# Answers to questions about medial layer graphs of self-dual regular and
chiral polytopes

Marston Conder
Department of Mathematics, University of Auckland,
Private Bag 92019, Auckland 1142, New Zealand
Email<EMAIL_ADDRESS>
Isabelle Steinmann
Eckhart Hall, University of Chicago, 5734 S University Ave,
Chicago, IL 60637, USA
Email<EMAIL_ADDRESS>
###### Abstract

An abstract $n$-polytope $\mathcal{P}$ is a partially-ordered set which
captures important properties of a geometric polytope, for any dimension $n$.
For even $n\geq 2$, the incidences between elements in the middle two layers
of the Hasse diagram of $\mathcal{P}$ give rise to the medial layer graph of
$\mathcal{P}$, denoted by $\mathcal{G}=\mathcal{G}(\mathcal{P})$. If $n=4$,
and $\mathcal{P}$ is both highly symmetric and self-dual of type
$\\{p,q,p\\}$, then a Cayley graph $\mathcal{C}$ covering $\mathcal{G}$ can be
constructed on a group of polarities of $\mathcal{P}$. In this paper we
address some open questions about the relationship between $\mathcal{G}$ and
$\mathcal{C}$ that were raised in a 2008 paper by Monson and Weiss, and
describe some interesting examples of these graphs. In particular, we give the
first known examples of improperly self-dual chiral polytopes of type
$\\{3,q,3\\}$, which are also among the very few known examples of highly
symmetric self-dual finite polytopes that do not admit a polarity. Also we
show that if $p=3$ then $\mathcal{C}$ cannot have a higher degree of $s$-arc-
transitivity than $\mathcal{G}$, and we present a family of regular
$4$-polytopes of type $\\{6,q,6\\}$ for which the vertex-stabilisers in the
automorphism group of $\mathcal{C}$ are larger than those for $\mathcal{G}$.

Keywords: Abstract polytope, regular polytope, chiral polytope, medial graph

##  1 Introduction

Abstract polytopes are combinatorial structures which mimic the general
properties of classical geometric polytopes, such as the Platonic solids, and
yet need not be convex or have natural geometric realisations. An abstract
polytope is simply a partially-ordered set, the elements of which are assigned
a rank representing their dimension, and with incidence satisfying certain
conditions that arise naturally in a geometric setting.

A maximal chain in this poset is called a flag, and from the defining
properties of an abstract polytope (given in Section 3), every automorphism of
such a poset is uniquely determined by its effect on any given flag. Some
$n$-polytopes, called equivelar polytopes, can be assigned a type
$\\{p_{1},\ldots,p_{n}\\}$ that encodes the structure of the 2-sections of the
polytope, which are isomorphic when they involve elements of the same ranks.

Two important classes of equivelar polytopes consist of the regular polytopes,
which possess the highest degree of symmetry (in that the automorphism group
is transitive on the flags), and the chiral polytopes, which have maximal
‘rotational’ symmetry but no reflectional symmetry (and so are not quite
regular). Many ‘classical’ polytopes and other early examples of polytopes are
regular, so regular polytopes have been studied extensively. A comprehensive
description of regular polytopes is given in McMullen and Schulte’s book
Abstract Regular Polytopes [16]. On the other hand, chiral $n$-polytopes do
not exist for $n=2$, and are challenging to find for all except small values
of $n\geq 3$, and cannot be convex. For these reasons, they are less well
understood. Nevertheless much information about chiral polytopes can be found
in a paper by Schulte and Weiss [21].

Following work by Monson and Weiss in [18], here we will investigate two
graphs associated with finite and self-dual regular or chiral $4$-polytopes of
type $\\{p,q,p\\}$. Our investigation will involve some aspects of the study
of finite symmetric (arc-transitive) graphs, which we will describe in Section
2.2.

The first graph of interest here is the medial layer graph $\mathcal{G}$,
which is constructed from the middle two layers of the Hasse diagram of such a
polytope $\mathcal{P}$. This graph $\mathcal{G}$ is bipartite and $p$-valent,
and by self-duality of $\mathcal{P}$, it admits automorphisms that swap its
two parts. Medial layer graphs were first studied (on the abstract level) in
[17], where it was shown that when $p=3$ the medial layer graph $\mathcal{G}$
is 3-arc-regular if $\mathcal{P}$ is regular, and 2-arc-regular if
$\mathcal{P}$ is chiral. As a partial converse, it was also shown how to
construct a group $G$ (called $\Gamma$ in [17]) from a finite connected 2- or
3-arc-regular cubic graph, such that if the generators of $G$ satisfy certain
conditions, then $G$ is isomorphic to the automorphism group of a finite self-
dual regular or chiral polytope of type $\\{3,q,3\\}$.

The second graph of interest is a Cayley graph $\mathcal{C}$ that covers the
medial layer graph, as constructed in [18]. This construction is possible
because (properly) self-dual regular or chiral $4$-polytopes of type
$\\{p,q,p\\}$ possess a polarity (a duality of order $2$) from which $p$
polarities can be created to form the Cayley group of $\mathcal{C}$. Monson
and Weiss described in [18] various connections between $\mathcal{G}$ and
$\mathcal{C}$ in detail, and also gave several interesting examples of these
graphs for $p=3$, $4$ and $5$.

In this paper we address some of the open questions raised by Monson and Weiss
near the end of their 2008 paper [18]. In particular, in Section 5 we exhibit
the first known examples of chiral polytopes of type $\\{3,q,3\\}$ that do not
possess a polarity that preserves the orbits of flags under the natural action
of the automorphism group of the polytope. (These are also two of the very few
known examples of highly symmetric self-dual finite polytopes that do not
admit any polarity. No such polytopes were known at the time their existence
was effectively questioned by Schulte and Weiss in their 1991 paper [21].) In
Section 6 we prove that if $p=3$, and $\mathcal{G}$ is $t$-arc-transitive and
$\mathcal{C}$ is $s$-arc-transitive, then $s\leq t$, and in Section 7 we
present an infinite family of directly-regular polytopes of type $\\{6,q,6\\}$
for which the vertex-stabilisers in the automorphism group of the Cayley graph
$\mathcal{C}$ are larger than those for the medial layer graph $\mathcal{G}$.

Before all that, in Sections 2, 3 and 4, we give relevant background on arc-
transitive graphs, abstract polytopes (including regular and chiral polytopes)
and their automorphism groups, and on medial layer graphs, and at the end we
make some final remarks in Section 8. More details are available in the second
author’s University of Auckland Masters thesis [22].

##  2 Arc-transitive graphs

###  2.1 Preliminary definitions

All graphs in this paper are assumed to be finite, undirected and simple (with
no loops or multiple edges).

An arc in a graph is an ordered edge, or equivalently, an ordered pair of
adjacent vertices, and a graph is called arc-transitive (or symmetric) if its
automorphism group is transitive on the set of all arcs. More generally, an
$s$-arc in a graph is a sequence $(v_{0},\ldots,v_{s})$ of vertices such that
every two consecutive $v_{i}$ are adjacent, and every three consecutive
$v_{i}$ are distinct. (In other words, an $s$-arc is a walk of length $s$ in
which an edge is never immediately followed by the reverse of that edge.)

A group $G$ of automorphisms of a graph $\Gamma$ is said to act $s$-arc-
transitively on $\Gamma$ if it acts transitively on the set of all $s$-arcs in
$\Gamma$, and $s$-arc-regularly on $\Gamma$ if $G$ acts regularly (that is,
sharply-transitively) on the $s$-arcs of $\Gamma$. The graph $\Gamma$ is then
called $s$-arc-transitive (respectively $s$-arc-regular) if ${\rm
Aut}(\Gamma)$ acts $s$-arc-transitively (resp. $s$-arc-regularly) on $\Gamma$.
In particular, 0-arc-transitive means vertex-transitive, and 1-arc-transitive
means arc-transitive (or symmetric).

Note that while arc-transitivity implies edge-transitivity, the converse does
not hold. Also an $s$-arc-transitive graph might not be $(s-1)$-arc-
transitive: the path graph on three vertices is edge-transitive and 2-arc-
transitive, yet is neither vertex- nor arc-transitive. But an $s$-arc-
transitive graph that is not a tree is $t$-arc-transitive whenever $0\leq
t\leq s$, and hence is vertex-transitive and therefore regular. Similarly,
$s$-arc-regularity implies $s$-arc-transitivity, but the converse is not true:
a graph might be $s$-arc-transitive for some $s$ but not be $t$-arc-regular
for any $t$, as shown by the $2$-arc-transitive complete graph $K_{5}$.

###  2.2 Symmetric cubic graphs

A focus of much of this paper will be on symmetric 3-valent graphs. These are
particularly well understood, thanks to the pioneering work in the 1940s and
50s by Tutte, who proved the following theorem in [23], bounding the order of
a vertex-stabiliser:

######  Theorem 2.1.

Let $\Gamma$ be a connected finite symmetric cubic graph. Then
$\mathrm{Aut}(\Gamma)$ acts regularly on the set of $s$-arcs of $\Gamma,$ for
some $s\in\\{1,2,3,4,5\\}$.

