ can be written as $\bar{w}h^{k}$. Thus, $\phi(w)$ acts
trivially on $S$ if and only if $k=0$ and $\bar{w}$ acts trivially on each
copy of $\Pi$ in $S$. The only elements $w$ with this property are those that
are trivial in $G$, and elements that are trivial in $G$ have this property
since products of conjugates of relators $r\in R$ have total exponent sum
zero. Thus, the group $G^{\prime}$ generated by $\overline{T}$ in
$\operatorname{Map}(S)$ is isomorphic to $G$.

Any element of $G^{\prime}$ that does not have total exponent sum zero with
respect to $\overline{T}$ is not in $\overline{\operatorname{Map}}_{c}(S)$,
since it must shift the surfaces $\Pi_{i}$. Remove finitely many copies of
$\Pi$ from the domain of $h$ to obtain a new shift $h^{\prime\prime}$, and
construct the group $G^{\prime\prime}=\langle\bar{t}h^{\prime\prime}\mid t\in
T\rangle\leq\operatorname{Map}(S)$. This group $G^{\prime\prime}$ is
isomorphic to $G$ for the same reason that $G^{\prime}\cong G$. By the same
reasoning as in Section 3.2, the complements of the supports of $G^{\prime}$
and $G’^{\prime}$ are non-homeomorphic. In particular, $G^{\prime}$ and
$G’^{\prime}$ are not conjugate. As in Lemma 3.8, this procedure produces
countably many non-conjugate embeddings of $G$ into $\operatorname{Map}(S)$.
Finally, no such embedding is contained in
$\overline{\operatorname{Map}_{c}(S)}$ by construction. ∎

It was suggested to the authors by Mladen Bestvina that one can get around
constructing the presentation in Lemma 5.1 for the indicable group $G$ by
working instead with the wreath product construction in Proposition 4.3. More
specifically, let $f\colon G\rightarrow\mathbb{Z}$ be a surjection from the
indicable group to $\mathbb{Z}$. Let $\Pi$ be a surface with exactly one
boundary component such that $G$ arises as a subgroup of
$\operatorname{Map}(\Pi)$, and let $S$ be a surface which admits a shift $h$
with domain $D_{\Pi}$. For $g\in G$, let $\bar{g}$ be the element which acts
as $g$ on each $\Pi_{i}$. Then, for $g\in G$, define a new map $\psi\colon
G\rightarrow G\wr\mathbb{Z}\leq\operatorname{Map}(S)$ via
$g\mapsto\bar{g}h^{f(g)}$. One readily checks that this map is an injective
homomorphism by observing that the restriction of the image of $G$ to
$\bigoplus_{-\infty}^{\infty}G$ is the diagonal subgroup, and so the action of
$\mathbb{Z}$ is trivial. The embedding in the proof of Theorem 5.2 is exactly
this map.

Theorem 5.2 applies to all subgroups constructed in Section 3. Another
interesting class of examples produces embeddings of pure mapping class groups
into a full mapping class group that are not induced by embeddings of the
underlying surfaces.

The following corollary is immediate from Theorem 5.2 and work of Aramayona,
Patel, and Vlamis [APV20, Corollary 6], which shows that the _pure_ mapping
class group of any surface with at least two nonplanar ends is indicable.

######  Corollary 5.3.

Let $\Pi$ be an infinite-type surface with at least two nonplanar ends and
exactly one boundary component. Given any surface $S$ that admits a shift
whose domain is $D_{\Pi}$, there exist uncountably many embeddings of
$\operatorname{PMap}(\Pi)$ into $\operatorname{Map}(S)$ that are not induced
by an embedding of $\Pi$ into $S$. In addition, none of these embeddings
preserve the notion of being compactly supported. When $\Pi$ is a
distinguished surface and $S\in\mathcal{C}(\Pi)$, countably many of these
embeddings are non-conjugate.

Corollary 5.3 is in line with a body of work aiming to find interesting
homomorphisms between big mapping class groups. It also gives a natural set of
examples of uncountable groups $G$ to which one can apply Theorem 5.2. We note
that determining which _full_ mapping class groups are indicable is an
important open question for both finite- and infinite-type surfaces. We now
give a few examples of indicable big mapping class groups.

