# Bergman-Einstein metrics on two-dimensional Stein spaces

Soumya Ganguly  Department of Mathematics, University of California San Diego,
La Jolla, CA, 92093, USA<EMAIL_ADDRESS>and  Shubham Sinha  Department of
Mathematics, University of California San Diego, La Jolla, CA, 92093, USA
<EMAIL_ADDRESS>
###### Abstract.

We show that the Bergman metric of the ball quotients $\mathbb{B}^{2}/\Gamma$,
where $\Gamma$ is a finite and fixed point free group, is Kähler-Einstein if
and only if $\Gamma$ is trivial. As a consequence, we characterize the unit
ball $\mathbb{B}^{2}$, among 2 dimensional Stein spaces with isolated normal
singularities, proving an algebraic version of Cheng’s conjecture for 2
dimensional Stein spaces.

##  1\. Introduction

The Bergman kernel and metric were introduced by S. Bergman [Ber33, Ber35] for
domains in $\mathbb{C}^{n}$ and these notions were generalized to complex
manifolds by S. Kobayashi in [Kob59]. The invariance of the Bergman metric
under biholomorphisms makes it a natural choice of metric in complex geometry
(see [Kra13] for details). Since their introduction, questions related to the
Bergman kernel and metric have unfurled new directions of research. In this
paper we investigate a classification problem of Stein spaces involving the
Bergman metric.

For any bounded strongly pseudoconvex domain there exists a natural complete
Kähler-Einstein metric, whose existence was proved by S.Y. Cheng and S.T. Yau
[CY80]. The Cheng-Yau metric can be constructed explicitly by solving a Monge-
Ampere type equation. This metric is also unique up to scaling by a constant.
Yau conjectured that the Cheng-Yau metric of a bounded pseudoconvex domain
coincides with its Bergman metric if and only if the domain is homogeneous
[Yau82]; recall that a domain in $\mathbb{C}^{n}$ is called homogeneous if its
automorphism group acts transitively on it. This is still an open conjecture
in complex geometry.

A homogeneous strongly pseudoconvex domain with a smooth boundary is
biholomorphic to the unit ball $\mathbb{B}^{n}$ [Won77]. Fu-Wong [FW97],
Nemirovski-Shafikov [NS06] and Huang-Xiao [HX21] proved an older conjecture of
Cheng [Che79] which states that Yau’s conjecture holds for strongly
psuedoconvex domains with smooth boundary. Huang-Li [HL20] then extended this
conjecture to Stein manifolds and proved the following: The only Stein
manifold with smooth and compact strongly pseudoconvex boundary for which the
Bergman metric is Kähler-Einstein is the unit ball $\mathbb{B}^{n}$ up to
biholomorphism. Furthermore, Huang-Xiao formulated the following conjecture
for Stein spaces with isolated normal singularities:

######  Conjecture 1 ([HX20]).

Let $\Omega$ be a normal Stein space with a compact, smooth, strongly
pseudoconvex boundary. Then the Bergman metric on the regular part of $\Omega$
is Kähler-Einstein if and only if $\Omega$ is biholomorphic to the unit ball
in a complex Euclidean space.

In this paper, we prove the following algebraic version of Cheng’s Conjecture
for normal Stein spaces in dimension two:

######  Theorem 1.

Let $\Omega$ be a two dimensional Stein space with isolated normal
singularities and a compact smooth boundary $\partial\Omega$. Assume that
$\partial\Omega$ is CR equivalent to an algebraic CR manifold in a complex
Euclidean space. The Bergman metric on the regular part of the Stein space
$\Omega$ is Kähler-Einstein if and only if $\Omega$ is biholomorphic to
$\mathbb{B}^{2}$.

An algebraic approximation theorem by Lempert from [Lem95] states that a
relatively compact domain in a reduced Stein space with only isolated
singularities is biholomorphic to a domain in an affine algebraic variety.
Thus the last theorem above gets reformulated as:

######  Theorem 2.

Let $G$ be a relatively compact domain in a 2-dimensional affine algebraic
variety $V$ such that $G$ has algebraic boundary. Assume that $V$ has at most
isolated normal singularities in $G$. Then the Bergman metric on the regular
part of $G$ is Kähler-Einstein if and only if $G$ is biholomorphic to
$\mathbb{B}^{2}$.

In Corollary 4.1 of [HL20], Huang and Li showed that it suffices to consider
$\Omega$ in Theorem 1 to be a ball quotient $\mathbb{B}^{n}/\Gamma$ where
$n\geq 2$, and $\Gamma\subset Aut(\mathbb{B}^{n})$ is finite and fixed point
free. Ebenfelt-Xiao-Xu in [EXX22] proved that such ball quotients
$\mathbb{B}^{n}/\Gamma$ do not admit a Bergman metric that is Kähler-Einsten
provided that $\Gamma$ is nontrivial and also abelian. In two dimensions we
prove the following theorem, which implies Theorems 1 and 2:

######  Theorem 3.

For any fixed point free, finite subgroup $\Gamma$ of $Aut(\mathbb{B}^{2})$,
the regular part of the ball quotient $\mathbb{B}^{2}/\Gamma$ has a Bergman
metric that is Kähler-Einstein if and only if $\Gamma=\\{id\\}$.

We emphasize that there is a contrasting result in one dimension. To be
precise, the Bergman metric on the regular part of any finite disk quotient
$\mathbb{B}^{1}/\Gamma$ is Kähler-Einstein [HL20].

The remaining sections of the paper are organized as follows: We provide some
known results and background material in Section 2. In Section 3, we
specialize to dimension 2 and obtain a necessary condition for the Bergman
metric on ball quotients to be Kähler-Einstein. In Section 4, we list all the
fixed point free, finite groups that we need to consider and find
representations of these groups. In the subsequent sections we explicitly show
that the necessary condition is not satisfied for the nontrivial groups in the
list, thus proving Theorem 3.

### Acknowledgement

The first author would like to thank Prof. Peter Ebenfelt, Prof. Ming Xiao,
Prof. Hang Xu, for giving the motivation and all the necessary guidance to
develop this project. The same author is also grateful to N. Ramachandran and
A. Patil for many enlightening conversations. The second author would like to
thank Prof. Dragos Oprea for the interest shown in this work. The first author
was partially supported by the NSF through the grants DMS-1800549, DMS-1900955
and DMS-2154368. The second author was supported by the NSF through grant DMS
1802228.