In particular, the Orbit-Stabiliser Theorem gives
$|\mathrm{Aut}(\Gamma)|=|V(\Gamma)|\,3\cdot 2^{s-1}$ for some $s\leq 5$, with
vertex-stabiliser of order $3\cdot 2^{s-1}=3,6,12,24$ or $48$.

Tutte’s work was taken further by Djoković and Miller, who in [13] divided the
collection of all connected finite symmetric cubic graphs into seven classes,
depending on the value of $s$ and on the existence or otherwise of an
involutory automorphism that reverses an arc. They proved that if $s=1,3$ or
$5$, then such an automorphism always exists, but could not do the same when
$s=2$ or $4$. Examples admitting no such involutory automorphism were
constructed by Conder and Lorimer in [9], thereby proving that all seven
classes (one each for $s=1,3$ and $5$, and two each for $s=2$ and $4$) are
non-empty. These classes are now denoted by $1$, $2^{1}$, $2^{2}$, $3$,
$4^{1}$, $4^{2}$ and $5$, with $s$ or $s^{1}$ indicating the class of
connected finite $s$-arc-regular cubic graphs admitting an involutory arc-
reversing involution, and with $s^{2}$ indicating the class (for $s=2$ or $4$)
of such graphs admitting no such involutory automorphism. Sometimes these
classes are denoted also by $1^{\prime}$, $2^{\prime}$, $2^{\prime\prime}$,
$3^{\prime}$, $4^{\prime}$, $4^{\prime\prime}$ and $5^{\prime}$, respectively,
in the obvious way.

Moreover, Djoković and Miller showed in [13] that for each of the seven
classes, the automorphism group of every graph $\Gamma$ in that class is a
homomorphic image of particular ‘universal’ group. This universal group is a
free product $V*_{A}E$ of the pre-image $V$ of the stabiliser (in ${\rm
Aut}(\Gamma)$) of a vertex $v$ and the pre-image $E$ of the stabiliser of an
edge $e=\\{v,w\\}$ incident to $v$, with the pre-image $A$ of the stabiliser
of the arc $(v,w)$ amalgamated. The subgroups $V$, $E$ and $A$ are finite, and
in each one of the seven classes, can be chosen independently of the given
graph $\Gamma$. In fact $V$ is isomorphic to $C_{3}$, $S_{3}$, $S_{3}\times
C_{2}$, $S_{4}$ and $S_{4}\times C_{2}$, for $s=1,2,3,4$ and $5$,
respectively, with $A$ being a Sylow $2$-subgroup (of index $3$) in each case,
and $E$ contains $A$ as a subgroup of index $2$, together with the pre-image
of an arc-reversing automorphism of order $2$ or $4$.

The seven universal groups were denoted by $A_{1}^{\prime}$, $A_{2}^{\prime}$,
$A_{2}^{\prime\prime}$, $A_{3}^{\prime}$, $A_{4}^{\prime}$,
$A_{4}^{\prime\prime}$ and $A_{5}^{\prime}$ in [13], but we will denote them
by $G_{1}$, $G_{2}^{\,1}$, $G_{2}^{\,2}$, $G_{3}$, $G_{4}^{\,1}$,
$G_{4}^{\,2}$ and $G_{5}$, as in [9]. Presentations for all seven groups are
obtainable from [13], and alternative (but equivalent) presentations for them
are given in [9].

Furthermore, if $\theta$ is any smooth homomorphism from one of these
universal groups $G_{s}$, $G_{s}^{\,1}$ or $G_{s}^{\,2}$ onto a finite group
$G$, with ‘smooth’ here meaning that $\theta$ preserves the orders of $V$, $E$
and $A$, then a connected finite symmetric cubic graph $\Gamma$ can be
constructed with $G$ being an $s$-arc-regular subgroup of ${\rm Aut}(\Gamma)$.
Conditions ensuring that $G={\rm Aut}(\Gamma)$ are given in [10], based on
observations made in any of [9, 13, 14]. The resulting correspondence between
appropriate quotients of the universal group and members of the relevant class
has enabled all graphs in each class to be determined up to a certain order,
extending the almost-complete collection of connected finite symmetric cubic
graphs of order up to $512$ compiled by hand by Foster, now known as the
Foster census [2]. With the help of various computational methods, this census
was completed and also extended to order up to $768$ by Conder and Dobcsányi
in [6], and subsequently by the first author up to order $10000$ in [5].

The Djoković-Miller classification was extended by Conder and Nedela in [10],
showing that the collection of all finite symmetric cubic graphs can be
divided into 17 classes, based on which types of arc-transitive group actions
the graph admits. A subgroup of the automorphism group of such a graph
$\Gamma$ may be said to act with type $s$ or $s^{1}$ if it acts regularly on
the $s$-arcs of $\Gamma$ and contains an arc-reversing automorphism of order
$2$, or with type $s^{2}$ if it so acts but without an arc-reversing
automorphism of order $2$.

For what follows later in this paper, there are two other important properties
of finite symmetric cubic graphs that need to be mentioned.

######  Proposition 2.2.

If $G$ is an arc-transitive group of automorphisms of a connected finite
symmetric cubic graph $\Gamma$ in class $4^{1}$, $4^{2}$ or $5$, then $G$
contains no subgroup acting with type $2^{1}$, $2^{2}$ or $3$. Equivalently,
if the automorphism group of a finite symmetric cubic graph contains a $2$\-
or $3$-arc-regular subgroup, then it contains no $4$\- or $5$-arc-regular
subgroup.

###### Proof.

This follows from observations made in [9, 13, 14]; see [10, Corollary 2.2]. ∎

For the other property, we give the presentations for $G_{2}^{\,1}$,
$G_{2}^{\,2}$ and $G_{3}$ from [9]:

$G_{2}^{\,1}=\langle\,h,p,a\ |\ h^{3}=a^{2}=p^{2}=1$, $apa=p$,
$php=h^{-1}\,\rangle$,

$G_{2}^{\,2}=\langle\,h,p,a\ |\ h^{3}=p^{2}=1$, $a^{2}=p$,
$php=h^{-1}\,\rangle$, and

$G_{3}=\langle\,h,p,q,a\ |\ h^{3}=a^{2}=p^{2}=q^{2}=1$, $apa=q$, $qp=pq$,
$ph=hp$, $qhq=h^{-1}\,\rangle$.

In these cases $V=\langle h,p\rangle$, $\,\langle h,p\rangle$ and $\langle
h,p,q\rangle$, respectively, with $A=\langle p\rangle$, $\,\langle p\rangle$
and $\langle p,q\rangle$, respectively, and with $a$ being the pre-image of an
automorphism reversing the arc $(v,w)$.

######  Proposition 2.3.

Let $G$ be a finite group that acts regularly on the $2$-arcs of the connected
cubic graph $\Gamma$. Then $\Gamma$ has a 3-arc-regular group of automorphisms
containing $G$ if and only if there is a group automorphism $\theta:G\to G$
which centralises the image of $V$ in $G$, and takes the image of $a$ to the
image of $ap$.

###### Proof.

See [9, Proposition 4.1], in which (by abuse of notation) $h$, $a$, $p$ and
$H$ are the images in $G$ of the generators $h$, $a$ and $p$ of $G_{2}^{\,1}$
or $G_{2}^{\,2}$ and the subgroup $V$. ∎

Next we describe some important background on regular and chiral polytopes,
helpful for what follows. More detailed information can be found on them in
[16] and [21].

##  3 Abstract polytopes and their automorphisms

###  3.1 Basic definition and properties

Abstract polytopes capture the combinatorial properties of classically defined
and studied geometric polytopes. Their definition is somewhat complicated, but
not difficult.

An abstract $n$-polytope (or abstract polytope of rank $n)$ is a partially
ordered set $\mathcal{P}$, endowed with a strictly monotone rank function
having range $\\{-1,0,1,...,n\\}$. The elements of $\mathcal{P}$ are called
faces, with those of rank $j$ called $j$-faces, and the faces of ranks $0,1$
and $n-1$ usually called the vertices, edges and facets of $\mathcal{P},$
respectively. The poset $\mathcal{P}$ must satisfy the following conditions,
which ensure that abstract polytopes share many of the combinatorial
properties frequently associated with the face lattices of convex polytopes.

First, $\mathcal{P}$ has a minimum face, $F_{-1}$, and a maximum face $F_{n}$,
and each maximal chain (called a flag) contains exactly $n+2$ faces. Two flags
are said to be adjacent if they differ by just one face, and if they differ
only in their $i$-faces, they are said to be $i$-adjacent. Also $\mathcal{P}$
must be strongly flag connected, which means that any two flags $\Phi$ and
$\Psi$ of $\mathcal{P}$ can be linked by a sequence of flags
$\Phi=\Phi_{0},\Phi_{1},\dots,\Phi_{k}=\Psi$, all containing $\Phi\cap\Psi$,
such that each two successive flags $\Phi_{i-1}$ and $\Phi_{i}$ are adjacent.
Finally, $\mathcal{P}$ must have a certain homogeneity property called the
diamond condition, namely that whenever $F$ and $G$ are faces of ranks $j-1$
and $j+1$ for some $j$ with $F\leq G$, there are exactly two faces $H$ of rank
$j$ such that $F\leq H\leq G$.

