# Discrete group actions on 3-manifolds and embeddable Cayley complexes

Agelos Georgakopoulos Supported by ERC grant 639046 and EPSRC grants
EP/V048821/1 and EP/V009044/1.  George Kontogeorgiou

###### Abstract

We prove that a group $\Gamma$ admits a discrete topological (equivalently,
smooth) action on some simply-connected 3-manifold if and only if $\Gamma$ has
a Cayley complex embeddable —with certain natural restrictions— in one of the
following four 3-manifolds: (i) $\mathbb{S}^{3}$, (ii) $\mathbb{R}^{3}$, (iii)
$\mathbb{S}^{2}\times\mathbb{R}$, (iv) the complement of a tame Cantor set in
$\mathbb{S}^{3}$.

Keywords: 3-manifold, Cayley complex, discrete action, Cantor sphere.

MSC 2020 Classification: 57M60, 57S25, 57S30, 57S17, 57K30, 05E45.

Dedicated to the memory of Mikis Theodorakis, whose oratorio Axion Esti rhymes
with ‘action $\mathbb{S}^{3}$’.

##  1 Introduction

The main result of this paper is that a group $\Gamma$ admits a discrete
topological action on some simply-connected 3-manifold if and only if $\Gamma$
has a Cayley complex embeddable —with certain natural restrictions— in one of
the following four 3-manifolds: (i) $\mathbb{S}^{3}$, (ii) $\mathbb{R}^{3}$,
(iii) $\mathbb{S}^{2}\times\mathbb{R}$, (iv) the complement of a tame Cantor
set in $\mathbb{S}^{3}$. Which of these four cases occurs is determined by the
number of ends of $\Gamma$ [22]. By a _discrete_ action we mean a faithful,
properly discontinuous, co-compact, topological action. All our manifolds and
actions are topological, but they can be smoothed by a classical result of
Bing and Moise [7, 28], and a recent result of Pardon [31, 30] (see Theorem
7.1).

A homeomorphic image $C$ of the Cantor set in $\mathbb{S}^{3}$ is called
_tame_ , if it is contained in a piecewise linear arc. It is known that if
$C^{\prime}$ is another tame Cantor set in $\mathbb{S}^{3}$, then
$\mathbb{S}^{3}-C$ is homeomorphic to $\mathbb{S}^{3}-C^{\prime}$; see [35]
and references therein. A topological space homeomorphic to $\mathbb{S}^{3}-C$
will be called a _Cantor 3-sphere_. Its importance is established by the
following result:

######  Theorem 1.1.

Let $M$ be a connected, simply connected, topological 3-manifold. Suppose $M$
admits a properly-discontinuous, co-compact action by homeomorphisms. Then $M$
is homeomorphic to one of the following four spaces: $\mathbb{S}^{3}$, or
$\mathbb{R}^{3}$, or $\mathbb{S}^{2}\times\mathbb{R}$, or the Cantor 3-sphere.

The special case of Theorem 1.1 when the action is smooth and free became
well-known to experts in the aftermath of the Thurston–Perelman Geometrization
Theorem [36, 32, 33]. A proof for the general case, which is due to others,
can be found in the Appendix.

###  1.1 Finite group actions on $\mathbb{S}^{2}$ and $\mathbb{S}^{3}$

The following is a classical theorem, essentially going back to a 1896 paper
of Maschke [24]; for modern references see e.g. [6, Theorem 1.16.] and [17].

######  Theorem 1.2 (Folklore).

A finite group admits a faithful action (by homeomorphisms or isometries) on
$\mathbb{S}^{2}$ if and only if it has a planar Cayley graph.

The finite case of our result is a 3-dimensional version of Theorem 1.2,
replacing $\mathbb{S}^{2}$ by $\mathbb{S}^{3}$. We will prove that a finite
group $\Gamma$ admits a faithful —topological, smooth, or isometric— action on
$\mathbb{S}^{3}$, if and only if it has a Cayley complex $X$ that embeds
topologically in $\mathbb{S}^{3}$ so that the canonical action
$\Gamma\curvearrowright X$ maps each chamber boundary to a chamber boundary.
Before clarifying the details, we recall that this class of groups is now
well-understood:

######  Theorem 1.3.

A finite group admits a faithful action by homeomorphisms/smooth
maps/isometries on $\mathbb{S}^{3}$ if and only if it is isomorphic to a
subgroup of the orthogonal group $O(4)$.

Indeed, using the Geometrization Theorem, Dinkelbach and Leeb [15] showed that
every finite smooth action on $\mathbb{S}^{3}$ is conjugate to an isometric
action. Pardon [31] complemented this by proving that every topological action
on $\mathbb{S}^{3}$ is the uniform limit of smooth actions. Since any isometry
of $\mathbb{S}^{3}$ extends to an element of $O(4)$, these two facts combined
establish that the finite groups admitting faithful, topological actions on
$\mathbb{S}^{3}$ coincide, up to isomorphism, with the finite subgroups of
$O(4)$, as anticipated by Zimmermann [39]. This forms a quite rich family of
groups, described explicitly in [13]. Theorem 1.3 is the culmination of a long
effort, the history of which is surveyed e.g. in [14, 16, 21, 38]. In
contrast, the groups of Theorem 1.2 were described by Maschke in 1986: they
are just the finite cyclic and dihedral groups, and 8 sporadic ones [24,
Figures 1–10], [19].

For an analogue of Theorem 1.2 for actions on $\mathbb{S}^{3}$, Cayley graphs
are unlikely to suffice: every finite graph embeds in $\mathbb{S}^{3}$. As we
will see, the key is to consider Cayley complexes instead. Define a
_generalised Cayley complex_ of a group $\Gamma$ to be a simply connected
2-dimensional cell-complex $X$, such that there is an action of $\Gamma$ on
$X$ that is regular on the 0-skeleton $X^{0}$. This generalises the standard
notion of Cayley complex, in that we allow the action to fix 1-cells and
2-cells.

Any embedding $\phi$ of a 2-dimensional cell-complex $X$ into $\mathbb{S}^{3}$
—more generally, into an orientable 3-manifold— induces a cyclic ordering
$\sigma_{e}$ of the 2-cells containing a given 1-cell $e$ of $X$, e.g. by
considering the clockwise cyclic order in which these 2-cells intersect a disc
locally perpendicular to $e$. This family of cyclic orderings
$\sigma(\phi):=\\{\sigma_{e}\\}_{e\in X^{1}}$ will be called a planar rotation
system, defined more carefully in Section 2.6. Importantly, $\sigma(\phi)$
contains all the information needed to determine which sets of 2-cells of $X$
bound a chamber of $\phi(X)$; these sets will be called _pre-chambers_. Given
a cellular group action $\Gamma\curvearrowright X$, e.g. when $X$ is a Cayley
complex of $\Gamma$, we say that $\sigma(\phi)$ is _$\Gamma$ -invariant_ if
$\Gamma\curvearrowright X$ preserves $\sigma(\phi)$ (see Section 2.7 for
details).

Our 3-dimensional analogue of Theorem 1.2 is the equivalence of items (i) and
(iii) of the following more general statement:

######  Theorem 1.4.

For a finite group $\Gamma$ the following are equivalent:

  1. (i)

$\Gamma$ is one of the groups of Theorem 1.3;

  2. (ii)

$\Gamma$ admits a generalised Cayley complex $X$ with a $\Gamma$-invariant
planar rotation system;

  3. (iii)

$\Gamma$ admits a generalised Cayley complex $X$ with an embedding
$\phi:X\to\mathbb{S}^{3}$ with $\Gamma$-invariant planar rotation system;

  4. (iv)

$\Gamma$ admits a generalised Cayley complex $X$ with an embedding
$\phi:X\to\mathbb{S}^{3}$ such that $\phi(X)$ is invariant under some faithful
topological action of $\Gamma$ on $\mathbb{S}^{3}$ which acts regularly on the
vertices of $X$.111In other words, (iv) states that $\phi$ is equivariant with
respect to actions of $\Gamma$ on $X$ and $\mathbb{S}^{3}$ with the desired
properties.

As an example, let $\Gamma$ be the cartesian product of two finite cyclic
groups $C_{k},C_{\ell}$, and consider its generalised Cayley complex $X$ with
respect to the standard presentation $\left<a,b\mid
a^{k},b^{\ell},[ab]\right>$. Thus $X$ is a quadrangulated torus $T$, united
with 2-cells bounding its essential cycles spanned by each one of the
generators $a,g$. It is easy to see a topological embedding of $X$ in
$\mathbb{S}^{3}$, with all $a$-coloured 2-cells inside $T$ and all
$b$-coloured 2-cells outside it. Notice that the planar rotation system
induced by this embedding is $\Gamma$-invariant, as required by item (iii).
The reader will be able to see a topological action of $\Gamma$ that preserves
this embedding, as postulated by item (iv), which trivially implies (i). The
implication (ii) $\to$ (iii) says that we could have specified the planar
rotation system abstractly, as a combinatorial set of cyclic orderings,
without mention to a particular embedding. This implication makes use of the
validity of the Poincaré conjecture via a result of Carmesin [12] (Theorem 3.1
below). In Section 3 we observe that Carmesin’s result fails in general for
infinite 2-complexes, but remains true for Cayley complexes, a fact that
relies on the Geometrization Theorem via Theorem 1.1.

###  1.2 Infinite discrete actions on open 3-manifolds

Theorem 1.4 is the specialisation of our main theorem to finite groups. We
call the four 3-manifolds featuring in Theorem 1.1 the _special_ 3-manifolds.
They also feature in our main theorem:

######  Theorem 1.5.

For a finitely generated group $\Gamma$ the following are equivalent:

  1. (i)

$\Gamma$ admits a faithful, properly discontinuous, co-compact, topological
action on a simply connected 3-manifold;

  2. (ii)

$\Gamma$ admits a generalised Cayley complex $X$ with a $\Gamma$-invariant
planar rotation system with finite pre-chambers;

  3. (iii)

$\Gamma$ admits a generalised Cayley complex $X$ with an embedding
$\phi:X\to\mathbb{S}^{3}$ such that $\sigma(\phi)$ is $\Gamma$-invariant and
has finite pre-chambers;

  4. (iv)

$\Gamma$ admits a generalised Cayley complex $X$ with an embedding $\phi$ into
a special 3-manifold $M$ such that $\phi(X)$ is invariant under some faithful,
properly discontinuous, co-compact, topological action of $\Gamma$ on $M$,
which acts regularly on the vertices of $X$. Moreover, $\phi(X)$ has finite
pre-chambers.

An analogous 2-dimensional statement, generalising Theorem 1.2, can be found
in [17, Theorem 1.1]. It says that the infinite groups acting discretely on a
planar surface are exactly the Kleinian function groups. They also coincide
with those groups admitting a Cayley graph with invariant _planar rotation
system_. Details can be found in [10, 17, 23].

We saw examples of discrete actions on $\mathbb{S}^{3}$ above. Groups acting
discretely on $\mathbb{R}^{3}$ include lattices in the Thurston geometries
homeomorphic to $\mathbb{R}^{3}$, such as euclidean and hyperbolic
crystalographic groups, and the discrete Heisenberg group. On the contrary,
$\mathbb{Z}^{4}$ cannot act discretely on $\mathbb{R}^{3}$, by a theorem of
Stallings saying that it is not isomorphic to any subgroup of a 3-manifold
group [8]. Groups acting discretely on $\mathbb{S}^{2}\times\mathbb{R}$
include $\mathbb{Z}$, and its cartesian product with any of the groups of
Theorem 1.2. An example of a group acting discretely on the Cantor 3-sphere is
the free group $F_{r}$ of rank $r\geq 2$. The universal cover of any closed
3-manifold with fundamental group $F_{r}$ provides an example, and such
manifolds can be easily obtained using connect sums of copies of
$\mathbb{S}^{2}\times\mathbb{S}^{1}$.