##  2\. Preliminaries

In this section we briefly discuss an ‘effective’ version of the Kähler-
Einstein equation for finite ball quotients. For details, we refer the reader
to [EXX22] (sections 2.2 and 4.1).

A Kähler-Einstein metric is locally the complex Hessian of a potential
function that obeys Einstein’s equation, i.e. the Ricci curvature is a
constant multiple of the metric tensor. Let $(M,g)$ be a Kähler manifold where
$g$ is induced locally by the Kähler form $\partial\bar{\partial}\log(\phi)$
such that $\phi>0$ is smooth. Then the Kähler-Einstein equation for $(M,g)$
is:

(2.1) $\displaystyle-\partial\bar{\partial}\log(\Phi)=c\
\partial\bar{\partial}\log(\phi),$

where $\Phi=det(g_{i\bar{j}})$ with
$g_{i\bar{j}}=\partial_{z_{i}}\partial_{\overline{z_{j}}}\log\phi(z,\bar{z})$.
It is well known that the constant $c$ has to be negative.

Recall that Bergman Kernel form on a manifold (or the regular part of a Stein
space) $M$ is an $(n,n)$ form and hence on a local chart the Bergman kernel
can be written as:

$\displaystyle K_{M}(z,\bar{w})=k_{M}(z,\bar{w})\ dz_{1}\wedge\cdots\wedge
dz_{n}\wedge d\bar{w}_{1}\wedge\cdots\wedge d\bar{w}_{n}.$

The Bergman metric is given by the Kähler form
$\omega=\partial\bar{\partial}\log k_{M}(z,\bar{z})$.

Let us fix $n\in\mathbb{N}$. The holomorphic automorphism group of the unit
ball, $Aut(\mathbb{B}^{n})$, is isomorphic to $PSU(n,1)$, the subgroup of the
automorphism group of $\mathbb{CP}^{n}$ with each element in the subgroup
fixing $\mathbb{B}^{n}\subset\mathbb{CP}^{n}$. $Aut(\mathbb{B}^{n})$ is
described explicitly in section 2.2 of [Rud08].

A subgroup $\Gamma$ of $Aut(\mathbb{B}^{n})$ is called fixed point free if any
non-identity element in $\Gamma$ has no fixed point on
$\partial\mathbb{B}^{n}$. By this fixed point free condition, the action of
$\Gamma$ on $\partial\mathbb{B}^{n}$ is properly discontinuous and hence
$\partial\mathbb{B}^{n}/\Gamma$ is a smooth manifold. We know from basic Lie
group theory that $U(n)$ is a maximal compact subgroup of
$Aut(\mathbb{B}^{n})$. This implies that any finite subgroup $\Gamma$ of
$Aut(\mathbb{B}^{n})$ equals $\psi^{-1}\cdot\tilde{\Gamma}\cdot\psi$ where
$\psi\in Aut(\mathbb{B}^{n})$ and $\tilde{\Gamma}$ is a subgroup of $U(n)$.
Note that the ball quotients $\mathbb{B}^{n}/\Gamma$ and
$\mathbb{B}^{n}/\tilde{\Gamma}$ are biholomorphic. Since the Bergman metric is
invariant under biholomorphisms, we may assume $\Gamma\subset U(n)$.

Let us now fix a finite, fixed point free, unitary group $\Gamma$ and let
$\Pi:\mathbb{B}^{n}\to\mathbb{B}^{n}/\Gamma$ be the quotient map. Note that a
function (form) $f_{\Gamma}$ on $\mathbb{B}^{n}/\Gamma$ is equivalent to a
function (form) $f$ on $\mathbb{B}^{n}$ invariant under $\Gamma$ i.e.
$f\circ\gamma=\gamma$ for all $\gamma\in\Gamma$. So we can express the Bergman
kernel form (hence the Bergman metric) on $\mathbb{B}^{n}/\Gamma$ in the
coordinates of $\mathbb{B}^{n}$ as a $\Gamma$ invariant form (or metric).
Also, the quotient map $\Pi$ is a holomorphic branched covering map and so is
a local biholomorphism between $\mathbb{B}^{n}\setminus\\{0\\}$ and the
regular part of $\mathbb{B}^{n}/\Gamma$. Being Kähler-Einstein is a local
property of a metric making it invariant under local biholomophisms. So the
Bergman metric on $\mathbb{B}^{n}/\Gamma$ is Kähler-Einstein if and only if
that metric, expressed in coordinates of $\mathbb{B}^{n}$ (i.e. the pullback
of that metric under $\Pi$) is Kähler-Einstein.

In [EXX22], the Bergman kernel form on $\mathbb{B}^{n}/\Gamma$ is explicitly
expressed in the coordinates of $\mathbb{B}^{n}$ as:

$\displaystyle
K_{\mathbb{B}^{n}/\Gamma}(z,\bar{w})=\frac{n!}{\pi^{n}}\phi(z,\bar{w})\
dz_{1}\wedge\cdots\wedge dz_{n}\wedge d\bar{w}_{1}\wedge\cdots\wedge
d\bar{w}_{n},$

where $\phi$ is the $\Gamma$ invariant function given by

$\displaystyle\phi(z,\bar{w}):=\sum_{\gamma\in\Gamma}\frac{\overline{\det{\gamma}}}{(1-<z,\overline{\gamma
w}>)^{n+1}}.$

where $<u,v>=\sum_{i=1}^{n}u_{i}v_{i}$ for any $u,v\in\mathbb{C}^{n}$. The
Bergman metric on $\mathbb{B}^{n}/\Gamma$ pulled back to $\mathbb{B}^{n}$ is
given locally by
$\partial_{z_{i}}\partial_{\overline{z_{j}}}\log\phi(z,\bar{z})$.