Note that by the diamond condition, for each flag $\Phi$ of $\mathcal{P}$ and
each $i\in\\{0,\dots,n-1\\}$ there exists a unique flag that is $i$-adjacent
to $\Phi$. We denote such a flag by $\Phi^{i}$.

Next, if $F$ and $G$ are faces of $\mathcal{P}$ with $F\leq G$, then the set
$\\{H\in\mathcal{P}\mid F\leq H\leq G\,\\}$ is called a section of
$\mathcal{P}$, and denoted by $G/F$. In particular, each facet $F_{n-1}$ may
be viewed as the section $F_{n-1}/F_{-1}=\\{H\in\mathcal{P}\mid H\leq
F_{n-1}\,\\}$. Also if $F_{0}$ is a vertex, then the section
$F_{n}/F_{0}=\\{H\in\mathcal{P}\mid F_{0}\leq H\,\\}$ is called the vertex-
figure of $\mathcal{P}\\!$ at $F_{0}$.

Note that every section $G/F$ of $\mathcal{P}$ is again an abstract polytope.
In particular, every rank 2 section $G/F$ between an $(i\\!-\\!2)$-face $F$
and an incident $(i+1)$-face $G$ of an abstract polytope $\mathcal{P}$ is
isomorphic to the face lattice of a (possibly infinite) polygon. If for each
$i$ the number of sides of this polygon is a constant, depending only on $i$
and not on $F$ or $G$, then we say that $\mathcal{P}$ is an equivelar
polytope. Also we define the Schläfli type of an equivelar $n$-polytope
$\mathcal{P}$ to be $\\{p_{1},p_{2},\dots,p_{n-1}\\}$, when each section
between an $(i-2)$-face and an $(i+1)$-face is an abstract $p_{i\,}$-gon. The
Platonic solids are equivelar polytopes, and have Schläfli types $\\{3,3\\}$,
$\\{4,3\\}$, $\\{3,4\\}$, $\\{3,5\\}$, and $\\{5,3\\}$.

We now consider symmetries, dualities and polarities of abstract polytopes.

An automorphism of a polytope $\mathcal{P}$ is an order- and rank-preserving
bijection from $\mathcal{P}$ to $\mathcal{P}$, and the group of all
automorphisms of $\mathcal{P}$ is denoted by ${\rm Aut}(\mathcal{P})$, or
often by $\Gamma(\mathcal{P})$. Note that by the diamond condition and the
strongly flag connected property, every automorphism is uniquely determined by
its effect on any given flag.

A duality $\delta$ is a bijection between two polytopes $\mathcal{P}$ and
$\mathcal{Q}$ which preserves incidence but reverses order, so that $F\leq G$
in $\mathcal{P}$ if and only if $F^{\delta}\geq G^{\delta}$ in $\mathcal{Q}$.
If such a bijection exists, then $\mathcal{P}$ and $\mathcal{Q}$ are duals of
each other. Every polytope $\mathcal{P}$ has a unique dual $\mathcal{P}^{*}$,
and if $\mathcal{P}\cong\mathcal{P}^{*}$, we say that $\mathcal{P}$ is self-
dual. If $\mathcal{P}$ is equivelar with type $\\{p_{1},\ldots,p_{n-1}\\}$,
then $\mathcal{P}^{*}$ is equivelar with type $\\{p_{n-1},\ldots,p_{1}\\}$, so
if $\mathcal{P}$ is self-dual, then $p_{i}=p_{n-i}$ for $1\leq i<n$.

Some self-dual polytopes admit a polarity (a duality of order 2). In such
cases we call the group of all automorphisms and dualities of $\mathcal{P}$
the extended group $\overline{\Gamma}(\mathcal{P})$ of $\mathcal{P}$, and this
contains the automorphism group $\Gamma(\mathcal{P})$ as a subgroup of index
2. Also if such a polytope $\mathcal{P}$ admits a polarity that preserves the
orbit of every flag of $\mathcal{P}$ under $\Gamma(\mathcal{P})$, then
$\mathcal{P}$ is said to be properly self-dual, while otherwise it is called
improperly self-dual. Note that if $\mathcal{P}$ is properly self-dual, then
all of its dualities preserve the flag orbits.

In the next few subsections we describe two important classes of highly
symmetric polytopes, namely regular and chiral polytopes.

###  3.2 Regular polytopes

An abstract $n$-polytope $\mathcal{P}$ is called regular if its automorphism
group $\Gamma(\mathcal{P})$ acts transitively on the flags of $\mathcal{P}$.
As every automorphism is uniquely determined by its effect on any given flag,
this definition implies that regular polytopes are precisely those whose
automorphism group has maximum possible order (equal to the number of flags).

Now let $\mathcal{P}$ be any regular $n$-polytope. Then every section of
$\mathcal{P}$ is a regular polytope, with automorphism group a subgroup of
$\Gamma(\mathcal{P})$. Moreover, any two sections of $\mathcal{P}$ defined by
faces of the same ranks are isomorphic, and so $\mathcal{P}$ is equivelar. Let
$\\{p_{1},\ldots,p_{n-1}\\}$ be its type, and also set some flag
$\Phi=\\{F_{-1},F_{0},\ldots,F_{n}\\}\in\mathcal{F}(\mathcal{P})$ as the base
flag of $\mathcal{P}$.

As $\Gamma(\mathcal{P})$ acts regularly on the flags of $\mathcal{P}$, it
follows that for $i=0,1,\ldots,n-1$ we can define $\rho_{i}$ to be the unique
automorphism sending the base flag $\Phi$ to its $i$-adjacent flag $\Phi^{i}$.
The resulting $n$ automorphisms are involutions, and are often referred to as
‘reflections’ of the polytope $\mathcal{P}$, in the same way that a reflection
of the $3$-cube about an edge $e=\\{v,w\\}$ fixes the vertex $v$ and the edge
$e$ while interchanging the two faces incident with $e$.

Moreover, as shown in [16], the $n$ involutions $\rho_{0},\ldots,\rho_{n-1}$
generate the automorphism group of $\mathcal{P}$, and satisfy (at least) the
Coxeter relations

$\rho_{i}^{2}=1\ \text{ for }0\leq
i<n\quad\text{and}\quad(\rho_{i}\rho_{j})^{m_{ij}}=1\ \text{where}\
m_{ij}=\begin{cases}1&\hbox{if }\ i=j\\\ 2&\hbox{if }\ |i-j|\geq 2{}\\\
p_{i}&\hbox{if }\ j=i+1\end{cases}$ (1)

as well as the following intersection conditions:

$\langle\,\rho_{i}\mid i\in I\,\rangle\cap\langle\,\rho_{i}\mid i\in
J\,\rangle=\langle\,\rho_{i}\mid i\in I\cap J\,\rangle\quad\text{for all
}I,J\subseteq\\{0,\ldots,n-1\\}.\quad$ (2)

For later use, it is also helpful to understand the stabilisers of faces of
$\mathcal{P}$ in $\Gamma(\mathcal{P})$. As is also shown in [16], the
stabiliser in $\Gamma(\mathcal{P})$ of the $i$-face $F_{i}$ of the base flag
$\Phi$ of $\mathcal{P}$ is

$\Gamma(\mathcal{P})_{F_{i}}=\langle\,\rho_{j}:0\leq j<n\mid j\neq
i\,\rangle.\qquad$ (3)

Next, for $i=1,\ldots,n-1$ we define $\sigma_{i}=\rho_{i-1}\rho_{i}$, and call
this a ‘rotation’, as $\sigma_{i}$ fixes the faces in
$\Phi\setminus\\{F_{i-1},F_{i}\\}$ and cyclically permutes both the $i$-faces
and the $(i-1)$-faces in the section $F_{i+1}/F_{i-2}$. These $n-1$ rotations
generate the rotation subgroup $\Gamma^{+}(\mathcal{P})$ of
$\Gamma(\mathcal{P})$, which has index at most 2 in $\Gamma(\mathcal{P})$, and
it follows from Equation (1) that they satisfy the following relations:

$\sigma_{j}^{p_{j}}=1\ \hbox{ for }1\leq j\leq
n-1\quad\hbox{and}\quad(\sigma_{j}\sigma_{j+1}\ldots\sigma_{k})^{2}=1\ \hbox{
for }1\leq j<k\leq n-1.\ $ (4)

We say that $\mathcal{P}$ is directly-regular if the index of
$\Gamma(\mathcal{P})^{+}$ in $\Gamma(\mathcal{P})$ is 2, and non-orientably
regular otherwise. All ‘classical’ regular polytopes (such as polygons and the
Platonic solids) are directly-regular, while the hemi-cube (obtained by
identifying antipodally opposite pairs of vertices of the cube) is non-
orientably regular.

Finally, note that a regular polytope $\mathcal{P}$ is self-dual if and only
if there exists a (unique) polarity $\delta\in\overline{\Gamma}(\mathcal{P})$
that preserves the base flag $\Phi$ but reverses all of its faces. Such a
polarity induces an involutory group automorphism of $\Gamma(\mathcal{P})$
taking $\rho_{j}$ to $\rho_{n-1-j}$ for all $j$, and this implies that
$\overline{\Gamma}(\mathcal{P})$ is isomorphic to the semi-direct product
$\Gamma(\mathcal{P})\rtimes C_{2}$.