We now discuss the tightness of the various conditions of Theorem 1.5. Item
(iv) provides the most detailed information about these groups, and it
trivially implies the other three (Section 6). The conditions in the other
three items are necessary in the sense that dropping any one of them would
violate at least one of the equivalences. To see that the condition of finite
pre-chambers is necessary for the implications (ii),(iii) $\to$ (iv),(i),
consider the group $\mathbb{Z}^{2}$. Its standard Cayley complex embeds in
$\mathbb{R}^{3}$ with invariant planar rotation system, but with two infinite
pre-chambers, and it is well-known that $\mathbb{Z}^{2}$ is not one of the
groups of Theorem 1.5 [2]. We do not have an explicit example of a group with
a Cayley complex that embeds in $\mathbb{S}^{3}$ with finite pre-chambers but
only with non-invariant planar rotation system, but expect that such a group
can be found using ideas of [17, §10].

Dropping some of our conditions in pairs can lead to interesting statements,
some of which are implicit in our proofs. A notable example is that the
implication (i) $\to$ (iii) generalises to group actions on non-simply
connected 3-manifolds, resulting to embedded 2-complexes with an appropriate
action by the same group; see Theorem 5.2. Dropping the condition of
finiteness of pre-chambers leads to an interesting class of groups; see
Section 6 for related open problems.

In items (i) and (iv) of Theorem 1.5, we can assume that the action is in
addition smooth.

This paper is structured as follows. Section 2 provides all the necessary
definitions. The implication (ii) $\to$ (iv) of Theorem 1.5 (and Theorem 1.4)
is proved in Section 3. Since the implication (iii) $\to$ (ii) is trivial,
this also establishes the implication (iii) $\to$ (iv). The implication (i)
$\to$ (iv) is proved in Section 5, while (iv) trivially implies the other
three statements. Section 6 puts the pieces together to conclude the proof of
Theorem 1.5.

In Section 4 we provide a separate proof of the implication (iii) $\to$ (iv)
of Theorem 1.4 avoiding the Poincaré conjecture. Thus we are able to prove the
equivalence of items (i), (iii) and (iv) of Theorem 1.4 without relying on
Perelman’s work. This is not the case for Theorem 1.5, where we do not see a
way to avoid using the Geometrization Theorem.

##  2 Definitions and preliminary results

###  2.1 Group actions

A 3-manifold is a topological space each point of which has a neighbourhood
homeomorphic to an open subset of $\mathbb{R}^{3}$.

A (topological) _action_ $\Gamma\curvearrowright M$ of a group $\Gamma$ on a
topological space $M$ is a homomorphism from $\Gamma$ into the group of
homeomorphisms of $M$. Given such an action, the images of a point $x\in M$
under $\Gamma\curvearrowright M$ form the _orbit_ of $x$.

An action $\Gamma\curvearrowright M$ is _faithful_ , if for every two distinct
$g,h\in\Gamma$ there exists an $x\in M$ such that $gx\neq hx$; or
equivalently, if for each $g\neq e\in\Gamma$ there exists an $x\in M$ such
that $gx\neq x$. It is _free_ if $gx\neq hx$ for every $g,h\in\Gamma$ and
$x\in M$. It is _transitive_ if for every $x,y\in M$ there is $g\in\Gamma$
with $gx=y$ (we will only encounter transitive actions on discrete spaces
$M$). Finally, $\Gamma\curvearrowright M$ is _regular_ if it is free and
transitive.

An action $\Gamma\curvearrowright M$ is _properly discontinuous_ , if for
every compact subspace $K$ of $M$, the set $\\{g\in\Gamma\mid gK\cap
K\neq\emptyset\\}$ is finite. It is _co-compact_ , if the quotient space
$M/\Gamma$ is compact. If $M$ is locally compact, then an equivalent condition
is that there is a compact subset $K$ of $M$ such that $\bigcup\Gamma K=M$.

###  2.2 Graphs

A (simple) _graph_ $G\ $is a pair $(V,E)$ of sets, where $V$ is called the set
of _vertices_ , and $E$ is a set of two-element subsets of $V$, called the set
of _edges_. We will write $uv$ instead of $\\{u,v\\}$ to denote an edge. A
_multi-graph_ is defined similarly, except that $E$ is a multi-set, and it can
have elements consisting of just one vertex.

We let $V(G)$ denote the set of vertices of a graph $G$, and $E(G)$ denote the
set of edges of $G$.

Every (multi-)graph $G\ =(V,E)$ gives rise to a $1$-complex, by letting $V$ be
the set of 0-cells, and for each $uv\in E$ introducing an arc with its
endpoints identified with $u$ and $v$. Thus we will sometimes interpret the
word _graph_ as a 1-complex. In particular, when discussing _embeddings_ of
graphs we will mean topological embeddings of 1-complexes.

A _generalised Cayley graph_ of a group $\Gamma$ is a graph $G$ endowed with
an action $\Gamma\curvearrowright G$ by isomorphisms which action is regular
on $V(G)$. This is analogous to our definition of a generalised Cayley complex
in the introduction. Again the difference to the standard notion of a Cayley
graph is that we allow the action to fix edges.

###  2.3 2-complexes

A _$2$ -complex_ is a topological space $X$ obtained as follows. We start with
a $1$-complex $X^{1}$ as defined above, called the _1-skeleton_ of $X$. We
then introduce a set $X^{2}$ of copies of the closed unit disc
$\mathbb{D}\subseteq\mathbb{R}^{2}$, called the _2-cells_ or _faces_ of $X$,
and for each $f\in X^{2}$ we attach $f$ to $X^{1}$ via a map
$\phi_{f}:\mathbb{S}^{1}\to X^{1}$, called the _attachment map_ , where we
think of $\mathbb{S}^{1}$ as the boundary of $\mathbb{D}$. Attaching here
means that we consider the quotient space where each point $x$ of
$\mathbb{S}^{1}\subset f$ is identified with $\phi_{f}(x)$. We let
$X^{0}:=V(X^{1})$ be the set of _vertices_ , or 0-cells, of $X$.

We say that $X$ is _regular_ , if $\phi_{f}$ is a homeomorphism onto its image
for every $f\in X^{2}$. We say that $X$ is _edge-regular_ , if each $\phi_{f}$
is injective on 1-cells, that is, $x\in X^{0}$ holds for every point $x\in X$
with more than one pre-image under $\phi_{f}$.

For $f\in X^{2}$, we write $f=[x_{1},\dots,x_{k}]$ if $x_{1},\dots,x_{k}$ is
the cyclic sequence of vertices appearing in the image of $\phi_{f}$.

To each 2-cell $f=[x_{1},\dots,x_{k}]\in X^{2}$, we associate two distinct
_directed 2-cells_ $f_{1},f_{2}$, also denoted by $\langle
x_{1},\dots,x_{k}\rangle$ and $\langle x_{k},\dots,x_{1}\rangle$ respectively.
Their _reverses_ are defined as $f_{1}^{-1}:=f_{2}$ and $f_{2}^{-1}:=f_{1}$.

If $X$ is not regular then it is always possible to produce a regular complex
$X^{\prime}$ homeomorphic to $X$ using the _barycentric subdivision_ defined
as follows. For each edge $e=uv\in X^{1}$, we subdivide $e$ by adding a new
vertex $m$ at its midpoint. For each occurrence of $e$ in a 2-cell $f$ of $X$,
we replace that occurence by the pair $um,mv$ or $vm,mu$ as appropriate. We
then triangulate each 2-cell $h=x_{1},\ldots,x_{k}$ of the resulting
$2$-complex by adding a new vertex $c$ in its interior, adding the edges
$cx_{1},\ldots,cx_{k}$ to the 1-skeleton, and replacing $h$ by the 2-cells
$[c,x_{1},x_{2}],[c,x_{2},x_{3}],\ldots,[c,x_{k},x_{1}]$.

Note that the barycentric subdivision $X^{\prime}$ of $X$ is a simplicial
complex, in particular a regular one, and its 1-skeleton is a simple graph.
Here, a 2-complex $X$ is _simplicial_ , if its 1-skeleton is a simple graph
and each $F\in X^{2}$ is of the form $f=[x_{1},x_{2},x_{3}]$ where
$x_{1},x_{2},x_{3}$ are distinct vertices.

For each $v\in X^{0}$, the _link graph_ $L_{X}(v)$ is the graph on the
neighbourhood of $v$ in $X^{1}$ with $uw\in E(L_{X}(v))$ if and only if
$u,v,w$ are consecutive in a 2-cell of $X$. Alternatively, we can define
$L_{X}(v)$ so that its vertices are the edges incident with $v$, and two edges
$vu,vw$ are joined by an edge of $L_{X}(v)$ whenever $u,v,w$ are consecutive
in a 2-cell of $X$. These two definitions yield isomorphic graphs when $X^{1}$
is a simple graph, and it is a matter of convenience to use the one or the
other. In general, link graphs are more naturally defined as multigraphs, as
$u,v,w$ may be consecutive in more than one 2-cells of $X$.

###  2.4 Embeddings and Chambers

An embedding of a space $X$ in a space $Y$ is homeomorphism between $X$ and a
subspace of $Y$.

For an embedding $\phi:X\rightarrow M$ of a $2$-complex $X$ into a 3-manifold
$M$, we call each connected component of $M\setminus\phi(X)$ a _chamber_. The
_boundary_ $\partial C$ of a chamber $C$ is the set of points $x\in\phi(X)$ in
the closure of $C$ that are not in the interior of $C$. Indeed, as $C$ is an
open set, $\partial C$ is disjoint from $C$. The following basic fact helps to
further explain the notion.

######  Proposition 2.1 ([18, Proposition 2.1]).

Let $\phi:X\rightarrow\mathbb{S}^{3}$ be an embedding of a finite, $2$-complex
$X$, such that every 0-cell and 1-cell of $X$ is contained in a 2-cell. Then
$\partial C$ is a union of 2-cells of $X$ for every $\phi$-chamber $C$.∎

###  2.5 Local flatness

We recall the standard notion of local flatness. An embedding
$\phi:\mathbb{S}^{2}\to M$, where $M$ is a 3-manifold, is _locally flat_ , if
for each $x\in\phi(\mathbb{S}^{2})$ there exists a neighbourhood $U_{x}$ of
$x$ such that the topological pair $(U_{x},U_{x}\cap\phi(X))$ is homeomorphic
to $(\mathbb{R}^{3},\mathbb{R}^{2})$, by which we mean that there is a
homeomorphism from $U_{x}$ to $\mathbb{R}^{3}$ mapping $U_{x}\cap\phi(X)$ to
$\mathbb{R}^{2}\subset\mathbb{R}^{3}$. (A _topological pair_ $(X,A)$ consists
of a topological space $X$ and a subspace $A\subseteq X$.)

We can extend the notion of local flatness to an embedding $\phi:X\to M$ of a
$2$-complex $X$ instead of $\mathbb{S}^{2}$: we say that $\phi$ is _locally
flat_ , if the restriction of $\phi$ to each homeomorphic image of
$\mathbb{S}^{2}$ in $X$ is locally flat, as defined above.

###  2.6 Rotation systems

A _rotation system_ of a graph $G$ is a family $(\sigma_{v})_{v\in V(G)}$ of
cyclic orderings of the edges incident with each vertex $v\in V(G)$. Every
embedding of $G$ on an orientable surface defines a rotation system, by taking
$\sigma_{v}$ to be the clockwise cyclic ordering in which the edges incident
to $v$ appear in the embedding. The rotation system $(\sigma_{v})_{v\in V(G)}$
is said to be _planar_ , if it can be defined by an embedding of $G$ in the
sphere $\mathbb{S}^{2}$.