From the preceding discussion we know that Bergman metric of our realization
of ball quotient $\mathbb{B}^{n}/\Gamma$ is Kähler-Einstein if and only if the
function $\phi$ obeys (2.1). In [EXX22], this criterion has been further
simplified to:

(2.2) $\displaystyle\Phi(z,\bar{z})=(n+1)^{n}\phi(z,\bar{z}).$

The well known identity
$\Phi(z,\bar{z})=J(\phi(z,\bar{z}))/\phi(z,\bar{z})^{n+1}$ simplifies (2.2)
into:

$\displaystyle J(\phi(z,\bar{z}))=(n+1)^{n}\phi(z,\bar{z})^{n+2},$

where $J$ is the Monge-Ampere type operator (which differs from the operator
$J$ introduced in [Fef76b, Fef76a] by a sign) given by

$\displaystyle J(\psi):=\det\begin{pmatrix}\psi&\psi_{\overline{z_{j}}}\\\
\psi_{z_{i}}&\psi_{z_{i}\overline{z_{j}}}\end{pmatrix}.$

Now, by explicit calculations in [EXX22] the authors have shown

$J(\phi)(z,\bar{z})=(n+1)^{n}\sum_{\gamma^{0},\gamma^{1},...,\gamma^{n}\in\Gamma}\det{A(\overline{\gamma^{0}},\overline{\gamma^{1}},...,\overline{\gamma^{n}})}\prod_{i=0}^{n}\frac{\overline{\det{\gamma^{i}}}}{(1-<z,\overline{\gamma^{i}z}>)^{n+2}},$

where $A(\gamma^{0},\gamma^{1},...,\gamma^{n})=(\xi_{0}(\gamma^{0})\ \
\xi_{1}(\gamma^{1})\ \cdots\ \xi_{n}(\gamma^{n}))$ is an $(n+1)\times(n+1)$
matrix with column vectors

$\displaystyle\xi_{0}(\gamma)(z,\bar{z})=\begin{pmatrix}1-<z,\gamma\bar{z}>\\\
(n+1)\gamma\bar{z}\end{pmatrix}\hskip 28.45274pt\text{and}\hskip
28.45274pt\xi_{j}(\gamma)(z,\bar{z})=\begin{pmatrix}z^{T}(\gamma)_{j}\\\
\frac{(\gamma)_{j}(1-<z,\gamma\bar{z}>)+(n+2)\gamma\bar{z}(z^{T}(\gamma)_{j})}{1-<z,\gamma\bar{z}>}\end{pmatrix},$

for $j\in\\{1,2,...n\\}$. Here, $(\gamma)_{j}$ is the j-th column vector of a
matrix $\gamma\in\Gamma$. Combining all the equations here we get a working
version of Kähler-Einstein equation for the ball quotient
$\mathbb{B}^{n}/\Gamma$:

(2.3)
$\displaystyle\sum_{\gamma^{0},\gamma^{1},...,\gamma^{n}\in\Gamma}\det{A(\overline{\gamma^{0}},\overline{\gamma^{1}},...,\overline{\gamma^{n}})}\prod_{i=0}^{n}\frac{\overline{\det{\gamma^{i}}}}{(1-<z,\overline{\gamma^{i}z}>)^{n+2}}=\left(\sum_{\gamma\in\Gamma}\frac{\overline{\det{\gamma}}}{(1-<z,\overline{\gamma
z}>)^{n+1}}\right)^{n+2}$

where $<z,\bar{z}>=|z|^{2}<1$.

##  3\. Kähler-Einstein equation in dimension 2

Let $\Gamma$ be a fixed point free subgroup of $U(2)$. In this section, we
obtain a condition on $\Gamma$ necessary for Bergman metric on
$\mathbb{B}^{2}/\Gamma$ to be Kähler-Einstein. In particular, we define an
expression $C(\Gamma)$ in Proposition 3.1 that must vanish.

######  Lemma 3.1.

$\Gamma$ be a subgroup of $U(n),n\geq 2$. If $\Gamma$ is fixed point free then
the only $\gamma\in\Gamma$ with $\gamma_{11}=1$ is identity.

###### Proof.

From the definition of ‘fixed point free’ above, one can prove that a unitary
group $\Gamma$ is ‘fixed point free’ if and only if any of the following
equivalent conditions hold: (i) any non-identity element in $\Gamma$ has no
eigenvalue equal to 1; (ii) $0\in\mathbb{B}^{n}$ is the only fixed point for
any non-identity element in $\Gamma$.

Since $\Gamma$ is a fixed point free subgroup of $U(n)$, if $\gamma_{11}=1$
for some $\gamma\in\Gamma$ then $\gamma_{1j}=0=\gamma_{j1}$ for
$j\in\\{2,3,...,n\\}$. In particular, 1 becomes an eigenvalue for $\gamma$. In
that case, the fixed point free condition on $\Gamma$ ensures that
$\gamma=id$. ∎

We specialize the Kähler-Einstein equation (2.3) in dimension $n=2$ and
substitute $z=(z_{1},0)$ and let $x=z_{1}\overline{z_{1}}$. The left hand side
of the equation (2.3) is given by

(3.1) $\displaystyle
f(x):=\sum_{\gamma^{0},\gamma^{1},\gamma^{2}\in\Gamma}\det{A(\overline{\gamma^{0}},\overline{\gamma^{1}},\overline{\gamma^{2}})}\prod_{i=0}^{2}\frac{\overline{\det{\gamma^{i}}}}{(1-\overline{\gamma^{i}_{11}}x)^{4}}$

and the right hand side of (2.3) equals $\phi(x)^{4}$ where

(3.2)
$\phi(x)=\sum_{\gamma\in{\Gamma}}\frac{\overline{\det{\gamma}}}{(1-<z,\overline{\gamma
z}>)^{3}}=\sum_{\gamma\in{\Gamma}}\frac{\overline{\det{\gamma}}}{(1-\overline{\gamma_{11}}x)^{3}}.$

We note that the determinant of the matrix
$A(\overline{\gamma^{0}},\overline{\gamma^{1}},\overline{\gamma^{2}})$ in
(3.1) is