###  3.3 Chiral polytopes

An abstract $n$-polytope $\mathcal{P}$ is said to be chiral if its
automorphism group $\Gamma(\mathcal{P})$ has two flag orbits, with two
adjacent flags always lying in different orbits. Such polytopes have maximal
rotational symmetry but admit no reflections, and come in pairs, with each
member of a pair being a ‘mirror image’ of the other (as the name suggests).

Chiral polytopes are neither as easy to find nor as easy to study as regular
polytopes. In particular, there are no finite chiral polytopes in Euclidean
3-space, and no convex chiral polytopes. There are plenty of rank $3$
(viewable also as orientably-regular but chiral maps), but it becomes more
challenging to construct examples as the rank increases.

Indeed for quite some time, the only known finite examples of chiral polytopes
had ranks $3$ and $4$, but just over 15 years ago, some finite examples of
rank $5$ were constructed by Conder, Hubard and Pisanski [8], and these were
followed by others of ranks $5$ to $8$ by Conder and Devillers (in unpublished
work). Soon after, Pellicer [19] proved the existence of finite chiral
polytopes of every rank $d\geq 3$.

Pellicer’s proof involved a clever construction, but this produced examples
that could have ‘extremely large’ order. More recently, Conder, Hubard and
O’Reilly-Regueiro [7] showed how to construct concrete examples of chiral
$n$-polytopes of type $\\{3,3,\dots,3,k\\}$ for some $k$, as extensions of
regular simplices, with all but finitely many alternating and symmetric groups
as their automorphism groups (for every integer $n\geq 5$). Meanwhile
Cunningham [12] gave an explanation for the large orders, proving that for
$n\geq 8$, every chiral $n$-polytope has at least $48(n-2)(n-2)!$ flags.

We now give some of the basic theory and properties of chiral polytopes, as
can be found in more detail in [16] and [21].

Let $\mathcal{P}$ be any chiral $n$-polytope, where $n\geq 3$. Then for any
given base flag $\Phi=\\{F_{-1},\ldots,F_{n}\\}$ of $\mathcal{P}$, there exist
automorphisms $\sigma_{1},\ldots,\sigma_{n-1}$ of $\mathcal{P}$ such that each
$\sigma_{i}$ fixes the faces in $\Phi\setminus\\{F_{i-1},F_{i}\\}$, and
cyclically permutes the $i$-faces and the $(i-1)$-faces in the section
$F_{i+1}/F_{i-2}$. Moreover, these automorphisms generate
$\Gamma(\mathcal{P})$, and can be chosen so that $\sigma_{i}$ takes $\Phi$ to
$(\Phi^{i})^{i-1}$, for $1\leq i<n$, and also $\mathcal{P}$ is equivelar of
type $\\{p_{1},\ldots,p_{n-1}\\}$, where each $p_{i}$ is the order of the
‘rotation’ $\sigma_{i}$. Again in this case the rotation group
$\Gamma(\mathcal{P})^{+}$ of $\mathcal{P}$ is the subgroup generated by
$\sigma_{1},\ldots,\sigma_{n-1}$, and so
$\Gamma(\mathcal{P})^{+}=\Gamma(\mathcal{P})$.

These $n-1$ rotations satisfy the relations in (4), and they also satisfy
certain intersection conditions analogous to those in (2), as follows. For
$0\leq i\leq j\leq n$, define

$\tau_{i,j}=\begin{cases}1&\text{if $\ 0=i<j\ $ or $\ i<j=n$,}\\\
\sigma_{i}&\text{if $\ 0<i=j<n$,}\\\
\sigma_{i}\sigma_{i+1}\ldots\sigma_{j-1}\sigma_{j}&\text{otherwise.}\end{cases}$

Then for all subsets $I$ and $J$ of $\\{-1,0,\ldots,n\\}$, these automorphisms
satisfy

$\langle\,\tau_{s+1,t}\mid s,t\in I\,\rangle\cap\langle\,\tau_{s+1,t}\mid
s,t\in J\,\rangle=\langle\,\tau_{s+1,t}\mid s,t\in I\cap J\,\rangle.\quad{}$
(5)

It is also easy to prove that every section of a chiral polytope is itself a
regular or chiral polytope.

The stabiliser in $\Gamma(\mathcal{P})$ of the face $F_{i}$ of the chosen flag
$\Phi$ of $\mathcal{P}$ is given by

$\Gamma(\mathcal{P})_{F_{i}}=\begin{cases}\langle\sigma_{2},\sigma_{3},\ldots,\sigma_{n-1}\rangle&\hbox{for
}\,i=0,\\\ \langle\\{\sigma_{j}\mid
j\not\in\\{i,i+1\\}\\}\cup\\{\sigma_{i}\sigma_{i+1}\\}\rangle&\hbox{for
}\,1\leq i\leq n-2,\\\
\langle\sigma_{1},\sigma_{2},\ldots,\sigma_{n-2}\rangle&\hbox{for
}\,i=n-1.\end{cases}$ (6)

Each of these coincides (in its description) with the stabiliser of the
corresponding face in the rotation subgroup for a regular polytope. In fact
there is quite a close relationship between the structure of the rotation
subgroup of a directly-regular polytope (with
$\Gamma(\mathcal{P})^{+}\neq\Gamma(\mathcal{P})$), and the automorphism group
of a chiral polytope, in that each group is generated by the ‘rotations’
$\sigma_{i}$. The principle difference is the existence or otherwise of
reflections, as summarised in the following theorem from [21], in which the
effect of the group automorphism $\varphi$ on the generators $\sigma_{i}$ is
the same as conjugation by $\rho_{0}$ in the regular case:

######  Theorem 3.1.

If $A$ is a finite group generated by $n-1$ elements
$\sigma_{1},\ldots,\sigma_{n-1}$ which satisfy the relations in (4) and
intersection conditions in (5), then $A$ is isomorphic to the rotation
subgroup $\Gamma(\mathcal{P})^{+}$ of some chiral or directly-regular polytope
$\mathcal{P}$. Moreover, $\mathcal{P}$ is directly-regular if and only if
there exists an involutary group automorphism $\varphi:A\to A$ taking
$\sigma_{1}$ to $\sigma_{1}^{-1}$, and $\sigma_{2}$ to
$\sigma_{1}^{\,2}\sigma_{2}$, and $\sigma_{i}$ to $\sigma_{i}$ for $3\leq
i<n$.

Finally in this subsection, we consider self-duality. Every self-dual regular
polytope is properly self-dual, but in contrast, chiral polytopes can be
properly or improperly self-dual, as defined in Subsection 3.1. By a pair of
theorems in [15], the kind of self-duality is determined by the following
theorem.

######  Theorem 3.2.

Let $\mathcal{P}$ be a chiral polytope with distinguished generators
$\\{\sigma_{1},\ldots,\sigma_{n-1}\\}$ with respect to its base flag $\Phi$.
Then
(a) $\mathcal{P}$ is properly self-dual if and only if there exists an
involutory automorphism of $\Gamma(\mathcal{P})$
taking $\sigma_{i}$ to $\sigma_{n-i}^{-1}$ for all $i\in\\{1,\ldots,n-1\\}$,
while
(b) $\mathcal{P}$ is improperly self-dual if and only if there exists a group
automorphism of $\Gamma(\mathcal{P})$
taking $\sigma_{i}$ to $\sigma_{n-i}^{-1}$ for $1\leq i\leq n-3$, and
$\sigma_{n-2}$ to $\sigma_{1}\sigma_{2}\sigma_{1}^{-1}$, and $\sigma_{n-1}$ to
$\sigma_{1}$.

###  3.4 Constructing directly-regular or chiral polytopes

There are many methods for constructing directly-regular or chiral polytopes,
and we briefly describe just three of them here.

The first method, which we use to construct improperly self-dual chiral
polytopes in Section 5, takes quotients of the orientation-preserving subgroup
(of index $2$) in an infinite string Coxeter group, as was done in [8].

A Coxeter group is a finitely-presented group with generators
$x_{1},\ldots,x_{n}$ subject to the relations $(x_{i}x_{j})^{m_{ij}}=1$ for
$1\leq i\leq j\leq n$, with $m_{i,i}=1$ (so that $x_{i}^{\,2}=1$) for all $i$,
and with $m_{i,j}\geq 2$ whenever $i\neq j$. If also $m_{i,j}=2$ whenever
$|i-j|\geq 2$, then this is called a string Coxeter group; and if
$p_{i}=m_{i,i+1}$ for $1\leq i<n$, then the group is often denoted by
$[p_{1},\ldots,p_{n-1}]$, and illustrated by the Coxeter-Dynkin diagram in
Figure 1.

Figure 1: Coxeter-Dynkin diagram for the string Coxeter group
$[p_{1},\ldots,p_{n-1}]$.

Now let $\Lambda$ be the string Coxeter group $[p_{1},\ldots,p_{n-1}]$, and
let $\Lambda^{+}$ be its ‘rotation subgroup’, namely the index $2$ subgroup
consisting of all elements of $\Lambda$ expressible as words of even length in
the canonical generators of $\Lambda$.