######  Examples 5.4.

Mann and Rafi build continuous homomorphisms from finite-index subgroups of
mapping class groups to $\mathbb{Z}^{k}$ and to $\mathbb{Z}$ in the proofs of
[MR19, Lemma 6.7 & Theorem 1.7], respectively. To find surfaces whose full
mapping class groups are indicable, we focus on the cases where the subgroup
has index $1$, a few of which we list below. We will define the homomorphism
to $\mathbb{Z}$ explicitly for example (1); the others are defined similarly.

  1. (1)

Let $\Pi$ be the surface with infinite genus whose end space is homeomorphic
to the two-point compactification of $\mathbb{Z}$, that is,
$E(\Pi)=\\{-\infty\\}\cup\mathbb{Z}\cup\\{\infty\\}$, where
$E^{g}(\Pi)=\\{\infty\\}$. Let $A\subset E(\Pi)$ be the subset of ends
corresponding to $-\mathbb{N}$, and let $B$ be the subset of ends
corresponding to $\\{0\\}\cup\mathbb{N}$. This surface is colloquially called
the bi-infinite flute with one end accumulated by genus, and it admits a shift
with domain $D_{\Sigma}$ for a punctured disk $\Sigma$. A homomorphism
$\ell\colon\operatorname{Map}(\Pi)\rightarrow\mathbb{Z}$ can be defined by

$\ell(\phi)=\left\lvert\\{x\in E\mid x\in A,~{}\phi(x)\in
B\\}\right\rvert-\left\lvert\\{x\in E\mid x\in B,~{}\phi(x)\in
A\\}\right\rvert.$

The map $\ell$ counts the difference in the number of punctures mapped from
negative to positive and punctures mapped from positive to negative. Note that
the shift map mentioned above evaluates to $1$ under $\ell$, so the map $\ell$
is surjective.

  2. (2)

Let $\Pi$ be a surface of any genus whose end space consists of a Cantor set
and $\\{-\infty\\}\cup\mathbb{Z}\cup\\{\infty\\}$, equipped with the same
topology as in part (1), where the end $\\{\infty\\}$ is identified with a
point in the Cantor set. The ends corresponding to
$\\{-\infty\\}\cup\mathbb{Z}\cup\\{\infty\\}$ must all be planar or all
nonplanar; the other Cantor set of ends can be planar or not. The homomorphism
to $\mathbb{Z}$ is defined as above, with sets $A=-\mathbb{N}$ and
$B=\\{0\\}\cup\mathbb{N}$.

  3. (3)

Let $\Pi$ be the surface with infinite genus and end space
$\mathbb{N}\cup\\{\infty\\}$, where only the ends corresponding to $1$ and
$\infty$ are nonplanar. This surface can be visualized as the ladder surface
with punctures accumulating to one end. Here we can similarly define a
homomorphism to $\mathbb{Z}$, which instead counts the number of genus that
are moved across a simple closed curve separating the ends in $E^{G}$.

The common thread in the examples above is that the two ends of the shift map
are of different topological types so that no element of
$\operatorname{Map}(\Pi)$ can exchange the two ends. This is the key fact
necessary to ensure that the map $\ell$ above is a well-defined homomorphism
of $\operatorname{Map}(\Pi)$ and not of a proper subgroup of
$\operatorname{Map}(\Pi)$.

Each of the examples above can be modified to have exactly one boundary
component. The third example can be extended to uncountably many more examples
by replacing one of the isolated planar ends with a disk punctured by any
closed subset of the Cantor set, of which there are uncountably many.

Moreover, in each case, $\Pi$ is like a distinguished surface in the sense
that if $S-D_{\Pi}$ has finitely many nonplanar ends in Cases (1) and (3),
then $\operatorname{Map}(S)$ can be used as the input for Theorem 5.2. In Case
(2), if $S-D_{\Pi}$ has finitely many nonplanar (resp. planar) ends when the
ends of $E(\Pi)$ corresponding to
$\\{-\infty\\}\cup\mathbb{Z}\cup\\{\infty\\}$ are nonplanar (resp. planar),
then $\operatorname{Map}(S)$ can be used as the input for Theorem 5.2.
Therefore, we can construct countably many non-conjugate embeddings of
$\operatorname{Map}(\Pi)$ into $\operatorname{Map}(S)$ in all such cases.

##  6\. Combination Theorem

In this section, we give a construction that takes as its input a set of
indicable subgroups of mapping class groups of surfaces with one boundary
component and outputs a new surface whose mapping class group contains a new
indicable subgroup of intrinisically infinite type built from the original
subgroups.