(3.3)
$\det{A(\overline{\gamma^{0}},\overline{\gamma^{1}},\overline{\gamma^{2}})}=\det{\begin{pmatrix}1-\overline{\gamma^{0}_{11}}x&\overline{\gamma^{1}_{11}}&\overline{\gamma^{2}_{12}}\\\
3\overline{\gamma^{0}_{11}}x&\frac{\overline{\gamma^{1}_{11}}(1-\overline{\gamma^{1}_{11}}x)+4(\overline{\gamma^{1}_{11}})^{2}x}{1-\overline{\gamma^{1}_{11}}x}&\frac{\overline{\gamma^{2}_{12}}(1-\overline{\gamma^{2}_{11}}x)+4\overline{\gamma^{2}_{11}}\overline{\gamma^{2}_{12}}x}{1-\overline{\gamma^{2}_{11}}x}\\\
3\overline{\gamma^{0}_{21}}x&\frac{\overline{\gamma^{1}_{21}}(1-\overline{\gamma^{1}_{11}}x)+4\overline{\gamma^{1}_{11}}\overline{\gamma^{1}_{21}}x}{1-\overline{\gamma^{1}_{11}}x}&\frac{\overline{\gamma^{2}_{22}}(1-\overline{\gamma^{2}_{11}}x)+4\overline{\gamma^{2}_{21}}\overline{\gamma^{2}_{12}}x}{1-\overline{\gamma^{2}_{11}}x}\end{pmatrix}}.$

######  Proposition 3.1.

Let $\Gamma$ be a fixed point free subgroup $U(2)$. If the Bergman metric on
the ball quotient $\mathbb{B}^{2}/\Gamma$ is Kähler-Einstein, then

$\displaystyle
C(\Gamma):=\sum_{\gamma\in{\Gamma\backslash\\{id\\}}}\frac{\det{\gamma}}{(1-\gamma_{11})^{4}}\bigg{(}1-3\operatorname{\mathrm{tr}}{\gamma}+\frac{4(\operatorname{\mathrm{tr}}{\gamma}-\det{\gamma}-1)}{(1-\gamma_{11})}\bigg{)}=0.$

###### Proof.

The Kähler-Einstein equation implies that $f(x)-\phi^{4}(x)=0$ for $0\leq
x<1$. Note that $f(x)-\phi^{4}(x)$ is a rational function in $x$, hence it
must equal zero as a rational function. Moreover, we may extend it to a
meromorphic function, and consider the Laurent series expansion at $x=1$. We
show that $f(x)-\phi^{4}(x)$ has a pole at 1 of order at most 8 with
$C(\Gamma)$ being the coefficient of $1/(x-1)^{8}$.

First, we find the principal part for Laurent expansion of the function
$\phi^{4}(x)$ around 1. For notational convenience, we work with conjugates of
$\phi^{4}(x)$ and $f(x)$. Using (3.2) and Lemma 3.1 we get:

(3.4)
$\overline{\phi^{4}(x)}=\left(\frac{1}{(1-x)^{3}}+\sum_{\gamma\in{\Gamma},\gamma\neq
id}\frac{\det{\gamma}}{(1-\gamma_{11}x)^{3}}\right)^{4}.$

Let us denote

$s(x):=\sum_{\gamma\in{\Gamma},\gamma\neq
id}\frac{\det{\gamma}}{(1-\gamma_{11}x)^{3}}.$

Using binomial expansion of (3.4) and Maclaurin expansion of $s(x)$ at 1, we
obtain the first few terms of the principal part of $\overline{\phi^{4}(x)}$:

$\displaystyle\frac{1}{(x-1)^{12}}-\frac{4s(1)}{(x-1)^{9}}-\frac{4s^{\prime}(1)}{(x-1)^{8}}+\cdots$
$\displaystyle=$
$\displaystyle\frac{1}{(x-1)^{12}}-\frac{4}{(x-1)^{9}}\sum_{\gamma\neq
id}\frac{\det{\gamma}}{(1-\gamma_{11})^{3}}-\frac{12}{(x-1)^{8}}\sum_{\gamma\in{\Gamma},\gamma\neq
id}\frac{\gamma_{11}.\det{\gamma}}{(1-\gamma_{11})^{4}}+\cdots$

The calculation for the Laurent expansion requires careful case work. We do
the computation in the next Lemma. We obtain the required result by
subtracting the principal part for $\overline{\phi^{4}(x)}$ and
$\overline{f(x)}$. ∎

######  Lemma 3.2.

The principal part for the Laurent expansion of
$\overline{f(x)}-\overline{\phi^{4}(x)}$ at 1 equals

$\displaystyle\frac{C(\Gamma)}{(x-1)^{8}}+\cdots$

###### Proof.

Recall the definition

$\displaystyle\overline{f(x)}=\sum_{\gamma^{0},\gamma^{1},\gamma^{2}\in\Gamma}\det{A(\gamma^{0},\gamma^{1},\gamma^{2})}\prod_{i=0}^{2}\frac{\det{\gamma^{i}}}{(1-\gamma^{i}_{11}x)^{4}},$

We are interested here to only find the coefficients of $1/(x-1)^{\ell}$ where
$\ell\geq 8$. Observe from (3.3) that
$\det(A(\overline{\gamma^{0}},\overline{\gamma^{1}},\overline{\gamma^{2}}))$
has a pole at 1 of order at most 2. Thus we only need to consider the cases
where at least two among the three matrices $\gamma^{0},\gamma^{1},\gamma^{2}$
are identity.

  * •

$\gamma^{0}=\gamma^{1}=\gamma^{2}=id$: In this case,
$\det(A(\gamma^{0},\gamma^{1},\gamma^{2}))=1$. So the contribution in $f(x)$
is $1/(x-1)^{12}$.