Then the automorphism group $\Gamma(\mathcal{P})$ of every regular polytope
$\mathcal{P}$ with Schläfli type $\\{p_{1},p_{2},\dots,p_{n-1}\\}$ is a
homomorphic image of $\Lambda$, with its rotation group
$\Gamma(\mathcal{P})^{+}$ being the image of $\Lambda^{+}$ . Conversely, if
$A$ is any smooth homomorphic image of $\Lambda$ (meaning that the orders of
the images of the canonical generators $x_{i}$ of $\Lambda$ and their pairwise
products $x_{i}x_{j}$ are preserved), and the images $\rho_{i-1}$ of the
$x_{i}$ satisfy the intersection conditions in (2), then there exists some
regular $n$-polytope $\mathcal{P}$ with Schläfli type
$\\{p_{1},p_{2},\dots,p_{n-1}\\}$ such that $A$ is isomorphic to
$\Gamma(\mathcal{P})$.

Similarly, the automorphism group $\Gamma(\mathcal{P})$ of every chiral
polytope $\mathcal{P}$ with Schläfli type $\\{p_{1},p_{2},\dots,p_{n-1}\\}$ is
a homomorphic image of $\Lambda^{+}$, and conversely, if $A$ is any smooth
homomorphic image of $\Lambda^{+}$, and the images $\sigma_{i}$ of the
generators $x_{i}x_{i+1}$ satisfy the intersection conditions given in
subsection 3.3 (in terms of the alternative generators $\tau_{i,j}$), then
there exists some chiral or directly-regular $n$-polytope $\mathcal{P}$ with
Schläfli type $\\{p_{1},p_{2},\dots,p_{n-1}\\}$ such that $A$ is isomorphic to
$\Gamma(\mathcal{P})^{+}$. The polytope $\mathcal{P}$ is directly-regular if
the epimorphism $\theta$ from $\Lambda^{+}$ to $A$ extends to an epimorphism
from $\Lambda$ to $\Gamma(\mathcal{P})$ as in Theorem 3.1, and otherwise it is
chiral. In the latter case, $\ker\theta$ is normal in $\Lambda^{+}$ but not in
$\Lambda$.

The command LowIndexNormalSubgroups in Magma can be used to find all normal
subgroups of up to a given index (within limits) in a finitely-presented
group, such as a string Coxeter group $\Lambda$ or its rotation group
$\Lambda^{+}$, and hence to find the automorphism groups of all regular and
chiral polytopes of small rank with up to a given number of flags. Such a
process uses checks on the intersection conditions and the extendability of
the normality (or otherwise) of the normal subgroup of $\Lambda^{+},$ as done
previously by the first author to produce the lists in [3, 4].

Once the automorphism group $\Gamma(\mathcal{P})$ of a regular or chiral
polytope $\mathcal{P}$ is found, we can use it to construct the polytope
itself. This can be achieved by identifying the $j$-faces of $\mathcal{P}$
with the right cosets of $\Gamma(\mathcal{P})_{F_{j}}$ in
$\Gamma(\mathcal{P})$ for all $j$, with the help of (3) or (6), and then the
partial ordering can be found via non-empty intersection of cosets; see [17].

The second method involves using connected finite 2- or 3-arc-regular cubic
graphs to construct 4-polytopes of type $\\{3,q,3\\}$. This is described in
detail in [17], and we give only a brief summary for the 3-arc-regular case,
with the 2-arc-regular case being similar.

If the connected cubic graph $\Gamma$ is 3-arc-regular, then for each vertex
$v$ there exists a unique non-trivial automorphism which fixes $v$ and its
three neighbours. (In fact this automorphism is the image of the generator $p$
in the universal group $G_{3}$ for 3-arc-regular cubic graphs given in
subsection 2.2.)

Now let $\rho_{2}$, $\rho_{0}$, $\rho_{3}$ and $\rho_{1}$ be such
automorphisms for the four vertices in a given 3-arc
$(v_{0},v_{1},v_{2},v_{3})$ of $\Gamma$, and let $\delta$ be the unique
involutory automorphism of $\Gamma$ that reverses this 3-arc. If $\rho_{0},$
$\rho_{1}$, $\rho_{2}$ and $\rho_{3}$ satisfy the intersection conditions in
(2), then the group generated by these four automorphisms and $\delta$ is
isomorphic to the extended automorphism group of a regular polytope
$\mathcal{P}$ of type $\\{3,q,3\\}$. (Moreover, the medial layer graph (as
defined in the next section) of this polytope $\mathcal{P}$ is isomorphic to
either $\Gamma$ or the canonical double cover of $\Gamma$, depending on
whether or not $\Gamma$ is bipartite.)

The third method constructs regular or chiral polytopes as covers of smaller
polytopes.

For example, if $\mathcal{Q}$ is a non-orientably regular $n$-polytope, then
there exists a directly-regular $n$-polytope $\mathcal{P}$ that is the
orientable double cover of $\mathcal{Q}$, constructible as follows. The
rotation group $\Gamma(\mathcal{Q})^{+}$ of $\mathcal{Q}$ is the epimorphic
image of the rotation subgroup $\Lambda^{+}$ of some string Coxeter group
$\Lambda$, and by non-orientability $\Gamma(\mathcal{Q})^{+}$ is equal to
$\Gamma(\mathcal{Q})$, and hence contains images of the $n$ reflecting
generators of $\Lambda$. Now if $\alpha$ is the involutory automorphism of
$\Gamma(\mathcal{Q})^{+}$ induced by conjugation by the image of one such
generator, then the semi-direct product
$\Gamma(\mathcal{Q})^{+}\rtimes\langle\alpha\rangle$ is the automorphism group
of a directly-regular polytope $\mathcal{P}$ with twice as many flags as
$\mathcal{Q}$, and with
$\Gamma(\mathcal{P})^{+}\cong\Gamma(\mathcal{Q})^{+}=\Gamma(\mathcal{Q})$.

Another approach takes a regular or chiral $n$-polytope $\mathcal{P}$, with
automorphism group $\Gamma(\mathcal{Q})$ obtainable as the quotient of some
string Coxeter group $\Lambda$ or its rotation subgroup $\Lambda^{+}$ by a
normal subgroup $J$, and then constructs a larger regular or chiral
$n$-polytope $\mathcal{Q}$ that covers $\mathcal{P}$, via some normal subgroup
$K$ of $\Lambda$ or its rotation subgroup $\Lambda^{+}$ such that $K$ is
contained in $J$. This approach was taken successfully to find chiral covers
of chiral polytopes in [11], and to find chiral covers of regular polytopes in
[24], with abelian covering group $J/K$ in each case.

##  4 Medial layer graphs

If $\mathcal{P}$ is a polytope with even rank $n$, then the two ‘middle’
layers of the Hasse diagram of $\mathcal{P}$ form the medial layer graph
$\mathcal{G}$ of $\mathcal{P}$. The vertices of this graph are the
$(\frac{n-2}{2})$-faces and $(\frac{n}{2})$-faces of $\mathcal{P}$, with two
vertices being adjacent whenever the corresponding two faces are incident in
$\mathcal{P}$. Obviously this graph $\mathcal{G}$ is bipartite, and if
$\mathcal{P}$ is equivelar with type $\\{p_{1},\ldots,p_{n-1}\\}$, then the
vertices of the two parts of $\mathcal{G}$ have valencies $p_{\frac{n-2}{2}}$
and $p_{\frac{n+2}{2}}$.

For the rest of this section, we will focus on equivelar 4-polytopes of type
$\\{p,q,p\\}$ that are either regular and self-dual, or chiral and properly
self-dual.

Let $\mathcal{P}$ be such a polytope, and let
$\Phi=\\{F_{-1},F_{0},F_{1},F_{2},F_{3},F_{4}\\}$ be its base flag. Note that
every 2-face of $\mathcal{P}$ is a $p$-gon, and there are $p$ of these around
each 1-face of $\mathcal{P}$, and in every 3-face of $\mathcal{P}$, there are
$q$ $p$-gons around each 0-face.

Because $\mathcal{P}$ is regular or chiral, its rotation group
$\Gamma(\mathcal{P})^{+}$ is generated by $\sigma_{1}$, $\sigma_{2}$ and
$\sigma_{3}$, satisfying the relations
$\sigma_{1}^{\,p}=\sigma_{2}^{\,q}=\sigma_{3}^{\,p}=(\sigma_{1}\sigma_{2})^{2}=(\sigma_{2}\sigma_{3})^{2}=(\sigma_{1}\sigma_{2}\sigma_{3})^{2}=1$
given by (4). Moreover, as shown by Schulte and Weiss in [21], the
intersection conditions in (5) can be replaced simply by

$\langle\sigma_{1}\rangle\cap\langle\sigma_{2}\rangle=\\{1\\},\quad\langle\sigma_{2}\rangle\cap\langle\sigma_{3}\rangle=\\{1\\}\quad\hbox{and}\quad\langle\sigma_{1},\sigma_{2}\rangle\cap\langle\sigma_{2},\sigma_{3}\rangle=\langle\sigma_{2}\rangle.$
(7)

If $\mathcal{P}$ is properly self-dual, admitting a polarity $\delta$ that
reverses the base flag $\Phi$, then $\delta$ satisfies the relations
$\,\delta^{2}=1,\ \delta^{-1}\sigma_{1}\delta=\sigma_{3}^{-1}\,$ and
$\,\delta^{-1}\sigma_{2}\delta=\sigma_{2}^{-1}$, and we may regard
$\overline{\Gamma}(\mathcal{P})$ as a subgroup of ${\rm Aut}(\mathcal{G})$.