######  Definition 6.1.

Given two subgroups $H_{1}$ and $H_{2}$ of groups $G_{1}$ and $G_{2}$,
respectively, the _free product of $G_{1}$ and $G_{2}$ with commuting
subgroups $H_{1}$ and $H_{2}$_ is

$(G_{1},H_{1})\star(G_{2},H_{2}):=(G_{1}\ast
G_{2})/\langle\langle[H_{1},H_{2}]\rangle\rangle.$

More generally, the free product of $G_{1},\ldots,G_{n}$ with commuting
subgroups $H_{1},\ldots,H_{n}$ is

$(G_{1},H_{1})\star\cdots\star(G_{n},H_{n}):=G_{1}\ast\cdots\ast
G_{n}/\langle\langle[H_{i},H_{j}]:i\neq j\rangle\rangle.$

These groups are a natural interpolation between free products (where the
$H_{i}$ are trivial) and direct products (where $H_{i}=G_{i}$ for all $i$).
Free products with commuting subgroups arise in many natural contexts. For
example, graph products of groups are a special kind of free product with
commuting subgroups, where $H_{i}=G_{i}$ for some indices $i$ and the
remaining $H_{j}$ are trivial.

We are interested in the case where the $G_{i}$ are indicable groups and the
$H_{i}$ are the kernels of the surjections to $\mathbb{Z}$.

######  Lemma 6.2.

Let $G_{1},\dots,G_{n}$ be indicable groups with surjective maps $f_{i}\colon
G_{i}\to\mathbb{Z}$, and let $H_{i}=\ker(f_{i})$. Then the group
$(G_{1},H_{1})\star\cdots\star(G_{n},H_{n})$ is also indicable.

###### Proof.

Let $T_{i}$ be a generating set for $G_{i}$. Then there is a map
$\phi\colon(G_{1},H_{1})\star\cdots\star(G_{n},H_{n})\to G_{1}$ defined by
$\phi(t)=1$ for each $t\in T_{i}$ with $i\neq 1$, and
$\phi(t^{\prime})=t^{\prime}$ for each $t^{\prime}\in T_{1}$. Here $1$ is the
identity element of $G_{1}$. This map $\phi$ is a homomorphism which restricts
to the identity on $G_{1}$. By post-composing $\phi$ with $f_{1}$, we obtain
the desired map $(G_{1},H_{1})\star\cdots\star(G_{n},H_{n})\to\mathbb{Z}$. ∎

$S_{1}$
---
$S_{2}$
---
$S_{1}$
---
$S_{1}$
---
$S_{1}$
---
$S_{1}$
---
$S_{2}$
---
$S_{2}$
---
$S_{2}$
---
$S_{2}$
---
$S_{1}$
---
$S_{2}$
---
$S_{2}$
---
$S_{2}$
---
$S_{2}$
---
$S_{2}$
---
$S_{2}$
---
$S_{2}$
---
$S_{2}$
---
$S_{1}$
---
$S_{1}$
---
$S_{1}$
---
$S_{1}$
---
$S_{1}$
---
$S_{1}$
---
$S_{1}$
---
Figure 14. The domains of the two multipushes $x_{a}$ (blue) and $x_{b}$ (red)
in the proof of Theorem 1.3 in the case that $\Gamma$ is the Cayley graph of
the free group generated by $a$ and $b$.

We are now ready to prove our main combination theorem, of which Theorem 1.3
is a special case.

######  Theorem 6.3.

For $i=1,\dots,n$, let $S_{i}$ be a distinguished surface, and let $\Pi$ be
obtained from $\\#_{n}S_{i}$ by capping off $n-1$ boundary components. Let $S$
be a Schreier surface in $\mathcal{C}(\Pi)$ for a triple $(G,T,H)$ with
$|T|=n$. Let $G_{i}$ be an indicable group that embeds in
$\operatorname{Map}(S_{i})$, fix a surjective map $f_{i}\colon
G_{i}\to\mathbb{Z}$ for each $i$, and let $H_{i}=\ker f_{i}$. There are
countably many non-conjugate embeddings of the indicable group
$(G_{1},H_{1})\star\cdots\star(G_{n},H_{n})$ into $\operatorname{Map}(S)$,
none of which lie in $\overline{\operatorname{Map}}_{c}(S)$.