  * •

$\gamma^{0}=\gamma^{1}=id$, $\gamma^{2}\neq id$: In this case,

$\displaystyle\det(A(\gamma^{0},\gamma^{1},\gamma^{2}))=\gamma^{2}_{22}+\frac{4\gamma^{2}_{21}\gamma^{2}_{12}x}{1-\gamma^{2}_{11}x}.$

So the contribution in the principal part of the Laurent expansion up to 8th
order is:

$\displaystyle\sum_{\gamma\in\Gamma\setminus\\{id\\}}\frac{\det{\gamma}}{(1-x)^{8}(1-\gamma_{11}x)^{4}}\bigg{(}\gamma_{22}+\frac{4\gamma_{21}\gamma_{12}x}{1-\gamma_{11}x}\bigg{)}$
$\displaystyle=\frac{1}{(x-1)^{8}}\sum_{\gamma\in\Gamma\setminus
id}\frac{\det{\gamma}}{(1-\gamma_{11})^{4}}\bigg{(}\gamma_{22}+\frac{4\gamma_{21}\gamma_{12}}{1-\gamma_{11}}\bigg{)}+\cdots$

  * •

$\gamma^{0}=\gamma^{2}=id$, $\gamma^{1}\neq id$: In this case,

$\displaystyle\det(A(\gamma^{0},\gamma^{1},\gamma^{2}))=(1-x)\bigg{(}\gamma^{1}_{11}+\frac{4(\gamma^{1}_{11})^{2}x}{1-\gamma^{1}_{11}}\bigg{)}-3x\gamma^{1}_{11}$

So the contribution in the principal part of the Laurent expansion up to 8th
order is:

$\displaystyle\sum_{\gamma\in\Gamma\setminus\\{id\\}}\frac{\det{\gamma}}{(1-x)^{8}(1-\gamma_{11}x)^{4}}\bigg{(}(1-x)(\gamma_{11}+\frac{4(\gamma_{11})^{2}x}{1-\gamma_{11}})-3x\gamma_{11}\bigg{)}$
$\displaystyle=\frac{1}{(x-1)^{8}}\sum_{\gamma\in\Gamma\setminus
id}\frac{-3\gamma_{11}\det{\gamma}}{(1-\gamma_{11})^{4}}+\cdots$

  * •

$\gamma^{2}=\gamma^{1}=id$, $\gamma^{0}\neq id$: In this case,

$\displaystyle\det(A(\gamma^{0},\gamma^{1},\gamma^{2}))=\frac{1+(3-4\gamma^{0}_{11})x}{1-x}$

So the contribution in the principal part of the Laurent expansion up to 8th
order is:

$\displaystyle\sum_{\gamma\in\Gamma\setminus\\{id\\}}\frac{(1+(3-4\gamma_{11})x)\det{\gamma}}{(1-x)^{9}(1-\gamma_{11}x)^{4}}$
$\displaystyle=\frac{1}{(x-1)^{9}}\sum_{\gamma\in\Gamma\setminus
id}\frac{-4\det{\gamma}}{(1-\gamma_{11})^{3}}-\frac{1}{(x-1)^{8}}\sum_{\gamma\in\Gamma\setminus
id}\frac{\det{\gamma}(3+12\gamma_{11})}{(1-\gamma_{11})^{4}}+\cdots$

After combining all these contributions together and using the expansion of
$\overline{\phi^{4}(x)}$, we observe that the coefficient of $(x-1)^{-12}$ and
$(x-1)^{-9}$ in $\overline{f(x)}-\overline{\phi^{4}(x)}$ equals zero and the
coefficient of $(x-1)^{-8}$ equals

$\displaystyle\sum_{\gamma\in{\Gamma\backslash\\{id\\}}}\frac{\det\gamma}{(1-\gamma)^{4}}\bigg{(}\gamma_{22}-3\gamma_{11}-3+\frac{4\gamma_{12}\gamma_{21}}{1-\gamma_{11}}\bigg{)}=C(\Gamma).$

∎

##  4\. Milnor’s Theorem and Group Representations

The following theorem by J. Milnor [Mil57] gives us all finite, fixed point
free groups having a unitary action on $\partial\mathbb{B}^{2}$. The
classification of such finite, fixed point free, unitary groups in higher
dimensions can be found in [Wol11] in the context of finding spaces of
constant curvature.

######  Theorem ([Mil57]).

The following is a list of all finite groups which act orthogonally on
$\mathbb{S}^{3}$ without fixed points:

  1. (1)

The group 1;

  2. (2)

$Q_{8n}=<X,Y:X^{2}=(XY)^{2}=Y^{2n}>,n\in\mathbb{N}$;

  3. (3)

$P_{48}=<X,Y:X^{2}=(XY)^{3}=Y^{4},X^{4}=id>$;

  4. (4)

$P_{120}=<X,Y:X^{2}=(XY)^{3}=Y^{5},X^{4}=id>$;

  5. (5)

$D_{2^{m}(2n+1)}=<X,Y:X^{2^{m}}=id,Y^{2n+1}=id,XYX^{-1}=Y^{-1}>$ where $m\geq
2,n\geq 1$;

  6. (6)

$P^{\prime}_{8.3^{m}}=<X,Y,Z:X^{2}=(XY)^{2}=Y^{2},ZXZ^{-1}=Y,ZYZ^{-1}=XY,Z^{3^{m}}=id>$
where $m\in\mathbb{N}$;

  7. (7)

Direct product of any of the groups above with a cyclic group of relatively
prime order.

These groups above are sufficient to consider because the real sphere
$\mathbb{S}^{3}$ forms the boundary of the two complex dimensional unit ball
$\mathbb{B}^{2}$ and we are interested in unitary action of a finite, fixed
point free group on the unit ball which is a special case of orthogonal action
(because $U(2)$ is embeddable in $SO(4)$).

######  Remark 4.1.

The representation of any of these groups above, in $U(2)$, has to be faithful
in this case, to obey the ‘fixed point free’ condition. Recall from Section 2
that a group $\Gamma\subset Aut(\mathbb{B}^{2})$ and its conjugate with an
element in $Aut(\mathbb{B}^{2})$ produce biholomorphic ball quotients. So it
is enough to consider the simplest unitary representations of the groups up to
conjugation.

In [EXX22], the authors have already taken care of the cyclic groups. In the
following sections we consider the remaining non-abelian groups in the list
and their direct products with cyclic groups.

###  4.1. Representation of the groups in $U(2)$:

######  Proposition 4.1.

Any embedding of the group $Q_{8n}=<X,Y:X^{2}=(XY)^{2}=Y^{2n}>$ in $U(2)$ is
represented (up to conjugation) by the matrix group generated by

$\displaystyle X=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix},\hskip
28.45274ptY=\begin{pmatrix}a&0\\\ 0&a^{-1}\end{pmatrix}$

where $a$ is a primitive $4n^{\text{th}}$ root of unity.