Let $X$ be an edge-regular $2$-complex, and let $\overleftrightarrow{E}(X)$
denote the set of _directions_ of its 1-cells, that is, the set of directed
pairs $\overrightarrow{xy}:=\left<x,y\right>$ such that $xy\in X^{1}$. Thus
every 1-cell gives rise to two elements of $\overleftrightarrow{E}(X)$. A
_rotation system_ of $X$ is a family
$(\sigma_{e})_{e\in\overleftrightarrow{E}(X)}$ of cyclic orderings
$\sigma_{e}$ of the 2-cells incident with each
$e=\overrightarrow{xy}\in\overleftrightarrow{E}(X)$, such that if
$e^{\prime}=\overrightarrow{yx}$, then $\sigma_{e^{\prime}}$ is the reverse of
$\sigma_{e}$. A rotation system $(\sigma_{e})_{e\in\overleftrightarrow{E}(X)}$
of $X$ induces a rotation system $\sigma^{v}$ at each of its link graphs
$L_{X}(v)$ by restricting to the directions of 1-cells emanating from $v$: for
every $u\in V(L_{X}(v))$ we let $\sigma^{v}_{u}$ be the cyclic order obtained
from $\sigma_{\overrightarrow{vu}}$ by replacing each 2-cell $f$ appearing in
the latter by the edge $uw$ where $w$ is the unique neighbour $u$ in
$V(L_{X}(v))$ such that $w,v,u$ appear consecutively in $f$.

A rotation system of a regular $2$-complex $X$ is _planar_ , if it induces a
planar rotation system on each of its link graphs. Note that once, we fix an
orientation, every locally flat embedding $\phi$ of $X$ into $\mathbb{S}^{3}$
or $\mathbb{R}^{3}$ defines a planar rotation system, by letting $\sigma_{e}$
be the cyclic order in which the images of the 2-cells incident with $e$
appear in $U_{x}$, where $x$ is any interior point of $\phi(e)$, and $U_{x}$
is as in the definition of local flatness (Section 2.5).

###  2.7 Invariant rotation systems

Suppose that a group $\Gamma$ acts on a $2$-complex $X$ by a faithful action
$\Gamma\curvearrowright X$. Let $\Sigma$ be the set of all rotation systems
$\boldsymbol{\sigma}=(\sigma_{e})_{e\in\overleftrightarrow{E}(X)}$ on $X$ as
defined in the previous subsection. Then we can let $\Gamma$ act on $\Sigma$
by elementwise multiplication as follows. Recall that $\sigma_{e}$ is formally
a ternary ‘betweenness’ relation on the set $F(e)$ of 2-cells containing $e$
for every 1-cell $e$. For $g\in\Gamma$, we define the product
$g\cdot\sigma_{e}:=\\{[ga,gb,gc]\mid[a,b,c]\in\sigma_{e}\\}$, which is a
cyclic ordering on $F(ge)$. This defines our action
$g\cdot\boldsymbol{\sigma}:=(g\cdot\sigma_{e})_{e\in E(X)}$ of $\Gamma$ on
$\Sigma$. We will say that $\boldsymbol{\sigma}$ is $\Gamma$-_invariant_ , if
$g\cdot\boldsymbol{\sigma}$ coincides with $\boldsymbol{\sigma}$ up to a
global change of orientation. To make this more precise, let
$\eta:\Gamma\to\mathbb{Z}_{2}$ be a homomorphism from $\Gamma$ to the group
$\mathbb{Z}_{2}$; we will use $\eta$ to carry the information of which
$g\in\Gamma$ preserve/reverse the orientation. We say that
$\boldsymbol{\sigma}$ is _invariant_ with respect to $\eta$, if

$g\cdot\sigma_{e}=(-1)^{\eta(g)}\sigma_{ge}$,  (1)

holds for every $e\in\overleftrightarrow{E}(X)$ and $g\in\Gamma$. We say that
$\boldsymbol{\sigma}$ is _$\Gamma$ -invariant_ if it is invariant with respect
to some homomorphism $\eta$. Note that $\eta$ is uniquely determined by
$\boldsymbol{\sigma}$ if it exists.

##  3 From Invariant Planar Rotation Systems to Invariant Cayley Complex
Embeddings

The implication (ii) $\to$ (iii) of Theorem 1.4 is an immediate sequence of
the following result of Carmesin:

######  Theorem 3.1 ([12]).

A finite, simply connected, simplicial 2-complex admits an embedding $\phi$ in
$\mathbb{S}^{3}$ if and only if it admits a planar rotation system $\sigma$.
Moreover, $\phi$ can be chosen so that $\sigma(\phi)$ coincides with
$\sigma$.222The second statement is not stated explicitly in [12] but it is
easily implied by the construction of the embedding.

Indeed, if $X$ is a generalised Cayley complex of $\Gamma$, and $\sigma$ a
$\Gamma$-invariant planar rotation system on $X$, then we can apply the
barycentric subdivision to turn $X$ into a simplicial 2-complex $X^{\prime}$,
extend $\sigma$ to $X^{\prime}$ in the obvious way, and apply Theorem 3.1 to
$X^{\prime}$ to obtain an embedding of $X^{\prime}$, which induces an
embedding $\phi:X\to\mathbb{S}^{3}$ with $\sigma(\phi)=\sigma$.

######  Remark 1.

If a locally-finite 2-complex $X$ admits an embedding $\phi$ in
$\mathbb{S}^{3}$ (or any 3-dimensional submanifold of $\mathbb{S}^{3}$), then
we may assume $\phi$ to be locally flat (as defined in Section 2.5). Indeed,
we can modify $\phi$ to make it piecewise-linear (PL) [25, Appendix C], and it
is easy to see that any PL embedding of $X$ in $\mathbb{S}^{3}$ is locally
flat.

The implication (ii) $\to$ (iii) of Theorem 1.5 is more difficult, because
Theorem 3.1 does not extend to infinite 2-complexes:

######  Theorem 3.2.

There is a locally-finite, simply connected, simplicial 2-complex which admits
a planar rotation system but does not admit an embedding in $\mathbb{S}^{3}$.

###### Proof.

It is known that there is a contractible, hence simply connected, open
3-manifold $W$ which does not embed in any compact 3-manifold, let alone in
$\mathbb{S}^{3}$; see e.g. [20] and references therein. Let $T$ be a
triangulation of such a manifold $W$. It is easy to see that if a simplicial
complex $X$ has a topological embedding into some oriented 3-dimensional
manifold, then it has a planar rotation system [12]. Letting $X$ be the
2-skeleton of $T$, we thus deduce that $X$ has a planar rotation system, since
it embeds in $W$.

We may assume without loss of generality that there is no 3-cell $C$ of $T$
the boundary $\partial C$ of which separates $W$, because even if $T$ does not
have this property its barycentric subdivision $T^{\prime}$ will, and we could
have chosen $T^{\prime}$ instead of $T$. Thus no $\partial C$ separates $X$.

Suppose now that $X$ admits an embedding $f$ in $\mathbb{S}^{3}$. By Remark 1,
we may assume that $f$ is locally flat. Then we can extend $f$ into an
embedding of $W$ in $\mathbb{S}^{3}$ as follows. For every 3-cell $C$ of $T$,
we observe that $\partial C\subset X$ is homeomorphic to $\mathbb{S}^{2}$,
hence it separates $\mathbb{S}^{3}$ into two components. One of these
components $A$ is disjoint from $f(X)$, because $\partial C$ does not separate
$X$ as mentioned above. Since $f$ is locally flat, $A$ is homeomorphic to
$\mathbb{R}^{3}$ by the generalised Schoenflies theorem [11, 26]. Thus we can
embed $C$ onto $A$. Doing so for each 3-cell $C$ of $T$, we obtain an
embedding of $W$ into $\mathbb{S}^{3}$, contradicting the choice of $W$. ∎

Despite the fact that Theorem 3.1 fails for infinite 2-complexes in general,
it does hold for Cayley-complexes:

######  Theorem 3.3.

Let $C$ be a finitely-presented Cayley complex admitting an invariant planar
rotation system $\sigma$ with finite pre-chambers. Then $C$ admits an
embedding $\phi$ into $\mathbb{S}^{3}$ such that $\sigma(\phi)$ coincides with
$\sigma$.

###### Proof.

We follow the lines of the proof of Theorem 3.1 in [12], the main difference
being that we apply Theorem 1.1 instead of the Poincaré conjecture. This
starts by defining a 3-manifold $M=T(C,\sigma)$, with 2-skeleton $C$ as
follows. The planar rotation system $\sigma$ induces a relation on the
(directed) 2-cells of $C$, where two directed 2-cells are _related_ if they
appear consecutively is $\sigma_{e}$ for some common 1-cell $e$. The
equivalence classes induced by this relation are called _pre-chambers_. The
intuition is that pre-chambers will coincide with the boundaries of chambers
of the embedding of $C$ that we are constructing. For each pre-chamber $S$
thus defined, we attach a solid surface $\hat{S}$ to $C$, so that the
attachment map —which is not always injective— maps the boundary of $\hat{S}$
onto $S$. This completes the definition of $M=T(C,\sigma)$, which Carmesin
proves to be an oriented topological 3-manifold [12, Lemma 4.5] (Carmesin
works with finite $C$, but this proof extends verbatim to the infinite case).
He then observes that $M$ is simply connected if $C$ is ([12, Lemma 4.6]);
indeed, any loop in $M$ can be homotoped to one in $C$ by the construction of
$M$. Finally, Carmesin observes that when $C$ is finite then $M$ is compact,
hence homeomorphic to $\mathbb{S}^{3}$ by the Poincaré conjecture! By
construction, $C$ is embedded in $M\cong\mathbb{S}^{3}$, and the rotation
system of this embedding coincides with $\sigma$.

It remains to consider the case where $C$ is potentially infinite. Notice that
the group of $C$ acts properly discontinuously and co-compactly on $C$, hence
on $M$. Here we used the fact that $\sigma$ is invariant in order to extend
the group action from $C$ to $M$. Again, $M$ is simply connected because $C$
is. Thus $M$ is homeomorphic to one of the 3-manifolds of Theorem 1.1, each of
which embeds in $\mathbb{S}^{3}$. Again, since $C$ embeds in $M$ with rotation
system $\sigma$ by construction, the statement follows.

∎

This establishes the implication (ii) $\to$ (iii) of Theorem 1.5. However, a
closer inspection of the last proof reveals that we can obtain the stronger
implication (ii) $\to$ (iv) (of both Theorem 1.5 and Theorem 1.4):

######  Corollary 3.4.

Let $C$ be a finitely-presented Cayley complex of a group $\Gamma$, admitting
a $\Gamma$-invariant planar rotation system $\sigma$ with finite pre-chambers.
Then $C$ admits an embedding $\phi:C\to M$ into a special 3-manifold $M$ such
that $\phi(C)$ is invariant under some properly discontinuous, co-compact
action $\Gamma\curvearrowright M$.

###### Proof.

Define $M=T(C,\sigma)$ as in the proof of Theorem 3.3. Notice that pre-
chambers are defined using $\sigma$, and so the action of $\Gamma$ on $C$
preserves pre-chambers. It follows that the action of $\Gamma$ on $C$ extends
to an action on $M$ by homeomorphisms. As already observed, $M$ is a special
3-manifold, and this action is properly discontinuous and co-compact. ∎

##  4 From Invariant Cayley Complex Embeddings to Group Actions

Since the implication (iii) $\to$ (ii) of Theorem 1.4 is trivial, the previous
section also establishes the implication (iii) $\to$ (iv). The aim of this
section is to re-prove this implication (iii) $\to$ (iv) by a more elementary
method that avoids the Poincaré conjecture. (We do not have a proof of the
analogous implication of Theorem 1.5 avoiding the Geometrization Theorem.)
Thus the reader can choose to skip this section.

The purpose of this section is a proof of the following theorem without using
the Poincaré conjecture.

######  Theorem 4.1.

Let $\Gamma$ be a finite group, let $X$ be a generalised Cayley complex of
$\Gamma$, and $\phi:X\to\mathbb{S}^{3}$ an embedding with $\Gamma$-invariant
rotation system $\sigma(\phi)$. Then there is a faithful topological action
$\Gamma\curvearrowright\mathbb{S}^{3}$ fixing $\phi(X)$ as a set, and acting
regularly on its vertices.