###### Proof.

We prove the theorem for $n=2$ for simplicity of notation, but the same proof
works for all $n$. Let $a$ and $b$ be two generators of $G$. By construction,
$S$ admits two multipushes $x_{a}$ and $x_{b}$, where each acts as
simultaneous pushes, as in Definition 3.3. See Figure 14.

By Lemma 5.1, each surjection $f_{i}\colon G_{i}\to\mathbb{Z}$ gives rise to a
presentation $G_{i}=\langle T_{i}\mid R_{i}\rangle$ such that every $r\in R$
has total exponent sum zero with respect to $T_{i}$ for $i=1,2$. Similarly to
Theorem 5.2, for each $g\in G_{i}$, define an element
$\bar{g}\in\operatorname{Map}(S)$, where $\bar{g}$ acts as $g$ simultaneously
on each copy of $\Pi$ in the domains of $x_{a}$ and $x_{b}$ in $S$. For
$i=1,2$, elements $g_{i}\in G_{i}$ act on the copies of $S_{i}$ in $\Pi$, and
the copies of $S_{1}$ and $S_{2}$ in each copy of $\Pi$ are disjoint. Thus,
$\bar{g}_{1}$ and $\bar{g}_{2}$ commute for any $g_{1}\in G_{1}$ and $g_{2}\in
G_{2}$. Let $\overline{T}_{1}=\\{\bar{t}x_{a}:t\in T_{1}\\}$ and let
$\overline{T}_{2}=\\{\bar{t}x_{b}:t\in T_{2}\\}$. We claim that the group
generated by $\overline{T}_{1}\cup\overline{T}_{2}$ in $\operatorname{Map}(S)$
is isomorphic to $(G_{1},H_{1})\star(G_{2},H_{2})$.

For a set $A$, we let $F_{A}$ denote the free group on generators $A$. Let
$\phi\colon F_{T_{1}\cup
T_{2}}\to\langle\overline{T}_{1}\cup\overline{T}_{2}\rangle\leq\operatorname{Map}(S)$
be the surjective map defined by $t\mapsto\bar{t}x_{a}$ for all $t\in T_{1}$
and $t\mapsto\bar{t}x_{b}$ for all $t\in T_{2}$. In order to show that
$\langle\overline{T}_{1}\cup\overline{T}_{2}\rangle\leq\operatorname{Map}(S)$
is isomorphic to $(G_{1},H_{1})\star(G_{2},H_{2})$, we must show that the
kernel of $\phi$ is generated by all relators in $R_{1}\cup R_{2}$ and the
commutator $[H_{1},H_{2}]$.

For any $t\in T_{1}\cup T_{2}$, the element $\bar{t}$ commutes with both
$x_{a}$ and $x_{b}$, and so we can write $\phi(w)=u\bar{w}$ where $u\in\langle
x_{a},x_{b}\rangle$. As in the proof of Theorem 5.2, if $r\in R_{i}$ for
$i=1,2$, then $\phi(r)$ is the identity element in $\operatorname{Map}(S)$,
and so $R_{1}\cup R_{2}\subset\ker(\phi)$. Next, given $w_{i}\in H_{i}$, we
claim that $\phi(w_{i})=\overline{w}_{i}$. This follows from the fact that the
subgroup $H_{i}$ is the kernel of $f_{i}$ so that $w_{i}$ has total exponent
sum zero in the generators $T_{i}$. Therefore, $\phi(w_{i})$ has total
exponent sum zero with respect to $\overline{T}_{i}$. Since we can write
$\phi(w_{i})=u\overline{w}_{i}$, for $u\in\langle x_{a}\rangle$ when $i=1$ and
$u\in\langle x_{b}\rangle$ for $i=2$, the total exponent sum zero condition
implies that $u$ is in fact trivial and $\phi(w_{i})=\overline{w}_{i}$. The
supports of $w_{1}$ and $w_{2}$ as elements of $\operatorname{Map}(\Pi)$ are
disjoint by the construction of $\Pi$ so that the supports of
$\overline{w}_{1}$ and $\overline{w}_{2}$ as elements $\operatorname{Map}(S)$
are disjoint. Thus, these elements commute and the image
$\phi(w_{1}w_{2}w_{1}^{-1}w_{2}^{-1})=\bar{w}_{1}\bar{w}_{2}\bar{w}_{1}^{-1}\bar{w}_{2}^{-1}$
is the identity in $\operatorname{Map}(S)$. It follows that
$[H_{1},H_{2}]\subset\ker(\phi)$.