###### Proof.

We may conjugate the group to assume that $Y$ is represented by a diagonal
matrix. Using the presentation, we observe that $Y$ has order $4n$ and $\det
Y\in\\{\pm 1\\}$, hence

$Y=\begin{pmatrix}a&0\\\ 0&\pm a^{-1}\end{pmatrix},$

where $a$ is primitive $4n^{\text{th}}$ root of unity. Taking $X$ as a general
element of $U(2)$, and using the relation $Y^{-1}X=XY$ and $\det X^{2}=1$, we
obtain that $X$ and $Y$ are given by

$X=\begin{pmatrix}0&b\\\ -\overline{b}&0\end{pmatrix},\hskip
28.45274ptY=\begin{pmatrix}a&0\\\ 0&a^{-1}\end{pmatrix},$

where $|b|^{2}=1$ and $a$ is primitive $4n^{\text{th}}$ root of unity. We may
further conjugate the group by the diagonal matrix
$\text{Diag}(\sqrt{b},\sqrt{\overline{b}})$ (where square root is defined
using any branch) to obtain the required representation. ∎

######  Proposition 4.2.

Any embedding of the group
$D_{2^{m}(2n+1)}=<X,Y:X^{2^{m}}=id,Y^{2n+1}=id,XYX^{-1}=Y^{-1}>$, where $m\geq
2,n\geq 1$, in $U(2)$ is represented (up to conjugation) by the matrix group
generated by

$\displaystyle X=\begin{pmatrix}0&1\\\ b&0\end{pmatrix},\hskip
28.45274ptY=\begin{pmatrix}a&0\\\ 0&a^{-1}\end{pmatrix}$

where $b$ is a primitive $2^{m-1}$th root of unity and $a$ is a primitive
$(2n+1)^{th}$ root of unity.

###### Proof.

We may assume that $Y$ is represented by a diagonal matrix. Moreover, the
given relations imply that $\det Y=1$ and has order $(2n+1)$, hence $Y$ is
given by the required matrix.

Taking $X$ as a general element in $U(2)$, the relation $XY=Y^{-1}X$ implies
that $X$ is an anti-diagonal matrix and is given by

$\displaystyle X=\begin{pmatrix}0&c\\\ b\bar{c}&0\end{pmatrix}.$

where $|c|=|b|=1$. We may assume $c=1$ by conjugating the group with diagonal
matrix $\text{Diag}(\sqrt{c},\sqrt{\bar{c}})$. Since the order of $X$ is
$2^{m}$, we obtain $b$ is a primitive $2^{m-1}$th root of unity. ∎

######  Proposition 4.3.

Any fixed point free embedding of the group $P_{48}$ (or $P_{120}$) in $U(2)$
is represented (up to conjugation) by the matrix group generated by

$\displaystyle X=\begin{pmatrix}c&d\\\ -\bar{d}&\bar{c}\end{pmatrix},\hskip
28.45274ptY=\begin{pmatrix}a&0\\\ 0&a^{-1}\end{pmatrix}$

where $a$ is a primitive $8^{th}$ (or $10^{th}$) root of unity,
$c=1/(a-\bar{a})$ and $|d|^{2}=1+c^{2}$.

###### Proof.

We use the group presentation $P_{48}=<X,Y:X^{2}=(XY)^{3}=Y^{4},X^{4}=id>$. We
may conjugate the group and assume that $Y$ is represented by a diagonal
matrix. Using the presentation of the group, we obtain that $Y$ has order $8$
and $\det X=\det Y\in\\{\pm 1\\}$.

Note that the matrix $X$ must not be a diagonal matrix, since $X$ does not
commute with $Y$. Thus the minimal polynomial of $X$ must have degree at least
2, hence it must equal the characteristic polynomial of $X$. Since $X$
satisfies $X^{4}-id=0$, $X$ has exactly two distinct eigenvalues
$\\{\lambda_{1},\lambda_{2}\\}$ from the set $\\{\pm 1,\pm i\\}$. Since $X$
acts on $\partial\mathbb{B}^{2}$ fixed point freely and $X$ is not the
identity matrix, its eigenvalue cannot be 1. Using $\det X\in\\{\pm 1\\}$, we
can conclude that $\\{\lambda_{1},\lambda_{2}\\}=\\{i,-i\\}$. In particular,
$\det Y=\det X=\lambda_{1}\lambda_{2}=1$.

We may thus assume that

$\displaystyle X=\begin{pmatrix}c&d\\\ -\bar{d}&\bar{c}\end{pmatrix}\hskip
28.45274pt\text{and}\hskip 28.45274ptY=\begin{pmatrix}a&0\\\
0&a^{-1}\end{pmatrix}.$

where $|c|^{2}+|d|^{2}=1$ and $a$ is primitive $8^{th}$ root of unity.

Observe that $Y^{4}=-id=X^{2}$ which implies $X=-X^{-1}$. It gives us the
identity $c+\bar{c}=0$. Furthermore, we use the relation $(XY)^{3}=Y^{4}=-I$
to obtain $XYX=-Y^{-1}X^{-1}Y^{-1}$. Equating entries of matrices in the last
identity, we obtain

$\displaystyle ac^{2}-|d|^{2}a^{-1}=-a^{-2}\bar{c}\hskip
28.45274pt\text{and}\hskip 28.45274ptacd+a^{-1}\bar{c}d=d.$

Note that $d\neq 0$ since $X$ is not diagonal. Thus the second identity
implies $c=1/(a-a^{-1})$.

The result for the group $P_{120}$ can be proved verbatim by letting $a$ be a
primitive $10^{\text{th}}$ root of unity. ∎

######  Proposition 4.4.

Any embedding of the group

$P^{\prime}_{8\cdot
3^{m}}=<X,Y,Z:X^{2}=(XY)^{2}=Y^{2},ZX=YZ,ZY=XYZ,Z^{3^{m}}=id>$

in $U(2)$ is represented (up to conjugation) by the matrix group generated by

$\displaystyle X=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix},\hskip
28.45274ptY=\begin{pmatrix}i&0\\\ 0&-i\end{pmatrix}\hskip
28.45274ptZ=\frac{\beta}{i-1}\begin{pmatrix}1&-i\\\ -1&-i\end{pmatrix}$

where $\beta$ is primitive $3^{m}$th root of unity.