The idea is to reduce this to the following result of [18]. We say that a
2-complex $X$ is _locally $k$-connected_, if each of its link graphs is
$k$-connected. Recall that a graph is _$k$ -connected_, if it has more than
$k$ vertices, and remains connected after removing any set of at most $k-1$
vertices.

######  Theorem 4.2 ([18, Theorem 1.3]).

Let $Y$ be a finite, simply-connected, locally 3-connected $2$-complex. Then,
for every two locally flat embeddings $\chi,\psi:Y\to\mathbb{S}^{3}$, there
exists a homeomorphism $\alpha:\mathbb{S}^{3}\to\mathbb{S}^{3}$ such that
$\psi=\alpha\circ\chi$.

Moreover, we may assume that $\alpha$ is determined by its restriction to
$\chi(Y)$.333The second sentence is not explicitly stated in [18], but it is
an immediate consequence of the construction of $\alpha$ given there. Indeed,
$\alpha$ is defined by extending $\psi\circ\chi^{-1}$ from $\chi(Y)$ to all of
$\mathbb{S}^{3}$ as follows. It is proved that every chamber of $\chi(Y)$ and
$\psi(Y)$ is bounded by a homeomorph of $\mathbb{S}^{2}$, and the generalised
Schoenflies theorem is then applied to map each chamber of $\chi$ to one of
$\psi$. Thus the second sentence follows by always choosing the same outcome
of the generalised Schoenflies theorem for a given homeomorphism between two
copies of $\mathbb{S}^{2}$ in $\mathbb{S}^{3}$.

Every finite Cayley complex automatically satisfies the simple connectedness
condition, and it is locally 1-connected (Lemma 4.3), but it is not
necessarily locally $3$-connected. With the lemmas that follow we will be able
to increase the local connectedness of a complex $X$ as in Theorem 4.1 by
extending it to a super-complex. This super-complex $X^{\prime}$ will inherit
the canonical action of $\Gamma$, and its rotation system will still be
$\Gamma$-invariant. This will allow us to apply Theorem 4.2 to $X^{\prime}$ to
prove Theorem 4.1.

We start with the following basic fact about finite Cayley complexes.

######  Lemma 4.3.

Every finite generalised Cayley complex $X$ is locally 1-connected, unless $X$
has fewer than 3 vertices.

This is well-known (see e.g. [12, Lemma 5.1]), but we provide a proof for
completeness:

###### Proof.

Recall that the 1-skeleton $X^{1}$ of $X$ is a generalised Cayley graph, and
so $X^{1}$ is 2-connected as it is finite. Thus any two incident edges of
$X^{1}$ are contained in a cycle.

To prove that the link graph $L=L(o)$ of the identity element $o$ of $X$ is
connected, pick two edges $e,f$ of $X$ incident with $o$. By the above remark,
there is a cycle $C$ in $X^{1}$ containing both $e,f$. Consider a van Kampen
diagram $K$ proving that $C$ is null-homotopic in $X$ using the 2-cells of
$X$. Let $K_{o}$ denote the set of 2-cells appearing in $K$ that contain $o$.
These 2-cells yield an $e$–$f$ path in $L$. Since $e,f$ where arbitrary edges
of $o$, this proves that $L$, and hence every link graph of $X$, is connected.
∎

Next, we show how to increase the local connectivity of an embedded 2-complex
from 1 to 2 by passing to a super-complex:

######  Lemma 4.4.

Let $X$ be a locally 1-connected $2$-complex, and let
$\phi:X\to\mathbb{S}^{3}$ be a locally flat embedding. Then there is a locally
$2$-connected $2$-complex $X^{\prime}$ containing $X$ as a topological
subspace, and a locally flat embedding
$\phi^{\prime}:X^{\prime}\to\mathbb{S}^{3}$, such that
$\phi^{\prime}(X)=\phi(X)$.

Moreover, any action $\Gamma\curvearrowright X$ with respect to which
$\sigma(\phi)$ is invariant extends to an action $\Gamma\curvearrowright
X^{\prime}$ with respect to which $\sigma(\phi^{\prime})$ is invariant.

Furthermore, $\pi_{1}(X^{\prime})\cong\pi_{1}(X)$.

Before giving the formal proof, let us explain the intuition by going one
dimension down. Recall that a _plane graph_ is a 1-complex embedded in
$\mathbb{S}^{2}$ or $\mathbb{R}^{2}$. Given a connected plane graph $G\ $that
has some cut-vertices, it is easy to extend $G\ $into a plane super-graph
$G^{\otimes}$ that is 2-connected by ‘fattening’ it, i.e. adding new vertices
and edges near each face-boundary; see Figure 1 and Definition 4.5 for
details. To prove Lemma 4.4, we will add new 2-cells to $X$ to ‘fatten’ it in
such a way that the effect on each of its link graphs will be the same as the
above modification of $G\ $into $G^{\otimes}$:

######  Definition 4.5.

Let $G$ be a finite, connected, plane graph. Let $G^{\prime\prime}$ be a plane
multigraph obtained from $G$ by adding two parallel edges
$e^{\prime},e^{\prime\prime}$ to each edge $e\in E(G)$, and embedding them so
that the circle $e^{\prime}\cup e^{\prime\prime}$ separates $e$ from the rest
of $G$. Then, for each $e$ of $G^{\prime\prime}$ with end-vertices $u,v$,
subdivide $e$ into a path of length 3 by placing two new vertices
$e_{u},e_{v}$ inside $e$. Finally, for each vertex $v$ of $G^{\prime\prime}$,
and each two edges $e,f$ incident with $v$ that appear consecutively in the
plane, add an edge between $e_{v}$ and $f_{v}$. We embed these edges in such a
way that they form a circle separating $v$ from any other vertex of $G$. Let
$G^{\otimes}$ denote the resulting plane graph; see Figure 1.

Figure 1: A portion of a graph $G$ (left half), and the corresponding part of
$G^{\otimes}$ (right half).

We remark that $G$ is a topological minor of $G^{\otimes}$.

######  Lemma 4.6.

Let $G$ be a connected, plane graph. Then $G^{\otimes}$ is $2$-connected.

###### Proof.

This is straightforward, and boils down to checking that neither an original
vertex of $G$, nor one of the new vertices, can be a cut-vertex of
$G^{\otimes}$. ∎

Using this we can now prove Lemma 4.4.

###### Proof of Lemma 4.4.

We may assume that $X$ is regular, for otherwise we can work with its
barycentric subdivision, which preserves all assumptions we have made on $X$
as well as $\pi_{1}(X)$.

We begin the construction of $X^{\prime}$ by ‘tripling’ each 2-cell of $X$ as
follows. For every $f\in X^{2}$, we introduce two new 2-cells $f^{-},f^{+}$
with the same boundary and attaching map as $f$, and embed $f^{-},f^{+}$ in
$\mathbb{S}^{3}$ locally flat and in such a way that their images bound a
3-ball that contains $f$ and is otherwise disjoint from $X$. Let $X^{\pm}$ be
the resulting $2$-complex, and $\phi^{\pm}:X^{\pm}\to\mathbb{S}^{3}$ the
resulting locally flat embedding.

Next, we modify $X^{\pm}$ into $X^{\prime}$ by engulfing each 1-cell $e\in
X^{1}$ inside a copy of $\mathbb{S}^{2}$. To make this more precise, we pick a
locally flat homeomorph $S_{e}$ of $\mathbb{S}^{2}$ in $\mathbb{S}^{3}$, such
that $S_{e}$ intersects $X^{0}$ at the endpoints of $e$, it intersects each
2-cell $f$ incident with $e$ along an arc, which we denote by $f_{e}$, and
$S_{e}$ is otherwise disjoint from $X^{\pm}$ and all other
$S_{e^{\prime}},e^{\prime}\neq e$. It is easy to find such an $S_{e}$ inside a
small neighbourhood of $e$.

To turn the resulting subspace of $\mathbb{S}^{3}$ into a 2-complex, we
declare $f_{e}$ to be an 1-cell for every pair $e,f$ as above, and we replace
$f$ by the two 2-cells into which $f_{e}$ dissects it (one of which 2-cells
will be further dissected by the other edges incident with $f$). Moreover, for
every two 2-cells $f,g$ that are consecutive in the rotation system of $e$,
the 1-cells $f_{e},g_{e}$ cut a ‘slice’ of $S_{e}$, which we also declare to
be a 2-cell of $X^{\prime}$, and denote it by $s_{feg}$. This completes the
construction of $X^{\prime}$, and the PL embedding $\phi^{\prime}$. As $X$ is
a subspace of $X^{\prime}$, we have $\phi^{\prime}(X)=\phi(X)$. Notice that
$X^{\prime}$ has the same vertex set as $X$, and so to check that $X^{\prime}$
is locally $2$-connected we just need to consider the effect of the newly
added cells to each link graph $L_{X}(v),v\in X^{0}$. It is straightforward to
check that the new link graph $L_{X^{\prime}}(v)$ coincides with
$L_{X}(v)^{\otimes}$ as in Definition 4.5 below. Thus $X^{\prime}$ is locally
$2$-connected by Lemma 4.6 below.

For the second statement, we first extend the action $\Gamma\curvearrowright
X$ to $X^{\pm}$ as follows. For every $\gamma\in\Gamma$ and $f\in X^{2}$, we
let $\gamma$ map the new 2-cells $f^{-},f^{+}$ bijectively to $(\gamma
f)^{-},(\gamma f)^{+}$. There are two ways to do so, and we choose the unique
option that retains the invariance of the rotation system
$\sigma:=\sigma(\phi)$, i.e. the choice that ensures that
$\gamma\sigma_{d}=(-1)^{\eta(\gamma)}\sigma_{\gamma d}$ for some, hence every,
directed edge $d$ incident with $f$, where $\eta:\Gamma\to\mathbb{Z}_{2}$ is a
homomorphism as in (1), witnessing the fact that $\sigma$ is invariant. This
ensures that the rotation system of $\phi^{\pm}$ is invariant with respect to
the resulting action $\Gamma\curvearrowright X^{\pm}$. Extending further to
$\Gamma\curvearrowright X^{\prime}$ is straightforward: we just let
$\gamma\in\Gamma$ map each $s_{feg}$ to $s_{(\gamma f)(\gamma e)(\gamma g)}$.

Finally, it is easy to prove
$\pi_{1}(X^{\prime})\cong\pi_{1}(X^{\pm})\cong\pi_{1}(X)$ by applying van
Kampen’s theorem whenever a new 2-cell is introduced, using the fact that
every new 2-cell forms a copy of $\mathbb{S}^{2}$ with existing 2-cells. ∎

Remark: Lemma 4.6 would remain true if instead of tripling each edge we just
doubled it. The reason we triple is that in Lemma 4.4 we have to triple each
2-cell in order to maintain the invariance of the action.

Next, we observe that we can increase the local connectivity further from 2 to
3, using a construction of [18]. It was shown in [18, §6] how given a locally
2-connected, simplicial, $2$-complex $X$, and an embedding
$\phi:X\to\mathbb{S}^{3}$, one can construct a super-complex ${\rm
fat}(X)={\rm fat}(X,\phi)$ with improved properties:

######  Lemma 4.7 ([18, Lemma 6.3]).

Suppose that $X$ is a locally 2-connected, simplicial, $2$-complex, and
$\phi:X\to\mathbb{S}^{3}$ is a locally flat embedding. Then ${\rm fat}(X)$ is
locally $3$-connected.

Moreover, $\phi$ extends to a locally flat embedding $\phi^{\prime}:{\rm
fat}(X)\to\mathbb{S}^{3}$ ([18, Lemma 6.1]). As the construction of ${\rm
fat}(X)$ is canonical, any group action $\Gamma\curvearrowright X$ extends to
an action $\Gamma\curvearrowright{\rm fat}(X)$. Finally, any loop in ${\rm
fat}(X)$ is homotopic to a loop in $X$ by the construction, and so ${\rm
fat}(X)$ is simply connected if $X$ is.