Next, following an approach taken by Monson and Weiss in [18], we may
construct a covering graph $\mathcal{C}$ of $\mathcal{G}$, which is a Cayley
graph on the group $G$ generated by the $p$ polarities of $\mathcal{P}$
defined by setting $\,\delta_{j}=\sigma_{1}^{\,1-j}\delta\sigma_{1}^{\,j-1}\,$
for $1\leq j\leq p$.

Note that these $p$ polarities are distinct, as they take the $2$-face $F_{2}$
of $\Phi$ to the $p$ distinct 1-faces in the section $F_{3}/F_{0}$. Some
interactions between them and the rotations $\sigma_{i}$ are given in the
following Lemma from [18], with subscripts treated mod $p$:

######  Lemma 4.1.

(a) $\,\delta_{j+k}=\sigma_{1}^{-k}\delta_{j}\sigma_{1}^{\,k};$ (b)
$\,\delta_{j}\delta_{k}=\sigma_{1}^{\,1-j}\sigma_{3}^{\,k-j}\sigma_{1}^{\,k-1};$
(c) $\,\sigma_{2}^{\,2}=\delta_{2}\delta_{3}\delta_{2}\delta_{1};$ (d)
$\,\sigma_{3}\delta_{j}\delta_{3}^{-1}=\delta_{1}\delta_{2}\delta_{j+1}\delta_{2}\delta_{1};$
(e)
$\,(\sigma_{2}\sigma_{3})^{-1}\delta_{j}\sigma_{2}\sigma_{3}=\delta_{3-j};$
(f)
$\,\sigma_{2}^{-1}\delta_{j}\sigma_{2}=\delta_{1}\delta_{2}\delta_{4-j}\delta_{2}\delta_{1}\,$
and
$\,\sigma_{2}\delta_{j}\sigma_{2}^{-1}=\delta_{2}\delta_{3}\delta_{4-j}\delta_{3}\delta_{2}$.

In particular, the group $G$ generated by the polarities
$\delta_{1},\delta_{2},\ldots,\delta_{p}$ is a normal subgroup of the extended
rotation group
$\overline{\Gamma}(\mathcal{P})^{+}=\langle\sigma_{1},\sigma_{2},\sigma_{3},\delta\,\rangle$.
The relationships between these and other subgroups of ${\rm
Aut}(\mathcal{G})$ associated with $\mathcal{P}$ are illustrated in Figure 2.

Figure 2: Inclusions between certain subgroups of $\mathrm{Aut}(\mathcal{G})$.

Now let $\mathcal{C}$ be the Cayley graph for $G$ with generating set
$\\{\delta_{1},\ldots,\delta_{p}\\}$, so that two vertices $g$ and $h$ are
adjacent in $\mathcal{C}$ if and only if $gh^{-1}=\delta_{i}$ for some
$i\in\\{1,\ldots,p\\}$. This graph is a connected symmetric $p$-valent cover
of $\mathcal{G}$, with the Cayley group $G$ acting transitively (indeed
regularly) on its vertices, so every vertex of $\mathcal{G}$ can be written in
the form $F_{2\hskip 1.0pt}g$ for some $g\in G$. Also let
$m=|G\cap\Gamma(\mathcal{P})^{+}_{F_{2}}|$, and recall that from equations (3)
and (6) that the stabiliser of $F_{2}$ in $\Gamma(\mathcal{P})^{+}$ is given
by

$\Gamma(\mathcal{P})^{+}_{F_{2}}=\begin{cases}\langle\sigma_{1},\sigma_{2}\sigma_{3}\rangle&\text{if
$\mathcal{P}$ is directly-regular or chiral, or}\\\
\langle\,\rho_{0},\rho_{1},\rho_{3}\,\rangle&\text{if $\mathcal{P}$ is non-
orientably regular.}\end{cases}$ (8)

The first possibility $\langle\sigma_{1},\sigma_{2}\sigma_{3}\rangle$ is
isomorphic to the dihedral group $D_{p}$ of order $2p$, since $\sigma_{1}$ has
order $p$ and both $\sigma_{2}\sigma_{3}$ and $\sigma_{1}\sigma_{2}\sigma_{3}$
are involutions. For the second possibility, where $\mathcal{P}$ is non-
orientably regular, the subgroup of $\Gamma(\mathcal{P})$ generated by
$\rho_{0}$ and $\rho_{1}$ is dihedral of order $2p$, because the section
$F_{2}/F_{-1}$ is a regular $p$-gon; see [16, Proposition 2B9]. Then since
$\rho_{3}$ commutes with $\rho_{0}$ and $\rho_{1}$, it follows that
$\langle\,\rho_{0},\rho_{1},\rho_{3}\,\rangle\cong D_{p}\times C_{2}$, of
order $4p$.

Next, we expand on the relationship between $\mathcal{C}$ and $\mathcal{G}$ by
noting observations made in [18]. Clearly there exists a surjective graph
homomorphism $\nu\\!:\mathcal{C}\to\mathcal{G}$ taking $g$ to $F_{2\hskip
1.0pt}g$ for all $g\in G$, and this is injective on the neighbourhood of every
vertex of $\mathcal{C}$. Moreover, if $\mathcal{P}$ is chiral or directly-
regular, then $\mathcal{C}$ is a $s$-fold covering of $\mathcal{G}$, where
$s=|G\cap\Gamma(\mathcal{P})^{+}_{F_{2}}|$, while if $\mathcal{P}$ is non-
orientably regular, then $\mathcal{C}$ is a $2s$-fold covering of
$\mathcal{G}$. Vertices $g$ and $h$ of $\mathcal{C}$ are taken to the same
vertex in $\mathcal{G}$ under $\nu$ if and only if $gh^{-1}\in
G\cap\Gamma(\mathcal{P})_{F_{2}}$. Also $1\leq s\leq 4p$, since
$\Gamma(\mathcal{P})^{+}_{F_{2}}$ has order $2p$ when $\mathcal{P}$ is
directly-regular or chiral, and order $4p$ when $\mathcal{P}$ is non-
orientably regular.

Finally, we have the following important theorem from [17, Theorems 2 and 5],
which exhibits the relationship between the properties of $\mathcal{P}$ and
$\mathcal{G}$ when $p=3$ (and $\mathcal{G}$ is cubic):

######  Theorem 4.2.

Let $\mathcal{P}$ be a finite self-dual regular or chiral polytope, with type
$\\{3,q,3\\}$. Then $\mathrm{Aut}(\mathcal{G})$ is isomorphic to the extended
group $\overline{\Gamma}(\mathcal{P})$ of $\mathcal{P}$, and the medial layer
graph $\mathcal{G}$ is $3$-arc-regular when $\mathcal{P}$ is regular, and
$2$-arc-regular when $\mathcal{P}$ is chiral.

##  5 Improperly self-dual chiral polytopes of type $\\{3,q,3\\}$

One of the final comments made by Monson and Weiss in [18] raised the question
of existence of improperly self-dual chiral polytopes of type $\\{p,q,p\\}$
(when $q$ is even).

We answer this question positively by giving two examples, namely two
improperly self-dual chiral polytopes of types $\\{3,6,3\\}$ and
$\\{3,18,3\\}$. We constructed these examples (with the help of Magma) by
finding quotients of the rotation subgroup of the string Coxeter group
$[3,q,3]$ for different fixed values of $q$, using the first method described
in subsection 3.4, and using Theorem 3.2 to check the resulting chiral
polytopes to see if any were improperly self-dual.

The automorphism group of the first example is a quotient $A$ of the rotation
subgroup of the Coxeter group $[3,6,3]$ with order $|A|=18522$, obtainable by
adding the relations

$(\sigma_{1}^{-1}\sigma_{2}^{\,2})^{4}\sigma_{1}\sigma_{2}^{-2}=(\sigma_{2}^{\,2}\sigma_{3}^{-1})^{3}\sigma_{2}^{\,2}\sigma_{3}\sigma_{2}^{-2}\sigma_{3}^{-1}=(\sigma_{2}^{-1}\sigma_{3}\sigma_{1}\sigma_{2}^{-1}\sigma_{3}\sigma_{2}\sigma_{1})^{2}=1.$

This polytope has $147$ vertices, $3087$ edges, $3087$ 2-faces, and $147$
3-faces, and its medial layer graph is a 2-arc-regular bipartite cubic graph
of order $6174$, with automorphism group of order $2|A|=37044$. As pointed out
by the referee, the 147 facets are isomorphic to the chiral toroidal map of
type $\\{3,6\\}_{(4,1)}$, and dually, the vertex-figures are isomorphic to the
one of type $\\{6,3\\}_{(4,1)}$.