###### Proof.

Observe that the subgroup generated by $X$ and $Y$ is isomorphic to $Q_{8}$.
Thus letting $Y$ be a diagonal matrix (using conjugation), we obtain the
required matrix representation for $X$ and $Y$.

Take a general matrix representation of $Z$ in $U(2)$ given by

$\displaystyle Z=\begin{pmatrix}a&b\\\
-e^{i\theta}\bar{b}&e^{i\theta}\bar{a}\end{pmatrix}$

where $|a|^{2}+|b|^{2}=1$. The relation $ZX=YZ$ implies $b=-ia$. Furthermore,
the relation $ZY=XYZ$ gives us $a=ie^{i\theta}\bar{a}$. Substituting back in
the above identities, we obtain

$\displaystyle Z=a\begin{pmatrix}1&-i\\\ -1&-i\end{pmatrix}.$

Note that $Z^{3}=(i-1)^{3}a^{3}\cdot id$. Let $\beta=a(i-1)$. Since $Z^{3}$
has order $3^{m-1}$, $\beta$ equals a $3^{m}$th primitive root of unity. ∎

######  Remark 4.2.

The analogous representation of the direct product of the groups $\Gamma$
considered above with $\mathbb{Z}/p\mathbb{Z}$ such that $gcd(p,|\Gamma|)=1$
can be easily obtained. Indeed, a generator of $\mathbb{Z}/p\mathbb{Z}$ must
be represented by the matrix $u\cdot id$, where $u$ is a primitive $p^{th}$
root of unity. This can be explicitly observed by using the property that the
matrix representing generator of $\mathbb{Z}/p\mathbb{Z}$ commutes with the
generators of $\Gamma$ and has order $p$.

##  5\. The groups $Q_{8n}$ and $D_{2^{m}(2n+1)}$

First we state an important lemma:

######  Lemma 5.1.

If $\Gamma$ is a nontrivial, fixed point free, finite subgroup of $SU(2)$ then
$\mathbb{B}^{2}/\Gamma$ does not have Kähler-Einstein Bergman metric. In
particular, it is true for $\Gamma=Q_{8n}$ or $D_{2^{m}(2n+1)}$.

###### Proof.

If $\Gamma$ is a subgroup of $SU(2)$ then Bergman kernel of the ball quotient,
$\mathbb{B}^{2}/\Gamma$ evaluated at the origin, $\phi(0,0)=\text{order of}\
\Gamma\neq 0$ . So by Corollary 5.4 of [HL20], the Bergman metric of
$\mathbb{B}^{2}/\Gamma$ is not Kähler-Einstein. This lemma also generalizes to
higher dimensional ball quotients. ∎

######  Remark 5.1.

The above lemma does not apply to all groups considered in the last section
because many of these groups cannot be realized as subgroups of $SU(2)$. If
$\Gamma$ is not a subgroup of $SU(2)$ then $\phi(0,0)=0$ and the method shown
above, fails. For example: the representations of direct products of
nontrivial cyclic groups with $\Gamma=Q_{8n}$ or $D_{2^{m}(2n+1)}$, obtained
in Section 4.1, are not embeddable in $SU(2)$.

######  Proposition 5.1.

Fix positive integers $n$ and $p\geq 2$ such that $gcd(p,8n)=1$. Let
$\Gamma\cong\mathbb{Z}/p\mathbb{Z}\times Q_{8n}$ be the matrix group generated
by $u\cdot id$, $X$ and $Y$, where $u$ is a primitive $p^{th}$ root of unity,
and $X$ and $Y$ are described in Proposition 4.1. Then

(5.1) $\displaystyle
C(\Gamma)=\sum_{\gamma\in{\Gamma\backslash\\{id\\}}}\psi(\gamma)\neq 0,$

where

$\psi(\gamma):=\frac{\det{\gamma}}{(1-\gamma_{11})^{4}}\bigg{(}1-3\operatorname{\mathrm{tr}}{\gamma}+\frac{4(\operatorname{\mathrm{tr}}{\gamma}-\det{\gamma}-1)}{(1-\gamma_{11})}\bigg{)}.$

###### Proof.

Using the presentation of the group $Q_{8n}$ given in Proposition 4.1, we
observe that

$\Gamma=\\{u^{j}Y^{k}X^{\ell}:0\leq j<p,0\leq k<2n,0\leq\ell<4\\}.$

The matrix $u^{j}Y^{k}X^{\ell}$ is diagonal for $\ell\in\\{0,2\\}$ and anti-
diagonal when $\ell\in\\{1,3\\}$.

Note that for $\gamma=u^{j}Y^{k}X^{\ell}$ where $\ell\in\\{1,3\\}$, we have
$\operatorname{\mathrm{tr}}\gamma=\gamma_{11}=0$ and $\det\gamma=u^{2j}$.
Since $p$ is odd and $u$ is primitive $p^{th}$ root of unity, for fixed
$\ell\in\\{1,3\\}$:

$\displaystyle\sum_{j,k}\psi(u^{j}Y^{k}X^{\ell})=\sum_{j,k}u^{2j}(1+4(-u^{2j}-1))=-2n\sum_{j=0}^{p-1}(3u^{2j}+4u^{4j})=0.$

We also observe that $X^{2}=-id=Y^{2n}$, thus $Y^{k}X^{2}=Y^{k+2n}$. For
$0\leq j\leq p-1$ and $0\leq k\leq 4n-1$, we have

$\displaystyle\psi(u^{j}Y^{k})$
$\displaystyle=\frac{u^{2j}}{(1-u^{j}a^{k})^{4}}\bigg{(}1-3u^{j}(a^{k}+a^{-k})+\frac{4(u^{j}(a^{k}+a^{-k})-u^{2j}-1)}{1-u^{j}a^{k}}\bigg{)}$
$\displaystyle=\frac{u^{2j}}{(1-u^{j}a^{k})^{4}}\big{(}1-3u^{j}(a^{k}+a^{-k})+4(u^{j}a^{-k}-1)\big{)}$
$\displaystyle=\frac{u^{2j}}{(1-u^{j}a^{k})^{4}}\big{(}-3-3u^{j}a^{k}+u^{j}a^{-k}\big{)}.$