We now have all the ingredients needed for the main result of this section:

###### Proof of Theorem 4.1.

By Remark 1, we may assume without loss of generality that $\phi$ is locally
flat. By Lemma 4.3, $X$ satisfies the conditions of Lemma 4.4, and we let
$X^{\prime}$ be the locally 2-connected 2-complex provided by the latter, and
$\phi^{\prime}:X^{\prime}\to\mathbb{S}^{3}$ the corresponding locally flat
embedding (the case where $X$ has fewer than 3 vertices is trivial). Since the
rotation system of $\phi$ is $\Gamma$-invariant by assumption, the second
sentence of Lemma 4.4 yields an action $\Gamma\curvearrowright X^{\prime}$
with respect to which $\sigma(\phi^{\prime})$ is $\Gamma$-invariant. By the
third sentence of Lemma 4.4 $X^{\prime}$ is simply connected since $X$ is. By
applying a barycentric subdivision (twice) if needed, we may assume that
$X^{\prime}$ is in addition a simplicial complex.

Next, we consider $Y:={\rm fat}(X^{\prime})={\rm
fat}(X^{\prime},\phi^{\prime})$, which is locally 3-connected by Lemma 4.7. By
the remarks following Lemma 4.7, we also obtain a locally flat embedding
$\chi:Y\to\mathbb{S}^{3}$, and an extension $\Gamma\curvearrowright Y$ of the
above action. Moreover, $Y$ is still simply connected.

We finish by applying Theorem 4.2 to $Y$, and pairs of embeddings of the form
$\chi,\chi\circ a$ for each $a\in\Gamma$. To make this precise, we recall that
$\Gamma$ acts on $\chi(Y)\subset\mathbb{S}^{3}$, and we want to extend each
$a\in\Gamma$ into a homeomorphism $h_{a}:\mathbb{S}^{3}\to\mathbb{S}^{3}$. We
let $h_{a}$ be the homeomorphism $\alpha$ obtained from Theorem 4.2 when
applied to the two embeddings $\chi$ and $\psi:=\chi\circ a$ of $Y$. In order
for this map $a\mapsto h_{a}$ to be an action on $\mathbb{S}^{3}$, we need it
to be a homomorphism from $\Gamma$ to $Aut(\mathbb{S}^{3})$. This will not be
the case in general if we let Theorem 4.2 output any $h_{a}$ satisfying
$\psi=h_{a}\circ\chi$, because for example $h_{a^{-1}}$ may differ from
$(h_{a})^{-1}$. But we can control the output of Theorem 4.2 by exploiting its
second statement. This ensures that $g\mapsto h_{a}$ is a homomorphism from
$\Gamma$ to $Aut(\mathbb{S}^{3})$ as desired, because restricting each $h_{a}$
to $Y$ recovers the action of $\Gamma$ on $Y$, which is a homomorphism. (This
idea is spelt out in more detail in [17, Lemma 5.6].)

Notice that $h_{a}\circ\chi=\psi=\chi\circ a$, i.e. $\chi(Y)$ is invariant
with respect to the action we just defined. ∎

##  5 From Group Actions to Invariant Embedded generalised Cayley complexes

In this section we prove the implication (i) $\to$ (iv) of Theorem 1.4. An
_embedded_ 2-complex in a 3-manifold $M$ is a homeomorphic image of a
2-complex in $M$. Given an action $\Gamma\curvearrowright M$, we say that an
embedded 2-complex $X$ is _$\Gamma$ -invariant_ if $\Gamma\curvearrowright M$
preserves $X$ setwise.

######  Theorem 5.1.

Let $\Gamma$ be a finite group, and $\Gamma\curvearrowright\mathbb{S}^{3}$ a
faithful action by homeomorphisms. Then $\Gamma$ admits an embedded,
$\Gamma$-invariant, generalised Cayley complex.

In fact we will prove the following more general statement, which yields a
generalisation of the implication (i) $\to$ (iv) of Theorem 1.5. We say that a
chamber $C$ of an embedded 2-complex $Y\subset M$ is _finitary_ , if $\partial
C$ is a finite subcomplex of $Y$.

######  Theorem 5.2.

Let $M$ be a 3-manifold, let $\Gamma$ be a finitely generated group, and
$\Gamma\curvearrowright M$ a faithful, properly discontinuous, co-compact
action by homeomorphisms. Then there is a $\Gamma$-invariant embedded
2-complex $Y\subset M$ with finitary chambers such that $\Gamma$ acts
regularly on $X^{0}$, and $\pi_{1}(Y)\cong\pi_{1}(M)$.

In particular, when $M$ is simply connected, then $\Gamma$ admits a
$\Gamma$-invariant generalised Cayley complex embedded in $M$.

Notice that the statement that $Y$ has finitary chambers implies in particular
that the vertices of $Y$ have no accumulation point in $M$.

Recall that $2$-complex $C$ is a generalised Cayley complex of $\Gamma$ if $C$
is simply connected and $\Gamma$ admits an action on $C$ that is regular on
$C^{0}$. We will construct such a complex embedded in $M$ in two steps. In the
first step we construct an embedded $2$-complex $X\subset M$ such that our
action $\Gamma\curvearrowright M$ is regular on the chambers of $X$ (Lemma
5.3). In the second step we perform local modifications on $X$ to transform
regularity on the chambers into regularity of the action on the vertices.

###  5.1 Step 1: Constructing an embedded $2$-complex with a regular action on
its chambers

The following lemma performs the first step of our construction of a
generalised Cayley complex of $\Gamma$ as mentioned above:

######  Lemma 5.3.

Let $M$ be a topological 3-manifold, and let $\Gamma\curvearrowright M$ be a
faithful, properly discontinuous, co-compact group action by homeomorphisms.
Then there is a $\Gamma$-invariant, embedded 2-complex $X\subset M$, such that
$\pi_{1}(X)\cong\pi_{1}(M)$, and $\Gamma$ acts regularly on the chambers of
$X$, each of which is finitary and homeomorphic to $\mathbb{R}^{3}$.

For the proof of this we will use the following basic fact:

######  Lemma 5.4.

Let $M$ be a topological 3-manifold, and let $X\subset M$ be an embedded
2-complex such that each chamber of $X$ is homeomorphic to $\mathbb{R}^{3}$,
and it is bounded by a finite subcomplex of $X$. Let $f\in X^{2}$ be a 2-cell
contained in the boundary of two distinct chambers. Then
$\pi_{1}(X)\cong\pi_{1}(X-f)$.

###### Proof.

Let $C_{1},C_{2}$ denote the two chambers having $f$ in their boundaries. We
can continuously deform $f$ via $C_{1}$ (or $C_{2}$) onto a continuous image
$f^{\prime}\subset\partial C_{1}-f$ of a topological disc using the fact that
$C_{1}$ is homeomorphic to a ball in $\mathbb{R}^{3}$, and $f\subset\partial
C_{1}$ is homeomorphic to a disc. We can use $f^{\prime}$ to show that the
circle $\partial f$ is 0-homotopic in $X-f$. Thus we have
$\pi_{1}(X)\cong\pi_{1}(X-f)$ by van Kampen’s theorem since $f$ is simply
connected. ∎

###### Proof of Lemma 5.3.

We may assume without loss of generality that our action
$\Gamma\curvearrowright M$ is smooth by Pardon’s Theorem 7.1.

It is known that for every such action, the quotient space $M/\Gamma$ —which
is a 3-orbifold, but the reader will not need to know what this means— admits
a triangulation $T$ [27, Proposition 1.2.1], which is _adapted_ to the action
in the sense that for each simplex $\sigma$ of $T$, the stabilisers under
$\Gamma$ of all pre-images of points in $\sigma$ are isomorphic to each other.
Since $\Gamma\curvearrowright M$ is co-compact, $M/\Gamma$ is compact, and
thus $T$ is finite. Let $\pi:M\to M/\Gamma$ be the quotient projection. Its
inverse $\pi^{-1}$ lifts $T$ to a triangulation $\tilde{T}$ of $M$, as proved
in [27, Lemma 1.2.2], which is $\Gamma$-invariant by construction. We think of
the 2-skeleton $\tilde{T}^{2}$ of $\tilde{T}$ as an embedded 2-complex in $M$.
It is straightforward to check that the chambers of $\tilde{T}$ are exactly
its 3-cells. It is easy to show that $\pi_{1}(\tilde{T}^{2})\cong\pi_{1}(M)$
by applying van Kampen’s theorem to the 3-cells of $\tilde{T}$.

Notice that the action $\Gamma\curvearrowright M$ is free on the 3-cells of
$\tilde{T}$, and therefore on the chambers of $\tilde{T}^{2}$, because it is
faithful. Indeed, if an element $g$ of $\Gamma$ fixed a 3-cell $C$ setwise,
then $g$ would have to fix $C$ pointwise. This would force $g$ to also fix the
3-cells incident with $C$, hence all of $M$ by its connectedness, implying
that $g$ can only be the identity of $\Gamma$.

If the action is not transitive on the chambers of $\tilde{T}^{2}$, then we
can find a subcomplex $X$ of $\tilde{T}^{2}$ which maintains the other desired
properties and such that $\Gamma$ acts transitively on the chambers of $X$, by
finding an appropriate fundamental domain of 3-cells of $\tilde{T}^{2}$ and
joining them into one chamber. To do so we introduce the following notion.

Say that two chamber-boundaries $C,D$ of a 2-complex $H$ are _adjacent_ , if
their boundaries share a 2-cell of $H$. Say that $C,D$ are _tight-connected_ ,
if there is a sequence $C_{1},\ldots C_{k}$ of chamber-boundaries such that
$C_{1}=C,C_{k}=D$, and $C_{i}$ is adjacent to $C_{i+1}$ for every $1\leq i<k$.
A _tight-component_ is a maximal tight-connected set of chamber-boundaries of
$H$. It is straightforward to check that

the boundaries of 3-cells of a triangulation of any connected 3-manifold form
a single tight-component. (2)

Let $F$ be a maximal tight-connected set of boundaries of 3-cells of
$\tilde{T}^{2}$ that contains at most one representative from each
$\Gamma$-orbit of 3-cells of $\tilde{T}$. The maximality of $F$, combined with
(2), easily implies that $F$ contains a representative from each
$\Gamma$-orbit of 3-cells, for otherwise we could add to $F$ a 3-cell adjacent
with one of its elements (this is a well-known idea, appearing e.g. in [5]).
Thus $F$ contains exactly one representative of each $\Gamma$-orbit of
3-cells, in other words, the action of $\Gamma$ on the translates of $F$ is
regular. Moreover, $F$ is finite since $\Gamma\curvearrowright M$ is co-
compact.

We claim that there is a set $D$ of 2-cells of $\bigcup F$ such that $\bigcup
F\backslash D$ has only one chamber $C$, and moreover $C$ is homeomorphic to
$\mathbb{S}^{3}$. Indeed, we can construct $D$ recursively as follows. As long
as $\bigcup F$ has more than one chamber (each homeomorphic to
$\mathbb{R}^{3}$) we can find two of them $C_{1},C_{2}$ sharing a 2-cell $f$
by tight-connectedness. By removing $f$ we join $C_{1},C_{2}$ into one
chamber, which is homeomorphic to $\mathbb{R}^{3}$ since both $C_{1},C_{2}$
are. (The boundary of the new chamber need not be homeomorphic to
$\mathbb{S}^{2}$, however.) It is easy to see that the tight-connectedness of
the chamber-boundaries of $F$ is preserved. Since $F$ is finite, this
recursion terminates leaving a single chamber, proving our claim.

Let $X\subset M$ be the 2-complex obtained from $\tilde{T}^{2}$ by removing a
set $D$ as above along with all its $\Gamma$-translates. Then $\bigcup F$ is
contained in one chamber of $X$, and it follows that $\Gamma$ acts regularly
on the chambers of $X$.