Lastly, we show that if $w\in F_{T_{1}\cup T_{2}}$ is in $\ker\phi$, then $w$
is in the group generated by $R_{1}\cup R_{2}\cup[H_{1},H_{2}]$. Fix any
nontrivial $w$ in $F_{T_{1}\cup T_{2}}$ such that $\phi(w)$ acts as the
identity on $S$, and write $\phi(w)=u\bar{w}$, where $u\in\langle
x_{a},x_{b}\rangle$. By Theorem 4.2, the group $\langle x_{a},x_{b}\rangle$ is
isomorphic to $\mathbb{F}_{2}$. If $u$ is non-trivial, then the support of $u$
is not contained in the union of the vertex surfaces by the discussion at the
end Section 4.1, while the support of $\bar{w}$ is contained in the union of
the vertex surfaces by definition. Thus, $w$ could not be in $\ker(\phi)$, a
contradiction. Therefore, $\phi(w)=\bar{w}$.

Since $w$ is nontrivial by assumption, $\bar{w}$ is a nontrivial word in the
free group generated by $\\{\bar{t}:t\in T_{1}\cup T_{2}\\}$. We will now show
that since $\phi(w)=\bar{w}$ is the trivial homeomorphism and, therefore, acts
trivially on each copy of $\Pi$, the element $w$ is a product of elements in
$[H_{1},H_{2}]$ and $R_{1}\cup R_{2}$.

There are natural maps from $F_{T_{i}}$ to $G_{i}$ and from $F_{T_{1}\cup
T_{2}}$ to $G_{1}*G_{2}$. Decompose $w$ as $w=c_{1}d_{1}\dots
c_{\ell}d_{\ell}$ for some $\ell\geq 1$, where $c_{j}\in F_{T_{1}}$ and
$d_{j}\in F_{T_{2}}$, and each $c_{j}$ and $d_{j}$ is non-trivial except
possibly $c_{1}$ and $d_{\ell}$. Then

$\displaystyle\phi(w)$ $\displaystyle=\phi(c_{1})\dots\phi(d_{\ell})$
$\displaystyle=x_{a}^{k_{1}}\bar{c}_{1}x_{b}^{k_{2}}\bar{d}_{1}\dots
x_{a}^{k_{2\ell-1}}\bar{c}_{\ell}x_{b}^{k_{2\ell}}\bar{d}_{\ell},$

so $u=x_{a}^{k_{1}}x_{b}^{k_{2}}\dots x_{a}^{k_{2\ell-1}}x_{b}^{k_{2\ell}}$.
Since $u$ is the trivial element in the free group $\langle
x_{a},x_{b}\rangle$, we must have $k_{1}=k_{2}=\cdots=k_{2\ell}=0$. In
particular, using the fact that the surjections $f_{i}\colon
G_{i}\to\mathbb{Z}$ send each element of $T_{i}$ to $1$, the image of each
$c_{j}$ under the natural maps defined above lies in $H_{1}<G_{1}$, and the
image of $d_{j}$ lies in $H_{2}<G_{2}$, for all $j=1,\dots,\ell$. Thus, the
image of $w$ in $G_{1}*G_{2}$ lies in $H_{1}*H_{2}$. Moreover, because
$\phi(w)\in\operatorname{Map}(S)$ acts as the identity on every copy of
$S_{1}$ and $S_{2}$ in $S$, the image of $w$ under the projections from
$H_{1}*H_{2}$ to $H_{i}\leq G_{i}$ is trivial for $i=1,2$.

If $w=c_{1}$, the image of $w$ in $G_{1}\ast G_{2}$ lies completely in
$H_{1}$, and the fact that $\phi(w)$ acts trivially on $S$, and in particular
on each copy of $S_{1}$ in $S$, means that $w\in R_{1}$. Similarly, if
$w=d_{1}$, then $w\in R_{2}$. On the other hand, if the image of $w$ is not
completely contained in one factor of $H_{1}\ast H_{2}$, then the fact the
projections to each $H_{i}$ are trivial, implies that $w\in[H_{1},H_{2}]$. In
all cases, we have proved that $w$ is in the group generated by $R_{1}\cup
R_{2}\cup[H_{1},H_{2}]$.