The automorphism group of the second example is a quotient $B$ of the rotation
subgroup of the Coxeter group $[3,18,3]$ with order $|B|=39366=2\cdot 3^{9}$,
obtainable by adding the relations

$\sigma_{2}^{\,5}\sigma_{1}\sigma_{2}^{-2}\sigma_{1}\sigma_{2}^{-1}\sigma_{1}\sigma_{2}^{-4}\sigma_{1}=(\sigma_{2}^{-1}\sigma_{3}\sigma_{1}\sigma_{2}^{-1}\sigma_{3}^{-1}\sigma_{1}\sigma_{2}^{-1}\sigma_{1})^{2}$
$=\sigma_{2}^{\,2}\sigma_{1}^{-1}\sigma_{3}\sigma_{2}\sigma_{1}\sigma_{2}^{-1}\sigma_{3}\sigma_{2}^{\,2}\sigma_{1}^{-1}\sigma_{3}\sigma_{2}\sigma_{1}^{-1}\sigma_{2}\sigma_{1}^{-1}=1.$

This polytope has $81$ vertices, $6561$ edges, $6561$ 2-faces, and $81$
3-faces, and its medial layer graph is a 2-arc-regular bipartite cubic graph
of order $13122$, with automorphism group of order $2|B|=78732$. Also we note
(following a suggestion by the referee) that the group $B$ has a non-central
normal subgroup $K$ of order $9$ such that the quotient $B/K$ (of order
$2\cdot 3^{7}$) is the rotation group of a directly regular polytope of type
$\\{3,18,3\\}$, covered by the given chiral polytope (with abelian covering
group isomorphic to $C_{3}\times C_{3}$).

Recall that self-dual regular polytopes and properly self-dual chiral
polytopes admit a polarity that reverses the base flag, while all dualities of
improperly self-dual chiral polytopes send the base flag to the reverse of a
flag in the other flag orbit, and hence improperly self-dual chiral polytopes
do not admit such a polarity. They can still admit other kinds of polarity,
and indeed it was mentioned by Schulte and Weiss in [21] that they did not
know at the time of any examples of highly symmetric self-dual polytopes that
do not admit a polarity.

The two examples above do not admit any polarities at all. (The first one
admits $18522$ dualities, of which $6174$ have order $4$ and $12348$ have
order $12$, while the second admits $39366$ dualities, of which $4374$ have
order $4$, and $8748$ have order $12$, and $26244$ have order $36$.) Among the
very few other such examples known are the two improperly self-dual chiral
4-polytopes with types $\\{4,4,4\\}$ and $\\{9,6,9\\}$ given in the first
author’s list of chiral polytopes with few flags [3], and the universal
polytope of type $\\{\\{4,4\\}_{(1,3)},\\{4,4\\}_{(1,3)}\\}$ mentioned by
Hubard and Weiss in [15].

##  6 Arc transitivity of $\mathcal{G}$ and $\mathcal{C}$

Monson and Weiss asked in [18] if the Cayley graph $\mathcal{C}$ they
associated with a self-dual regular or chiral polytope of type $\\{p,q,p\\}$
could have a higher degree of $s$-arc-transitivity than the medial layer graph
$\mathcal{G}$ that it covers – or in other words, if $\mathcal{G}$ is $t$-arc-
transitive and $\mathcal{C}$ is $s$-arc-transitive, could $s$ be greater than
$t$? They were particularly interested in the case $p=3$, where Theorem 2.1
implies that $s\leq 5$, and Theorem 4.2 implies that $t=2$ or $3$. They
suspected that $s$ cannot be 4 or 5, and we will prove it.

Suppose that $\mathcal{G}$ is $t$-arc-transitive and $\mathcal{C}$ is $s$-arc-
transitive. We will show that if the polytope is chiral, then $s=t=2$, and if
the polytope is regular, then $2\leq s\leq 3=t$. To do this, we construct a
subgroup of $\mathrm{Aut}(\mathcal{C})$ that acts 2-arc-regularly on
$\mathcal{C}$, and show that this has index at most $2$ in
$\mathrm{Aut}(\mathcal{C})$, and that if this index is $2$ then $\mathcal{P}$
is regular.

Recall from Section 4 that the Cayley group $G$ of $\mathcal{C}$ is generated
by $\\{\delta_{1},\delta_{2},\delta_{3}\\}$, and is normal in
$\overline{\Gamma}(\mathcal{P})^{+}=\langle\sigma_{1},\sigma_{2},\sigma_{3},\delta\rangle$.
Again with a slight abuse of notation, let $h$ and $p$ be the group
homomorphisms of $G$ defined by conjugation by $\sigma_{1}$ and
$\sigma_{1}\sigma_{2}\sigma_{3}$, respectively, and let $a$ be the involutory
automorphism of $\mathcal{C}$ defined by right multiplication by $\delta_{1}$
($=\delta$).

By Lemma 4.1 we see that $h$ induces a $3$-cycle on the Cayley set
$\\{\delta_{1},\delta_{2},\delta_{3}\\}$ for $\mathcal{C}$, while $p$ induces
a $2$-cycle (fixing $\delta_{1}$), so $h$ and $p$ induce graph automorphisms
of $\mathcal{C}$, fixing the vertex $1_{G}$. Moreover, as
$\sigma_{1}\sigma_{2}\sigma_{3}$ commutes with $\delta_{1}$ and conjugates
$\sigma_{1}$ to $\sigma_{1}^{-1}$, we find that $p^{a}=p$ and $h^{p}=h^{-1}$.
It follows that $h$, $p$ and $a$ satisfy the defining relations for
$G_{2}^{\,1}$ from [9] that were given after Proposition 2.2, and so they
generate a subgroup $T$ of $\mathrm{Aut}(\mathcal{C})$ that acts 2-arc-
regularly on $\mathcal{C}$, with vertex-stabiliser $\langle h,p\rangle\cong
S_{3}$.

Moreover, by Proposition 2.2 this implies that $\mathrm{Aut}(\mathcal{C})$
does not contain a $4$\- or $5$-arc-regular subgroup, and so $\mathcal{C}$
cannot be $4$\- or $5$-arc-regular, and hence is $2$\- or $3$-arc-regular. In
particular, the index of $T$ in $\mathrm{Aut}(\mathcal{C})$ is at most $2$.

Now if $\mathcal{P}$ is regular, then we know that $\mathcal{G}$ is $3$-arc-
regular, and therefore $\mathcal{C}$ cannot have a higher degree of $s$-arc-
regularity than $\mathcal{G}$.

On the other hand, we can show that if $\mathcal{P}$ is chiral (which implies
by Theorem 4.2 that $\mathcal{G}$ is 2-arc-regular), then $\mathcal{C}$ cannot
be $3$-arc-regular. For if we assume the contrary, then by Proposition 2.3,
there exists a group automorphism $\theta$ of $\langle h,p,a\rangle=T$ that
fixes $h$ and $p$, and takes $a$ to $ap$. Accordingly, $\theta$ induces an
automorphism $\psi$ of
$\langle\sigma_{1},\sigma_{1}\sigma_{2}\sigma_{3},\delta\rangle$ that fixes
$\sigma_{1}$ and $\sigma_{1}\sigma_{2}\sigma_{3}$, and takes $\delta$ to
$\delta\sigma_{1}\sigma_{2}\sigma_{3}$. From this and the fact that
$\delta\sigma_{3}\delta=\sigma_{1}^{-1}$ we find that

$\delta_{2}^{\,\psi}=(\sigma_{1}^{-1}\delta\sigma_{1})^{\psi}=\sigma_{1}^{-1}(\delta\sigma_{1}\sigma_{2}\sigma_{3})\sigma_{1}=\sigma_{1}^{-1}\delta(\sigma_{1}\sigma_{2}\sigma_{3})\sigma_{1}=\delta\sigma_{3}(\sigma_{3}^{-1}\sigma_{2}^{-1}\sigma_{1}^{-1})\sigma_{1}=\delta\sigma_{2}^{-1},$

and then since
$\sigma_{3}=\delta\sigma_{1}^{-1}\delta=\delta\sigma_{1}^{-1}\delta\sigma_{1}\sigma_{1}^{-1}=\delta\delta_{2}\sigma_{1}^{-1}$,
it follows that

$\sigma_{3}^{\,\psi}=(\delta\delta_{2}\sigma_{1}^{-1})^{\psi}=(\delta\sigma_{1}\sigma_{2}\sigma_{3})(\delta\sigma_{2}^{-1})(\sigma_{1}^{-1})=(\sigma_{3}^{-1}\sigma_{2}^{-1}\sigma_{1}^{-1})\sigma_{2}^{-1}\sigma_{1}^{-1}=\sigma_{3}^{-1}(\sigma_{1}\sigma_{2})^{-2}=\sigma_{3}^{-1},$

and so the $\psi$-image of $\sigma_{2}$ is

$\sigma_{2}^{\,\psi}=(\sigma_{1}^{-1}(\sigma_{1}\sigma_{2}\sigma_{3})\sigma_{3}^{-1})^{\psi}=\sigma_{1}^{-1}\sigma_{1}\sigma_{2}\sigma_{3}\,\sigma_{3}=\sigma_{2}\sigma_{3}^{\,2}.$

In particular, $\psi$ induces the same automorphism of
$\Gamma(\mathcal{P})^{+}=\langle\sigma_{1},\sigma_{2},\sigma_{3}\rangle$ as
would conjugation by the reflection $\rho_{3}$ in the case where $\mathcal{P}$
is regular, namely taking $(\sigma_{1},\sigma_{2},\sigma_{3})$ to
$(\sigma_{1},\sigma_{2}\sigma_{3}^{\,2},\sigma_{3}^{-1})$. But the rotation
subgroup of a chiral polytope admits no such automorphism, which is a
contradiction. It follows that $\mathcal{C}$ cannot be $3$-arc-regular, as
claimed.