By construction, each chamber of $X$ is still homeomorphic to $\mathbb{R}^{3}$
and finitary.

Notice that whenever we removed a 2-cell $f$ of $\tilde{T}^{2}$ we joined two
chambers $C_{1},C_{2}$ into one, and so we did not change $\pi_{1}$ by Lemma
5.4. Thus $\pi_{1}(X)\cong\pi_{1}(\tilde{T}^{2})$, which coincides with
$\pi_{1}(M)$ as noticed above. ∎

###  5.2 Step 2: From regularity on the chambers to regularity on the vertices

Having constructed an embedded 2-complex $X\subset M$ such that the action
$\Gamma\curvearrowright M$ of Theorem 5.1 or 5.2 is regular on the chambers of
$X$, our aim now is to modify $X$ locally so that the action becomes regular
on the vertices. Most of the work will go into making the action free on the
vertices, because having done so we will be able to use the standard trick of
contracting a fundamental domain to achieve transitivity. To formulate this
trick in our setup, given a graph $G$, and a subgroup $\Gamma$ of the
automorphism group $Aut(G)$ of $G$, we call a subgraph $H\subseteq G$ a
_fundamental domain_ for $\Gamma$, if $H$ contains exactly one vertex from
each $\Gamma$-orbit.

######  Lemma 5.5 (Babai’s Contraction Lemma [5]).

Let $G\ $be a connected graph, and suppose a group $\Gamma\leq Aut(G)$ acts
freely on the vertex set $V(G)$. Then there is a connected subgraph $D\subset
G$ that is a fundamental domain for the action, and the graph $G/D$ obtained
by contracting each $\Gamma$-image of $D$ into a point is a generalised Cayley
graph of $\Gamma$. In particular, $\Gamma$ acts transitively on $V(G/D)$.

The graph we will later apply Lemma 5.5 to is the 1-skeleton of our 2-complex.

Thus it remains to transform the freeness of the action on the chambers
arising from Lemma 5.3 into freeness on the vertices. This is carried out by
the following result.

######  Theorem 5.6.

Let $M$ be a topological 3-manifold, and let $\Gamma\curvearrowright M$ be
properly discontinuous group action by homeomorphisms. Let $X\subset M$ be a
$\Gamma$-invariant, embedded 2-complex, such that $\pi_{1}(X)\cong\pi_{1}(M)$,
the action of $\Gamma$ on the chambers of $X$ is regular, and each chamber is
finitary and homeomorphic to $\mathbb{R}^{3}$. Then there is a
$\Gamma$-invariant, embedded 2-complex $F(X)\subset M$ with finitary chambers,
such that the action of $\Gamma$ on $V(F(X))$ is free, and
$\pi_{1}(F(X))\cong\pi_{1}(M)$.

Our formal definition of the complex $F(X)$ that achieves this takes some
time, but the idea is rather simple: given a vertex $x$ of $X$ stabilised by
$\Gamma\curvearrowright M$, we notice that we can pick a set $O_{x}$ of nearby
points inside the chambers incident with $x$ such that the action of $\Gamma$
on $O_{x}$ is regular, because $\Gamma$ acts freely on the chambers. The idea
is to blow up each 2-cell, 1-cell, and 0-cell of $X$ into a homeomorph of
$\mathbb{S}^{2}$, in order to modify $X$ into a complex $F(X)$ with vertex set
$\bigcup_{x\in X^{0}}O_{x}$. An example is shown in Figure 2: if $X$ is the
standard cubic lattice embedded in $\mathbb{R}^{3}$ (top left), then a portion
of $F(X)$ is displayed in the bottom right of the figure.

The reader will lose nothing by assuming that $M=\mathbb{S}^{3}$ and $\Gamma$
is finite throughout this section; this is enough for proving Theorem 1.4, and
this assumption makes no difference for any the proofs in this section.

|
---|---
|
Figure 2: An example $F(X)$, when $X$ is the cubic lattice (top left). Each
link graph is an octahedron (top right), and so each pineapple of $F(X)$ is a
truncated cuboctahedron (bottom left). The pineapples are arranged as in the
bottom right figure, which shows four of them in the front.

We now prepare for the formal definition of $F(X)$. Given an embedded
2-complex $X\subset M$, and a homeomorphic image $S$ of $\mathbb{S}^{2}$ in
$M$ which is locally flat, we say that $S$ is _adapted_ to $X$ if

  1. (i)

for every 1-cell $e\in X^{1}$ the intersection $S\cap e$ is either a single
point, or all of $e$, or empty;

  2. (ii)

for every 2-cell $f\in X^{2}$, the intersection $S\cap f$ is either an arc
between two points of the boundary of $f$ (either 0-cells, or interior points
of 1-cells), or empty, and

  3. (iii)

$S$ separates $M$ into two components.

If $S$ is adapted to $X$, and $A$ is one of the two components into which $S$
separates $M$, then we can obtain an embedded 2-complex $X_{A}$ from $X$ by
removing $X\cap A$ and adding $S$ to $X$; to make this more precise, we define
the _$A$ -truncation_ of $X$ to be the embedded 2-complex $X_{A}$ obtained as
follows.

  1. (i)

For every 1-cell $e\in X^{1}$ that intersects $S$ at a point $p$, we declare
$p$ to be a 0-cell of $X_{A}$, we declare the subarc of $e$ lying outside $A$
to be a 1-cell of $X_{A}$, and discard the subarc of $e$ lying inside $A$.

  2. (ii)

For every 2-cell $f\in X^{2}$ intersecting $S$ along an arc $P$ between two
points $x,y$ of the boundary of $f$, we declare $P$ to be a 1-cell of $X_{A}$
—its end-points $x,y$ must be 0-cells of $X_{A}$ by (i). Notice that
$f\backslash(S\cup A)$ is homeomorphic to a disc $f^{\prime}$, and we declare
$f^{\prime}$ to be a 2-cell of $X_{A}$, discarding $f$.

  3. (iii)

The points $p$ and arcs $P$ as in (i)–(ii)subdivide $S$ into topological
discs, which we declare to be 2-cells of $X_{A}$.

  4. (iv)

For every cell $C$ of $X$ that does not intersect $S$, we keep $C$ in $X_{A}$
if it lies outside $A$, and discard it if it is contained in $A$.

We now construct the 2-complex $F(X)$ featuring in Theorem 5.6 by a
combination of such truncations.

######  Definition 5.7.

Given an embedded 2-complex $X\subset M$, we construct another embedded
2-complex $F(X)\subset M$ as follows.

  1. (i)

We blow each 2-cell $f\in X^{2}$ up like a _mango_ ; that is, we replace $f$
by two ‘parallel’ copies $f^{\prime},f^{\prime\prime}$ with the same boundary
and attachment map, and embed $f^{\prime},f^{\prime\prime}$ into $M$ so that
one of the sides of the 2-sphere $S_{f}:=f^{\prime}\cup f^{\prime\prime}$
contains $f$ and is otherwise disjoint from $X$. Moreover, we ensure that
$S_{f}$ is locally flat, and disjoint from $g^{\prime}\cup g^{\prime\prime}$
for every $g\neq f\in X^{2}$ except possibly for intersections along $X^{1}$;
in other words, our mangos do not cross each other. Let $X_{1}$ denote the
resulting 2-complex. We call $f^{\prime}\cup f^{\prime\prime}$ the _mango_ of
$f$, and imagine its side containing $f$ as one.

  2. (ii)

Next, we blow each 1-cell $e\in X_{1}^{1}$ up like a _banana_. To define this
formally, let $S_{e}$ be a homeomorph of $\mathbb{S}^{2}$ in $M$ such that the
end-vertices of $e$ lie on $S_{e}$, one side $A_{e}$ of $S_{e}$ contains the
interior of $e$ and is otherwise disjoint from the 1-skeleton of $X_{1}$, and
$S_{e}$ is adapted to $X_{1}$ and locally flat. It is easy to find such
$S_{e}$, and to ensure that they are pairwise disjoint except possibly at
their vertices. We apply the $A_{e}$-truncation of $X_{1}$ for each $e\in
X_{1}^{1}$ to obtain a new embedded 2-complex $X_{2}$. We call $S_{e}$ the
_banana_ of $e$.

  3. (iii)

Finally, we blow each vertex $v\in X_{2}^{0}=X^{0}$ up like a _pineapple_ ;
that is, we pick a homeomorph $S_{v}$ of $\mathbb{S}^{2}$ in $M$, such that
one side $A_{v}$ of $S_{v}$ contains $v$ but no other vertices of $X_{2}$, and
$S_{v}$ is adapted to $X_{2}$ and locally flat. Moreover, we choose the
$S_{v},v\in X_{2}^{0}$ small enough that they are pairwise disjoint. We apply
the $A_{v}$-truncation of $X_{2}$ for each $v\in X_{2}^{0}$ to obtain the
desired 2-complex $F(X)$. We call $S_{v}$ the _pineapple_ of $v$.

######  Remark 2.

The 1-skeleton of the pineapple of $v$ can be obtained from the link graph of
$v$ by doubling each edge by a parallel one, and then blowing up each vertex
of the resulting plane graph into a cycle of vertices of degree 3.

Example: When $X$ is the cubic lattice in $\mathbb{R}^{3}$, the link graph of
each vertex is isomorphic to the 1-skeleton of the octahedron. Each pineapple
of $F(X)$ is a truncated cuboctahedron. They are arranged as shown in Figure
2.

######  Remark 3.

There is an alternative, more abstract, way to define $F(X)$. A _flag_ of $X$
is a 4-tuple $(c_{0},c_{1},c_{2},c_{3})$ where $c_{i}$ is an $i$-cell of $X$,
and $c_{i}$ is incident with $c_{i-1}$ for $1\leq i\leq 4$, with the
convention that the 3-cells of $X\subset M$ are its chambers. We can identify
the set of 0-cells of $F(X)$ with the set of flags of $X$. We connect two
flags with an 1-cell of $F(X)$ whenever they differ in exactly one coordinate.
We can 4-colour the 1-cells of $F(X)$ using the coordinate at which its end-
vertices differ as a colour. The 2-cells of $F(X)$ are bounded by the
2-coloured cycles with respect to this colouring. Again Figure 2 can serve as
an example. This definition generalises in any dimension.

The letter $F$ in our notation $F(X)$ stands for ‘fruit’, but also for ‘flag’.

Using van Kampen’s theorem it will be easy to deduce that $F(X)$ preserves the
fundamental group of $X$:

######  Lemma 5.8.

For $F(X)$ as in Definition 5.7, we have $\pi_{1}(F(X))\cong\pi_{1}(X)$.

###### Proof.

Notice that if we contract each pineapple $S_{v}$ in the construction of
$F(X)$ to a point, we obtain a 2-complex homeomorphic to $X_{2}$. Thus
$\pi_{1}(F(X))\cong\pi_{1}(X_{2})$ by van Kampen’s theorem. Similarly,
squeezing each banana $S_{e}$ in $X_{2}$ back to an edge with the same
endpoints as $e$, results into a 2-complex homeomorphic to $X_{1}$, and so
$\pi_{1}(X_{2})\cong\pi_{1}(X_{1})$. Finally, squashing each mango $S_{f}$ of
$X_{1}$ onto a disc with the same boundary as $f$ results in a homeomorph of
$X$, yielding $\pi_{1}(X_{1})\cong\pi_{1}(X)$. ∎

In order to be able to use $F(X)$ to prove Theorem 5.6, we need to construct
it more carefully so that $\Gamma\curvearrowright M$ extends to an action on
$F(X)$. This would be easy if $\Gamma$ acted freely on $X$, but in general we
need to take some care to ensure that the stabiliser of each 2-cell, 1-cell or
0-cell fixes the corresponding mango, banana, or pineapple, respectively. We
will be able to achieve this by choosing a chamber $C$ of $X$ and using its
closure $\overline{C}$ as a fundamental domain. More precisely, we will prove

######  Lemma 5.9.

Let $X$ and $\Gamma$ be as in the statement of Theorem 5.6, let $C$ be a
chamber of $X$, and let $D$ be the sub-complex of $X$ bounding $C$. Then there
is a homeomorphic copy $F\subset M$ of $F(D)$, such that
$\Gamma(F\cap\overline{C})$ is homeomorphic to $F(X)$.