We have shown that $(G_{1},H_{1})\star(G_{2},H_{2})$ embeds in
$\operatorname{Map}(S)$. As in the proof of Theorem 1.2, by removing finitely
many copies of $\Pi$ from the domains of $x_{a}$ and $x_{b}$, we obtain
countably many non-conjugate embeddings of $(G_{1},H_{1})\star(G_{2},H_{2})$
into $\operatorname{Map}(S)$. By construction, no such embedding is contained
in $\overline{\operatorname{Map}}_{c}(S)$. ∎

###  6.1. Constructing right-angled Artin groups

In this subsection, we describe how to use Theorem 1.3 to produce certain
right-angled Artin groups $A_{\Lambda}$ that embed in big mapping class groups
in countably many non-conjugate ways. The groups $A_{\Lambda}$ are never
completely contained in $\overline{\operatorname{Map}}_{c}(S)$ and can never
act by isometries for any hyperbolic metric on the infinite-type surface.

Theorem 1.3 produces embeddings of free products with commuting subgroups into
$\operatorname{Map}(S)$. In general, the free product of $G_{1}$ and $G_{2}$
with commuting subgroups $H_{1}$ and $H_{2}$ will not be finitely presented,
even when the groups $G_{i}$ are finitely presented. For example, consider the
indicable group $\mathbb{F}_{2}=\langle a,b\rangle$ with the map to
$\mathbb{Z}$ defined by $a\mapsto 1$ and $b\mapsto 0$. It is an exercise to
see that the kernel $K$ of this map is not finitely generated, see Exercise 7
of Section 1.A in [Hat02]. Therefore, if $G_{1}=G_{2}=\mathbb{F}_{2}$ and
$H_{1}=H_{2}=K$, then $(G_{1},H_{1})\star(G_{2},H_{2})$ is a finitely
generated but infinitely presented group.

However, there are instances where the free product of indicable groups with
commuting subgroups is a recognizable finitely presented group. Let $H_{i}$ be
right-angled Artin groups with defining graphs $\Delta_{i}$ for $i=1,\dots,n$,
and let $G_{i}=\mathbb{Z}\times H_{i}$. The group
$(G_{1},H_{1})\star\dots\star(G_{n},H_{n})$ is the right-angled Artin group
defined by the graph shown in Figure 15.

$\Delta_{1}$
---
$\Delta_{2}$
---
$\Delta_{3}$
---
$\Delta_{4}$
---
$\Delta_{5}$
---
Figure 15. The lines between $\Delta_{i}$ and $\Delta_{j}$ signify that each
vertex of $\Delta_{i}$ is adjacent to every vertex in $\Delta_{j}$.

######  Examples 6.4.

We now give some explicit examples of right-angled Artin groups arising as the
free product with commuting subgroups $(G_{1},H_{1})\star(G_{2},H_{2})$. In
each graph, the blue vertices correspond to generators of $G_{1}$ and the
orange correspond to generators of $G_{2}$.

  1. (1)

Taking $G_{1}=\mathbb{Z}^{m}$ and $G_{2}=\mathbb{Z}^{n}$, we produce the
right-angled Artin groups defined by the following graphs. Specifically,
Figure 16 shows the defining graphs for
$(\mathbb{Z}^{2},\mathbb{Z})\star(\mathbb{Z}^{2},\mathbb{Z})$,
$(\mathbb{Z}^{3},\mathbb{Z}^{2})\star(\mathbb{Z}^{2},\mathbb{Z})$,
$(\mathbb{Z}^{3},\mathbb{Z}^{2})\star(\mathbb{Z}^{3},\mathbb{Z}^{2})$ and a
general schematic for the group
$(\mathbb{Z}^{m+1},\mathbb{Z}^{m})\star(\mathbb{Z}^{n+1},\mathbb{Z}^{n})$.

$K_{m}$
---
$K_{n}$
---
Figure 16.

  2. (2)

Taking $G_{1}=\mathbb{Z}\times\mathbb{F}_{n}$, $H_{1}=\mathbb{F}_{n}$ and
$G_{2}=\mathbb{Z}^{2}$, with $H_{2}$ being one of the $\mathbb{Z}$ factors, we
produce the right-angled Artin groups defined by the following graphs.
Specifically, Figure 17 shows the defining graphs for $n=2$, $n=3$, $n=4$, and
a schematic for general $n$.

Figure 17.

  3. (3)

Taking $G_{1}=\mathbb{Z}\times\mathbb{F}_{m}$, $H_{1}=\mathbb{F}_{m}$ and
$G_{2}=\mathbb{Z}\times\mathbb{F}_{n}$, $H_{2}=F_{n}$ we produce the right-
angled Artin group defined by the graphs shown in Figure 18. These are the
defining graphs for $m=n=2$, $m=n=3$, and a schematic for general $m,~{}n$.