Thus we have the following:

######  Theorem 6.1.

Let $\mathcal{P}$ be a properly self-dual finite regular or chiral polytope
with type $\\{3,q,3\\}$ for some $q$. If $\mathcal{G}$ is $t$-arc-regular and
$\mathcal{C}$ is $s$-arc-regular, then $s\leq t$.

We complete this section by showing that both cases occur, by giving an
example of a self-dual regular $4$-polytope $\mathcal{P}$ for which
$\mathcal{C}$ is 2-arc-regular, and another for which $\mathcal{C}$ is 3-arc-
regular.

The regular 4-simplex (or 5-cell) is a directly-regular polytope of type
$\\{3,3,3\\}$ that has ten triangular 2-faces and five tetrahedral 3-faces,
and automorphism group $S_{5}$ of order $120$. Its medial layer graph
$\mathcal{G}$ is isomorphic to the 3-arc-regular Desargues graph F020B (of
order $20$) in the Foster census [2] and listed as C20.2 at [5], while the
associated Cayley graph $\mathcal{C}$ is isomorphic to the 2-arc-regular graph
F120B (of order $120$) in the Foster census [2] and listed as C120.2 at [5].

In contrast, the 24-cell (or octaplex) is a directly-regular polytope of type
$\\{3,4,3\\}$ that has 96 triangular 2-faces and 24 octahedral 3-faces, and
automorphism group of order $1152$. Its medial layer graph $\mathcal{G}$ and
the associated Cayley graph $\mathcal{C}$ are both isomorphic to the 3-arc-
regular graph F192A in the Foster census [2] and listed as C192.3 at [5].

Curiously, the medial layer graph $\mathcal{G}$ and Cayley graph $\mathcal{C}$
for the properly self-dual chiral polytope of type $\\{3,8,3\\}$ listed at [3]
are both isomorphic to another 2-arc-regular graph of order $192$, namely the
graph F192C in [2] and listed as C192.2 at [5].

##  7 Larger vertex-stabilisers for ${\mathcal{C}}$ than for ${\mathcal{G}}$

The theory of 3-valent arc-transitive graphs is well-developed (as may be seen
from its brief partial description in subsection 2.2, or in more detail in
[13] or [9], for example), helped by the fact that Tutte’s theorem (Theorem
2.1) bounds the order of the vertex-stabilisers in the 3-valent case. For
higher valencies it can be much more complicated, especially for non-prime
valencies such as $4$ (for which the vertex-stabilisers have unbounded order).
In particular, for such higher valencies not every arc-transitive finite graph
is $s$-arc-regular for some $s$.

Nevertheless a comparison of the automorphism groups of the medial layer graph
$\mathcal{G}$ and its covering Cayley graph $\mathcal{C}$ for a regular or
chiral $4$-polytope of type $\\{p,q,p\\}$ can be made in terms of the orders
of their vertex-stabilisers. In the case $p=3$ we know that this order is no
greater for $\mathcal{C}$ than for $\mathcal{G}$, by Theorem 6.1, because
those orders are $3\cdot 2^{s-1}$ and $3\cdot 2^{t-1}$ with $s\leq t$. But we
have an infinite family of examples with $p=6$ for which the analogous
property does not hold, constructible as follows.

Let $\Lambda$ be the group obtainable from any string Coxeter group $[6,q,6]$
by deleting the relation $(x_{2}x_{3})^{q}=1$, and adding the two extra
relations $((x_{1}x_{2})^{2}x_{3})^{2}=((x_{3}x_{4})^{2}x_{2})^{2}=1$.

In this group, the element $(x_{2}x_{3})^{3}$ generates a normal subgroup $K$
of index $216$, which by the Reidemeister-Schreier process (implemented as the
Rewrite command in Magma) is free of rank $1$ and hence is infinite cyclic.
Moreover, $\Lambda/K$ is a smooth quotient of $\Lambda$, satisfying the
intersection conditions (2), and in which the image of $x_{2}x_{3}$ has order
$3$. It follows that $\Lambda/K$ is the automorphism group of a regular
polytope with type $\\{6,3,6\\}$, which (as noted by the referee) is the
universal polytope $\\{\\{6,3\\}_{(1,1)},\\{3,6\\}_{(1,1)}\\}$ with $216$
flags given in [16, 11C14].

Now for any multiple of $3$, say $q=3k$, let $N_{q}$ be the subgroup of
$\Lambda$ generated by $(x_{2}x_{3})^{q}$. Then $N_{q}$ is characteristic in
$N$ and therefore normal in $\Lambda$, with smooth quotient $\Lambda/N_{q}$ of
order $216k=72q$. It is not difficult to prove that $\Lambda/N_{q}$ satisfies
the intersection conditions (2), and hence is the automorphism group of a
directly-regular polytope $\mathcal{P}_{q}$ of type $\\{6,q,6\\}$. Also it is
clear from the quotient group relations that $\mathcal{P}_{q}$ is self-dual.
In fact for $k=1$ to $9$, this is the self-dual regular $4$-polytope of type
$\\{6,3q,6\\}$ with automorphism group of order $72q$ listed at [4].

Moreover, it can be shown that if $q$ is odd, then:
(a) the medial layer graph $\mathcal{G}$ of $\mathcal{P}_{q}$ has order $6q$,
(b) the automorphism group of $\mathcal{G}$ has order order $2^{2q+2}3^{2q}q$,
(c) the associated Cayley graph $\mathcal{C}$ has order $36q$, and
(d) the automorphism group of $\mathcal{C}$ has order order $2^{4q+3}3^{4q}q$.
The proofs of these facts are straightforward but time-consuming, and so we
omit them. Details are given in Chapter 6 of the second author’s Masters
thesis [22], in which it is shown that in fact $\mathcal{G}$ is isomorphic to
the graph $C(3,2q,1)$ and $\mathcal{C}$ is isomorphic to the graph
$C(3,4q,2)$, in a family of vertex-transitive graphs $C(p,r,s)$ with large
vertex-stabilisers studied by Praeger and Xu in [20].

In particular, the stabiliser in ${\rm Aut}(\mathcal{G})$ of a vertex of
$\mathcal{G}$ has order $2^{2q+1}3^{2q-1}$, while the stabiliser in ${\rm
Aut}(\mathcal{C})$ of a vertex of $\mathcal{C}$ has order $2^{4q+1}3^{4q-2}$,
which is larger than $2^{2q+1}3^{2q-1}$ for all $q$, and very much larger when
$q$ is large.

We also found a similar family of nonorientably-regular polytopes of type
$\\{4,q,4\\}$ with $q=6t$, for which the vertex-stabilisers for the Cayley
graph ${\mathcal{C}}$ are larger than those for the medial layer graph
${\mathcal{G}}$. For each of these, the automorphism group of the polytope is
a smooth quotient of the finitely-presented group obtained from the $[4,6t,4]$
Coxeter group with the following extra relations:
$[x_{1},x_{2}x_{3}x_{2}x_{3}x_{2}]=[x_{4},x_{3}x_{2}x_{3}x_{2}x_{3}]=(x_{2}x_{1}x_{2}x_{3}x_{4}x_{3})^{2}$
$=x_{1}x_{2}x_{3}(x_{1}x_{2})^{2}x_{4}x_{3}x_{2}(x_{3}x_{4})^{2}=(x_{1}x_{2}x_{3})^{3}(x_{2}x_{3})^{3t-3}=1$.
Here in each case the graphs ${\mathcal{G}}$ and ${\mathcal{C}}$ are
isomorphic to the graphs $C(2,q,2)$ and $C(2,2q,4)$ from the same family of
graphs $C(p,r,s)$ studied in [20].

##  8 Final remarks

We complete this paper by mentioning a few open questions that naturally
arise.

First, for which values of $q$ do there exist improperly self-dual chiral
polytopes of type $\\{3,q,3\\}$? We found examples with $q=6$ and $q=18$, but
we see no reason why further examples could not exist. Similarly, it would be
interesting to find examples of improperly self-dual chiral polytopes of type
$\\{3,q,3\\}$ that admit polarities. This is guaranteed when $q$ is odd, but
could they exist also when $q$ is even? We know of no such examples.

The referee suggested also looking for improperly self-dual chiral polytopes
of type $\\{p,q,p\\}$ for prime $p>3$.

Next, if $\mathcal{P}$ is a regular polytope of type $\\{3,q,3\\}$ for which
the medial layer graph $\mathcal{G}$ is 3-arc-regular, then the associated
Cayley graph can be either 2-arc-regular or 3-arc-regular. Are there necessary
and sufficient conditions on $\mathcal{P}$ for each of these two
possibilities?

Finally, a natural question to ask is whether Theorem 6.1 holds also for
regular and/or chiral polytopes of type $\\{p,q,p\\}$ for larger values of
$p$. Again, we know no such examples.

Acknowledgements

The authors also acknowledge the helpful use of Magma [1] in conducting
experiments, testing conjectures and verifying many of the details of their
construction and proof of the new discoveries presented in this paper. Also
the first author is grateful to New Zealand’s Marsden Fund for its support
(via grant UOA 2030).

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