Here, $\Gamma A$ denotes the image of a set $A\subset M$ under the action
$\Gamma\curvearrowright M$, and $F(D)$ is given by Definition 5.7.

Before proving Lemma 5.9, let us see how it implies Theorem 5.6.

###### Proof of Theorem 5.6.

Given $X$ as in the statement, we construct $F(X)$ as in Definition 5.7. By
Lemma 5.8, $F(X)$ is simply connected since $X$ is. Easily, $F(X)$ has
finitary chambers since $X$ does.

Let $D$ be the sub-complex of $X$ bounding a chamber $C$ of $X$. Then Lemma
5.9 yields an embedded 2-complex $F\subset M$ such that
$F^{\prime}:=\Gamma(F\cap\overline{C})$ is homeomorphic to $F(X)$. Notice that
$F^{\prime}$ is $\Gamma$-invariant by definition.

It remains to check that the action of $\Gamma$ on $V(F^{\prime})$ is free.
This is true because each vertex of $F^{\prime}$ lies in the interior of a
chamber of $X$, and $\Gamma$ acts freely on the chambers of $X$ by assumption.

∎

It remains to prove Lemma 5.9. To construct the desired copy $F$ of $F(D)$, we
will first design the intersection of $F$ with each 2-cell of $F(D)$. To do
so, we need to remember how the bananas and pineapples of $F(X)$ intersect
each 2-cell of $X$ (the mangos do not); these intersections are described in
the following definition, but they are easier to see in Figure 3.

Figure 3: A topological 5-gon with a slice pattern. The $E_{i}$ are depicted
in red (if colour is shown), and the $V_{i}$ in blue.

A _topological $n$-gon_ is a regular 2-complex $P$ containing exactly one
2-cell $f$, and such that $P^{1}$ is homeomorphic to $\mathbb{S}^{1}$ (and
coincides with the boundary of $f$).

######  Definition 5.10.

Let $P$ be a _topological $n$-gon_, with vertices $v_{1},\ldots v_{n}$, and
edges $v_{i}v_{i+1\pmod{n}}$. A _slice pattern_ on $P$ consists of two sets
$\mathcal{E}:=\\{E_{1},\ldots,E_{n}\\}$ and
$\mathcal{V}:=\\{V_{1},\ldots,V_{n}\\}$ of arcs on $P$, such that (Figure 3)

  1. (i)

the end-points of $E_{i}$ are $v_{i}$ and $v_{i+1\pmod{n}}$;

  2. (ii)

the end-points of $V_{i}$ are interior points of the edges
$v_{i-1\pmod{n}}v_{i}$ and $v_{i}v_{i+1\pmod{n}}$;

  3. (iii)

each $E_{i}$ and $V_{i}$ meets the 1-skeleton of $P$ at its end-points only;

  4. (iv)

the elements of $\mathcal{E}$ are pairwise disjoint, and so are the elements
of $\mathcal{V}$, and

  5. (v)

$V_{i}$ is disjoint from $E_{j}$ unless $j=i$ or $j=i-1\pmod{n}$.

An _automorphism_ of a topological $n$-gon $P$ is a homeomorphism of $P$
mapping each vertex to a vertex (and hence each edge to an edge). To prove
Lemma 5.9 we will apply the following lemma to each 2-cell of $D$; this helps
us by pushing the difficulty one dimension down.

######  Lemma 5.11.

Let $P$ be a topological $n$-gon, and $h:P\to P$ an automorphism such that
$h^{2}$ is the identity. Then there is a slice pattern
$(\mathcal{E},\mathcal{V})$ of $P$ preserved by $h$. That is, $h$ maps each
element of $\mathcal{E}$ to an element of $\mathcal{E}$, and each element of
$\mathcal{V}$ to an element of $\mathcal{V}$.

###### Proof.

Notice that $h$ must fix some arc $A$ joining two boundary points of $P$, and
exchange the two components into which $A$ separates $P$. We can thus pick
‘half’ a slice pattern on the quotient polygon $P/h$, and lift it back to $P$
to obtain a slice pattern of $P$. We have assumed here that $h$ in not the
identity, in which case the statement is trivial. ∎

###### Proof of Lemma 5.9.

Notice that although $\Gamma\curvearrowright M$ is free on the chambers of
$X$, some of the 2-cells of $D$ may have a non-trivial stabiliser in $\Gamma$.
However, for every such 2-cell $f\in D^{2}$, there is at most one non-identity
element $h_{f}$ of $\Gamma$ fixing $f$, because $f$ is in the boundary of at
most two chambers of $X$, and no non-identity element of $\Gamma$ fixes a
chamber. For the same reason, $h_{f}$ must be an involution. Applying Lemma
5.11 with $P=\overline{f}$ we obtain a slice pattern
$(\mathcal{E},\mathcal{V})$ of $\overline{f}$ preserved by $h_{f}$. Choosing
such a slice pattern for one representative $f$ of each $\Gamma$-orbit of
2-cells of $D$, and translating it to the other representatives via the action
of $\Gamma$, we obtain a family $(\mathcal{E}_{f},\mathcal{V}_{f})_{f\in
D^{2}}$ of slice patterns of all 2-cells of $D$, which family is _compatible_
with our action $\Gamma\curvearrowright M$, i.e.
$(\mathcal{E}_{f},\mathcal{V}_{f})=(\mathcal{E}_{gf^{\prime}},\mathcal{V}_{gf^{\prime}})$
whenever $gf^{\prime}=f$ for some $f,f^{\prime}\in D^{2}$ and some
$g\in\Gamma$. Notice that this implies that the set of $\Gamma$-translates of
this family is $\Gamma$-invariant.

To find the desired copy $F$ of $F(D)$, we can start by picking the mangoes
arbitrarily as in Definition 5.7 (i). (The half-mango outside $\overline{C}$
will be irrelevant.) For each $e\in D^{1}$, pick the corresponding banana
$S_{e}$ (Definition 5.7 (ii)) so that its intersection with each 2-cell of $D$
is contained in one of the $\mathcal{E}_{f}$ of the above family of slice
patterns. Similarly, for each $v\in D^{0}$, pick the corresponding pineapple
$S_{v}$ (Definition 5.7 (iii)) so that its intersection with each 2-cell of
$D$ is contained in one of the $\mathcal{V}_{f}$. This completes the
construction of $F$.

It remains to check that $\Gamma(F\cap\overline{C})$ is homeomorphic to $F(X)$
as claimed. To see this, notice that $F\cap\overline{C}$ contains one half of
each mango, a slice of each banana, and a sector of each pineapple of $F$.
Moreover, when acted upon by $\Gamma$, these portions combine well to produce
a homeomorph of $F(X)$. Indeed, for any $g,h\in\Gamma$, the translates
$g(F\cap\overline{C})$ and $h(F\cap\overline{C})$ are disjoint except possibly
at the boundaries of $g\overline{C}$ and $h\overline{C}$, where they meet
along the $\Gamma$-invariant family of slice patterns chosen above.

∎

We are now ready to complete the main result of this section.

###### Proof of Theorems 5.1 and 5.2.

Apply Lemma 5.3 to obtain an embedded 2-complex $X\subset M$ such that
$\pi_{1}(X)\cong\pi_{1}(M)$ and $X$ satisfies all the assumptions of Theorem
5.6. Then apply Theorem 5.6 to this $X$, to turn it into an embedded 2-complex
$F(X)$ such that the action of $\Gamma$ on $V(F(X))$ is free.

Suppose that the action of $\Gamma$ on $V(F(X))$ is not transitive. Let
$D\subset F(X)^{1}$ be a connected fundamental domain for the action as
provided by Lemma 5.5. Notice that $D$ is finite because
$\Gamma\curvearrowright M$ is co-compact. Let $T$ be a spanning tree of $D$,
and let $U$ be an open neighbourhood of $T$ in $M$ homeomorphic to
$\mathbb{R}^{3}$. Easily, we can pick $U$ small enough that its
$\Gamma$-translates are pairwise disjoint. Contracting each $\Gamma$-translate
of $U$ into a point, we obtain a manifold homeomorphic to $M$, still acted
upon by $\Gamma$, into which manifold the 2-complex $Y:=F(X)/T$ is embedded
$\Gamma$-invariantly. These contractions preserve the property that all
chambers are finitary. By the choice of $T$, the action of $\Gamma$ on $V(Y)$
is regular. Moreover,
$\pi_{1}(Y)\cong\pi_{1}(F(X))\cong\pi_{1}(X)\cong\pi_{1}(M)$. In particular,
when $M$ is simply connected, then $Y$ is a generalised Cayley complex of
$\Gamma$. ∎

Notice that when $M$ is simply connected in Theorem 5.2, then it is a special
3-manifold by Theorem 1.1. Thus we have proved the implication $\ref{m
i}\rightarrow\ref{m iv}$ of Theorems 1.4 and 1.5.

######  Remark 4.

Since $X$, and hence $F(X)$, are regular 2-complexes by construction, the
2-complex $Y$ that the above proof provides is edge-regular (as defined in
Section 2.3). We do not know if we can always obtain a regular $Y$ in Theorems
5.1 and 5.2.

##  6 Concluding remarks

Our proofs of Theorems 1.4 and 1.5 are now complete. The implication $\ref{m
iv}\rightarrow\ref{m i}$ is trivial because every special 3-manifold is simply
connected. The implications $\ref{m iv}\rightarrow\ref{m iii}\rightarrow\ref{m
ii}$ are trivial because every special 3-manifold embeds in $\mathbb{S}^{3}$.
The implications (ii) $\to$ (iv) and $\ref{m i}\rightarrow\ref{m iv}$ have
been proved in Sections 3 and 5, respectively.

As mentioned in the introduction, there are groups admitting a Cayley complex
embeddable in $\mathbb{R}^{3}$ with invariant planar rotation system, but only
if we allow infinite pre-chambers. Examples include $\mathbb{Z}^{2}$, and more
generally any fundamental group of an orientable closed surface [1].

######  Question 6.1.

Which infinite groups admit a Cayley complex embeddable in $\mathbb{R}^{3}$
with invariant planar rotation system?

This class of groups contains the fundamental groups of closed 3-manifolds (as
proved by Theorem 1.4) and closed surfaces; more generally, it contains all
Kleinian function groups [17, §12]444This is not stated explicitly in [17],
but the proof of Theorem 1.3 there constructs an embedding of a Cayley complex
in $\mathbb{R}^{3}$.. It is easy to see that it also contains groups of the
form $F\times\mathbb{Z}$ where $F$ is free. It would be interesting to clarify
the relationship between the groups of Question 6.1 and the Kleinian groups.

######  Question 6.2.

Is there a finite set $X$ of 3-manifolds, such that each of the groups of
Question 6.1 admits a discrete action on an element of $X$?

This $X$ should contain the three special open 3-manifolds of Theorem 1.1. It
should also contain $\mathbb{R}^{2}\times S^{1}$, because of $\mathbb{Z}^{2}$
and other surface groups. Moreover, $X$ should contain $C\times\mathbb{R}$
where $C$ stands for the Cantor 2-sphere, i.e. $\mathbb{S}^{2}$ with a Cantor
set removed; we include $C\times\mathbb{R}$ to let groups of the form
$F\times\mathbb{Z}$ act. These 5 manifolds could suffice as far as we can
tell.