$m$
---
$n$
---
Figure 18.

Free products with commuting subgroups can also be used to construct free
products of right-angled Artin groups with $\mathbb{F}_{n}$. We demonstrate
this in the case $n=2$ with the example below, using the fact that if
$\Lambda^{\prime}$ is an induced subgraph of $\Lambda$, then the right-angled
Artin group $A_{\Lambda^{\prime}}$ is a subgroup of the right-angled Artin
group $A_{\Lambda}$.

$>$
---
$\Lambda$
---
$\Lambda^{\prime}$
---
Figure 19. The induced subgraph $\Lambda^{\prime}$ of $\Lambda$ corresponds to
the right-angled Artin group
$A_{\Lambda^{\prime}}=A_{P_{4}}\ast\mathbb{F}_{2}$. This group can be found as
a subgroup of the mapping class group of any surface in the collection
$\mathcal{B}$.

######  Example 6.5.

Let $A_{\Delta}$ be a right-angled Artin group. Let $G_{1}=\mathbb{Z}\times
A_{\Delta}$ and $G_{2}=G_{3}=\mathbb{Z}^{2}$. Denote by $\Lambda$ the defining
graph for
$(G_{1},A_{\Delta})\star(\mathbb{Z}^{2},\mathbb{Z})\star(\mathbb{Z}^{2},\mathbb{Z})$
given by Figure 15. Now, let $\Lambda^{\prime}$ be the induced subgraph on the
vertices of $\Lambda$ which are not adjacent to the copy of $\Delta$ in
$\Lambda$, and the copy of $\Delta$. The corresponding right-angled Artin
group is $A_{\Lambda^{\prime}}=A_{\Delta}\ast\mathbb{F}_{2}$. See Figure 19
for an example where $\Delta=P_{4}$, the path graph on four vertices.

We now show how to apply Theorem 1.3 to produce right-angled Artin subgroups
of mapping class groups.

######  Corollary 6.6.

For any surface $S\in\mathcal{B}_{\infty}$ and any graph $\Lambda$ from 6.4
and Example 6.5, there are countably many non-conjugate embeddings of the
right-angled Artin group $A_{\Lambda}$ in $\operatorname{Map}(S)$, such that
no embedded copy is contained in $\overline{\operatorname{Map}}_{c}(S)$ and
such that no embedded copy is contained in the isometry group for any
hyperbolic metric on $S$.

###### Proof.

We need to show that each of the groups $G_{i}$ from 6.4 and Example 6.5 arise
as subgroups of mapping class groups of surfaces with one boundary component
and that, in each case, the subgroup $H_{i}$ is the kernel of a surjection
$G_{i}\to\mathbb{Z}$. The result will then follow immediately from Theorem
1.3.

The group $\mathbb{Z}^{m}$ can be realized as a subgroup of the mapping class
group of a surface $S$ by considering the group generated by Dehn twists about
$m$ disjoint simple closed curves on $S$. The group
$\mathbb{Z}\times\mathbb{F}_{2}$ is generated by suitably high powers of $2$
independent partial pseudo-Anosov elements and a Dehn twist with disjoint
support from both. Because $\mathbb{F}_{n}$ is a subgroup of $\mathbb{F}_{2}$
for any $n$, the group $\mathbb{Z}\times\mathbb{F}_{n}$ is a subgroup of
$\mathbb{Z}\times\mathbb{F}_{2}$. Thus, each of the groups $G_{i}$ in 6.4 and
6.5 can be found as subgroups of $\operatorname{Map}(\Pi)$, where $\Pi$ is a
distinguished surface of sufficient topological complexity. Since every
surface $S\in\mathcal{B}_{\infty}$ (see Definition 3.7) is built from a
surface $\Pi$ of infinite type, this condition on complexity is always
satisfied.

Finally, in each example, the group $G_{i}$ is the direct product of
$\mathbb{Z}$ and $H_{i}$, and so $H_{i}$ is the kernel of the projection map
onto the first factor. Thus, Theorem 1.3 applies, completing the proof of the
corollary. ∎

## References

  * [ACCL20] Santana Afton, Danny Calegari, Lvzhou Chen, and Rylee Alanza Lyman.  Nielsen realization for infinite-type surfaces.  2020\.  Preprint. arXiv: 2002.09760.
  * [AIM] AIM Problem List: Surfaces of infinite type.  available at http://aimpl.org/genusinfinity.
  * [All06] Daniel Allcock.  Hyperbolic surfaces with prescribed infinite symmetry groups.  Proc. Amer. Math. Soc., 134(10):3057–3059, 2006.
  * [ALM21] Javier Aramayona, Christopher J. Leininger, and Alan McLeay.  Homomorphisms between mapping class groups.  2021\.  Preprint. arXiv:2101.07188.
  * [ALS09] Javier Aramayona, Christopher J Leininger, and Juan Souto.  Injections of mapping class groups.  Geometry & Topology, 13(5):2523–2541, 2009.
  * [AMP] Carolyn Abbott, Nicholas Miller, and Priyam Patel.  Infinite-type loxodromic isometries of the relative arc graph.  In preparation.
  * [APV] Tarik Aougab, Priyam Patel, and Nicholas G. Vlamis.  Isometry groups of infinite genus hyperbolic surfaces.  Math. Ann. To appear. arXiv:2007.01982.
  * [APV20] Javier Aramayona, Priyam Patel, and Nicholas G. Vlamis.  The first integral cohomology of pure mapping class groups.  Int. Math. Res. Not. IMRN, (22):8973–8996, 2020.
  * [AS13] Javier Aramayona and Juan Souto.  Homomorphisms between mapping class groups.  Geometry & Topology, 16(4):2285–2341, 2013.
  * [BDR20] Juliette Bavard, Spencer Dowdall, and Kasra Rafi.  Isomorphisms between big mapping class groups.  Int. Math. Res. Not. IMRN, (10):3084–3099, 2020.
  * [CLM12] Matt T. Clay, Christopher J. Leininger, and Johanna Mangahas.  The geometry of right-angled Artin subgroups of mapping class groups.  Groups Geom. Dyn., 6(2):249–278, 2012.
  * [FM12] Benson Farb and Dan Margalit.  A primer on mapping class groups, volume 49 of Princeton Mathematical Series.  Princeton University Press, Princeton, NJ, 2012.
  * [FPR22] Elizabeth Field, Priyam Patel, and Alexander J. Rasmussen.  Stable commutator length on big mapping class groups.  Bulletin of the London Mathematical Society, 54(6):2492–2512, 2022\.
  * [Gro77] Jonathan L Gross.  Every connected regular graph of even degree is a schreier coset graph.  Journal of Combinatorial Theory, Series B, 22(3):227–232, 1977\.
  * [Hat02] Allen Hatcher.  Algebraic topology.  Cambridge University Press, Cambridge, 2002.
  * [Iva84] N. V. Ivanov.  Algebraic properties of the Teichmüller modular group.  Dokl. Akad. Nauk SSSR, 275(4):786–789, 1984.
  * [Ker23] B. v. Kerékjártó.  Vorlesungen über Topologie. I.  Spring, Berlin, 1923.
  * [LL20] Justin Lanier and Marissa Loving.  Centers of subgroups of big mapping class groups and the Tits alternative.  Glasnik Matematicki, 55:85–91, 06 2020.
  * [Lub95] Alexander Lubotzky.  Cayley graphs: eigenvalues, expanders and random walks.  1995\.
  * [McC85] John McCarthy.  A “Tits-alternative” for subgroups of surface mapping class groups.  Trans. Amer. Math. Soc., 291(2):583–612, 1985.
  * [MKS04] Wilhelm Magnus, Abraham Karrass, and Donald Solitar.  Combinatorial group theory.  Dover Publications, Inc., Mineola, NY, second edition, 2004.  Presentations of groups in terms of generators and relations.
  * [MR19] Kathryn Mann and Kasra Rafi.  Large scale geometry of big mapping class groups.  2019\.  Preprint. arXiv: 1912.10914.
  * [PM07] A. O. Prishlyak and K. I. Mischenko.  Classification of noncompact surfaces with boundary.  Methods Funct. Anal. Topology, 13(1):62–66, 2007.
  * [PV18] Priyam Patel and Nicholas G. Vlamis.  Algebraic and topological properties of big mapping class groups.  Algebr. Geom. Topol., 18(7):4109–4142, 2018.
  * [Ric63] Ian Richards.  On the classification of noncompact surfaces.  Trans. Amer. Math. Soc., 106:259–269, 1963.