One could enquire more generally about the class of groups admitting a Cayley
complex embeddable in $\mathbb{R}^{3}$ with no further restrictions, though we
do not expect an easy alternative description. An example of such a group is
the Baumslag–Solitar group $BS(1,2)$. Its standard Cayley complex defined by
$\left<a,b\mid bab^{-1}=a^{2}\right>$ embeds in the cartesian product of a
binary tree and $\mathbb{R}$, as well-known figures show. It has been proved
that $BS(1,2)$ cannot be mapped in a nondegenerate way into the fundamental
group of an orientable 3-manifold [34], and so it is not one of the groups of
Theorem 1.5.

##  7 Appendix: The universal covers of 3-manifolds and orbifolds

In this section we provide a proof of Theorem 1.1. We emphasize that our only
contribution to this proof was to ask the experts about it and put the pieces
together.

For our proof of Theorem 1.1, and also in Section 5, we make use of Pardon’s
theorem that topological actions on a 3-manifold can be smoothed:

######  Theorem 7.1 ([31, 30]).

Every properly discontinuous action of a finitely generated group $\Gamma$ on
a 3-manifold $M$ by homeomorphisms is the uniform limit of smooth actions of
$\Gamma$ on $M$.

We will also need the following consequence of the Orbifold theorem of
Boileau, Leeb & Porti:

######  Theorem 7.2 ([9, COROLLARY 1.3]).

Every compact connected 3-orbifold which does not contain any bad
2-suborbifolds is the quotient of a compact 3-manifold by a finite group
action.

###### Proof of Theorem 1.1.

Suppose first that $M$ admits an action $\Gamma\curvearrowright M$ as above,
which is in addition free. Then $Q:=M/\Gamma$ is a closed 3-manifold, and $M$
is its universal cover because it is simply connected. Moreover,
$\Gamma\cong\pi_{1}(Q)$.

We may assume without loss of generality that $Q$ is orientable by replacing
$\Gamma$ by its subgroup of orientation-preserving elements. Thus $Q$ is a
closed, orientable, connected, 3-manifold. The fact that such a $Q$ has
special universal cover is apparently well-known to experts. We reproduce a
proof by Ian Agol [3].

If $\pi_{1}(Q)$ is finite, then its universal cover $M$ is homeomorphic to
$\mathbb{S}^{3}$ by the validity of the Poincaré conjecture.

If $\pi_{1}(Q)$ is infinite and $\pi_{2}(Q)$ is trivial, then we claim that
the universal cover $M$ is homeomorphic to $\mathbb{R}^{3}$. Indeed, in this
case, $Q$ has a geometric decomposition by the Geometrization Theorem [36, 32,
33]. If the decomposition is trivial, then $Q$ is modelled on one of the six
Thurston geometries homeomorphic to $\mathbb{R}^{3}$, and hence the universal
cover is $\mathbb{R}^{3}$. Otherwise, we apply the Virtually Haken conjecture,
proved by Agol [4], which asserts that every compact, irreducible 3-manifold
$Q$ with infinite $\pi_{1}(Q)$ is finitely covered by a Haken manifold
$Q^{\prime}$. The reader does not need to know what a Haken manifold is, all
we need is a result of Waldhausen proving that any Haken manifold $Q^{\prime}$
has universal cover homeomorphic to $\mathbb{R}^{3}$ [37, Theorem 8.1]. Since
$Q^{\prime}$ covers $Q$, we deduce that $M$ is homeomorphic to
$\mathbb{R}^{3}$ as claimed.

If $\pi_{1}(Q)$ is infinite and $\pi_{2}(Q)$ non-trivial, then it may be that
$Q$ is modelled on the $\mathbb{S}^{2}\times\mathbb{R}$ geometry and $Q$ is
homeomorphic to $RP3\\#RP3$ or $\mathbb{S}^{2}\times\mathbb{S}^{1}$. In this
case the number of ends of $\pi_{1}(Q)$, and $M$, is 2, and $M$ is
homeomorphic to $\mathbb{S}^{2}\times\mathbb{R}$. Otherwise, $Q$ is a non-
trivial connect sum by Papakyriakopoulos’ sphere theorem [29], and we claim
that the universal cover $M$ is a Cantor 3-sphere. Indeed, the connect
summands have universal cover either $\mathbb{S}^{3}$, or
$\mathbb{S}^{2}\times\mathbb{R}$, or $\mathbb{R}^{3}$ by the above discussion.
When forming connect sums, we remove open balls from each summand manifold,
and glue their sphere boundaries together. The universal cover is obtained by
gluing the universal covers of each summand punctured along balls, either
finitely many in $\mathbb{S}^{3}$, or infinitely many in
$\mathbb{S}^{2}\times\mathbb{R}$ or $\mathbb{R}^{3}$. It is easy to see that
such manifolds are built out of thrice punctured spheres, and hence the
universal cover can be decomposed into thrice punctured spheres. It is not
hard to see that such a manifold is homeomorphic to the Cantor 3-sphere.

It remains to consider the case where the action $\Gamma\curvearrowright M$ is
non-free, which we will be able to reduce to the free case. By Theorem 7.1, we
may assume that $\Gamma\curvearrowright M$ is smooth. Thus the quotient
$O:=M/\Gamma$ is endowed with the structure of an orbifold. Theorem 7.2
implies that $O$ is finitely covered by a manifold $M^{\prime}$. The universal
cover of $M^{\prime}$ covers $O$, and so it coincides with $M$ by the
uniqueness of a simply connected cover. Recall that $\pi_{1}(M^{\prime})$ acts
on its universal cover $M$ freely, properly discontinuously, and co-compactly.
We have reduced to the free case, and so we can deduce that $M$ is special in
all cases. ∎

## Acknowledgement

We thank Ryan Budney, Louis Funar, John Pardon, and Henry Wilton for
discussions around Theorem 1.1, and Ian Agol for providing most of the proof.
We thank Johannes Carmesin, Bruno Zimmermann and Mathoverflow user Sam Nead
for other helpful discussions.

## References

  * [1] A 3-manifold with fundamental group isomorphic to a surface group.  Mathematics Stack Exchange.  https://math.stackexchange.com/q/734107.
  * [2] Closed 3-manifolds with free abelian fundamental groups.  MathOverflow.  https://mathoverflow.net/q/31904.
  * [3] I. Agol.  How wild can an open topological 3-manifold be if it has a compact quotient?  MathOverflow.  https://mathoverflow.net/q/422343.
  * [4] I. Agol.  The virtual Haken conjecture. With an appendix by Agol, Daniel Groves, and Jason Manning.  Documenta Mathematica, 18:1045–1087, 2013.
  * [5] L. Babai.  Some applications of graph contractions.  J. Graph Theory, 1:125–130, 1977.
  * [6] L. Babai.  Automorphism groups, isomorphism, reconstruction.  In Graham, R. L. (ed.) et al., Handbook of combinatorics. Vol. 2\. Amsterdam: Elsevier (North-Holland), pages 1447–1540. 1995.
  * [7] R. H. Bing.  An alternative proof that 3-manifolds can be triangulated.  Annals of Mathematics, 69:37–65, 1959.
  * [8] M. Boileau.  Around 3-manifold groups.  Winter Braids Lecture Notes, 5(II):1–26, 2018.
  * [9] M. Boileau, B. Leeb, and J. Porti.  Geometrization of 3-Dimensional Orbifolds.  Annals of Mathematics, 162(1):195–290, 2005.
  * [10] B. H. Bowditch.  Notes on Maskit’s planarity theorem.  Enseign. Math., 68(1):1–24, 2022.
  * [11] M. Brown.  A proof of the generalized Schoenflies theorem.  Bull. Amer. Math. Soc, 66:74–76, 1960.
  * [12] J. Carmesin.  Embedding simply connected 2-complexes in 3-space II.  Preprint 2017.
  * [13] John H. Conway and Derek A. Smith.  On Quaternions and Octonions | Their Geometry, Arithmetic, and Symmetry.  CRC Press, 2003.
  * [14] James F. Davis and R. James Milgram.  A survey of the spherical space form problem.  Harwood Academic Publishers, 1985.
  * [15] J. Dinkelbach and B. Leeb.  Equivariant ricci flow with surgery and applications to finite group actions on geometric 3-manifolds.  Geometry and Topology, 13:1129–1173, 2009.
  * [16] A. Edmonds.  Transformation groups and low-dimensional manifolds.  In Group actions on manifolds, Contemporary Mathematics, pages 339–366, 1983.
  * [17] A. Georgakopoulos.  On planar cayley graphs and kleinian groups.  Transactions of the American Mathematical Society, 373:4649–4684, 2020.
  * [18] A. Georgakopoulos and J. Kim.  2-complexes with unique embeddings in 3-space.  To appear in Bulletin of the LMS.
  * [19] Jonathan L. Gross and Thomas W. Tucker.  Topological Graph Theory.  Courier Corporation, January 2001.
  * [20] S. Gu.  Contractible open manifolds which embed in no compact, locally connected and locally 1-connected metric space.  Algebraic & Geometric Topology, 21(3):1327–1350, August 2021.
  * [21] I. Hambleton.  Topological spherical space forms.  In Handbook of Group Actions, Clay Math. Proc., volume 3, pages 151–172. Beijing-Boston: ALM, 2015.
  * [22] H. Hopf.  Enden offener Räume und unendliche diskontinuierliche Gruppen.  Comment. Math. Helv., 16:81–100, 1944.
  * [23] H. Levinson and B. Maskit.  Special embeddings of Cayley diagrams.  J. Combin. Theory (Series B), 18(1):12–17, 1975.
  * [24] H. Maschke.  The representation of finite groups, especially of rotation groups of three and four dimensional space, by Cayley’s color diagrams.  Amer. J. Math, 18:156–194, 1896.
  * [25] J. Matoušek, M. Tancer, and U. Wagner.  Hardness of embedding simplicial complexes in $R^{d}$.  13(2):259–295, 2011.
  * [26] B. Mazur.  On embeddings of spheres.  Bull. Amer. Math. Soc, 65:59–65, 1959.
  * [27] I. Moerdijk and D. A. Pronk.  Simplicial Cohomology of Orbifolds.  Indagationes Mathematicae, 10(2):269–293, 1999.
  * [28] E.E. Moise.  Affine Structures in 3-Manifolds: V. The Triangulation Theorem and Hauptvermutung.  Annals of Mathematics, 56:96–114, 1952.
  * [29] C. D. Papakyriakopoulos.  On Dehn’s Lemma and the Asphericity of Knots.  Annals of Mathematics, 66(1):1–26, 1957.
  * [30] J. Pardon.  From topological actions on $R^{3}$ to isometric actions.  MathOverflow.  https://mathoverflow.net/q/420591 (version: 2022-04-18).
  * [31] J. Pardon.  Smoothing finite group actions on three-manifolds.  Duke Mathematical Journal, 170:1043–1084, 2021.
  * [32] G. Perelman.  The entropy formula for the Ricci flow and its geometric applications.  arXiv:math/0211159, 2002.
  * [33] G. Perelman.  Ricci flow with surgery on three-manifolds.  arXiv:math/0303109, 2003.
  * [34] P. B. Shalen.  Three-manifolds and Baumslag–Solitar groups.  Topology and its Applications, 110(1):113–118, 2001.
  * [35] R. Skora.  Cantor sets in $S^{3}$ with simply connected complements.  Topology and its Applications, 24(1):181–188, December 1986.
  * [36] W. P. Thurston.  Three dimensional manifolds, Kleinian groups and hyperbolic geometry.  Bull. Amer. Math. Soc. (N.S.), 6(3):357–381, 1982.
  * [37] F. Waldhausen.  On Irreducible 3-Manifolds Which are Sufficiently Large.  Annals of Mathematics, 87(1):56–88, 1968.
  * [38] B.P. Zimmermann.  On finite groups acting on spheres and finite subgroups of orthogonal groups.  arXiv:1108.2602, 2011.
  * [39] B.P. Zimmermann.  On topological actions of finite groups on ${S}^{3}$.  Topology Appl., 236:59–63, 2018.