Note that $u^{j}a^{k}$ runs over all $N:=4np$ roots of unity. Let $0\leq r\leq
N-1$ be the unique remainder modulo $N$ determined by the Chinese remainder
theorem satisfying

$r\equiv\begin{cases}&0\mod 4n\\\ &2\mod p\end{cases}.$

Let $\xi=u^{j}a^{k}$ be an $N^{th}$ root of unity, then the choice of $r$
implies $\xi^{r}=u^{2j}$ and

$\psi(u^{j}Y^{k})=\frac{\xi^{r}}{(1-\xi)^{4}}(-3-3\xi+\xi^{r-1}).$

Thus we obtain

$\displaystyle C(\Gamma)=\sum_{\xi\neq
1}\frac{-3\xi^{r}-3\xi^{r+1}+\xi^{2r-1}}{(1-\xi)^{4}},$

where $\xi$ runs over $N^{th}$ roots of unity. Using Lemma 5.2 given below, we
obtain that $720\cdot C(\Gamma)$ is an integer congruent to
$-3f(r)-3f(r+1)+f(2r-1)$ modulo $N$ where
$f(r):=-30r^{4}+240r^{3}-660r^{2}+720r-251$. Observe that $N$ is even, and
$f(r)$, $f(r-1)$ and $f(2r-1)$ are odd. Therefore, $720\cdot C(\Gamma)$ is
odd, implying $C(\Gamma)\neq 0$.

∎

######  Lemma 5.2.

For any positive integer $r$, the sum

$\displaystyle 720\cdot\sum_{\xi\neq 1}\frac{\xi^{r}}{(1-\xi)^{4}},$

is an integer congruent to $f(r)=-30r^{4}+240r^{3}-660r^{2}+720r-251\mod N$,
where $\xi$ runs over $N^{th}$ roots of unity.

###### Proof.

Let $r\in\\{1,\dots,N\\}$. Consider the generating function

$\displaystyle\sum_{n\geq 1}z^{n}\sum_{\xi\neq 1}\frac{\xi^{r}}{(1-\xi)^{n}}$
$\displaystyle=z\sum_{\xi\neq 1}\frac{\xi^{r}}{1-z-\xi}$
$\displaystyle=1+z\sum_{\xi}\frac{\xi^{r}}{1-z-\xi}$
$\displaystyle=1+\frac{Nz(1-z)^{r-1}}{(1-z)^{N}-1}.$

A careful calculation (or using computer) shows that the coefficient of
$z^{4}$ in the above power series equals

(5.2) $\displaystyle
F(N,r)=\frac{N^{4}}{720}-\left(r^{2}-4r+\frac{11}{3}\right)\frac{N^{2}}{24}+\binom{r-1}{3}\frac{N}{2}-\frac{r^{4}}{24}+\frac{r^{3}}{3}-\frac{11r^{2}}{12}+r-\frac{251}{720}.$

The lemma follows by multiplying the above equation by $720$. For
$r\notin\\{1,\dots,N\\}$, we may pick an integer $q$ such that
$r+qN\in\\{1,\dots,N\\}$. We obtain the proof by substituting $r+qN$ in place
of $r$ in (5.2) and taking modulo $N$. ∎

######  Proposition 5.2.

Fix positive integers $n$, $m\geq 2$ and $p\geq 2$ such that
$gcd(p,2^{m}(2n+1))=1$. Let $\Gamma\cong\mathbb{Z}/p\mathbb{Z}\times
D_{2^{m}(2n+1)}$ be matrix group generated by $u\cdot id$, $X$ and $Y$, where
$u$ is a primitive $p^{th}$ root of unity and $X$ and $Y$ are described in
Proposition 4.2. Then

$\displaystyle
C(\Gamma)=\sum_{\gamma\in{\Gamma\backslash\\{id\\}}}\psi(\gamma)\neq 0,$

where $\psi$ is defined in (5.1).

###### Proof.

Using the presentation of $D_{2^{m}(2n+1)}$ given in Proposition 4.2, we
obtain

$\Gamma=\\{u^{j}Y^{k}X^{\ell}:0\leq j<p,0\leq k<2n+1,0\leq\ell<2^{m}\\}.$

When $\ell$ is odd, $\gamma=u^{j}Y^{k}X^{\ell}$ is anti-diagonal, thus
$\operatorname{\mathrm{tr}}\gamma=\gamma_{11}=0$ and
$\det\gamma=(-b)^{\ell}u^{j}$. Since $p$ is odd, summing

$\psi(u^{j}Y^{k}X^{\ell})=(-b)^{\ell}u^{2j}(1-4(1+(-b)^{\ell}u^{2j}))$

over $0\leq j\leq p-1$ equals zero. Thus only the terms with even exponents of
$X$ contribute to $C(\Gamma)$.

Observe that $X^{2}=b\cdot id$ where $b$ is $2^{m-1}$th root of unity. If we
let $\ell=2s$ then $u^{j}Y^{k}X^{\ell}=u^{j}b^{s}Y^{k}$. For a fixed $k$, we
may replace the sum of

$\sum_{\begin{subarray}{c}0\leq j<p\\\ 0\leq
s<2^{m-1}\end{subarray}}\psi(u^{j}b^{s}Y^{k})=\sum_{0\leq
j<2^{m-1}p}\psi(\tilde{u}^{j}Y^{k})$

where $\tilde{u}$ is a primitive $2^{m-1}p$ th root of unity. For $0\leq
j<2^{m-1}p$ and $0\leq k<2n+1$, we have

$\displaystyle\psi(\tilde{u}^{j}Y^{k})$
$\displaystyle=\frac{\tilde{u}^{2j}}{(1-\tilde{u}^{j}a^{k})^{4}}\big{(}-3-3\tilde{u}^{j}a^{k}+\tilde{u}^{j}a^{-k}\big{)}.$

Note that $\tilde{u}^{j}a^{k}$ runs over all $N:=2^{m-1}p(2n+1)$ roots of
unity. Let $0\leq r\leq N-1$ be the unique remainder modulo $N$ determined by
the Chinese remainder theorem satisfying

$r\equiv\begin{cases}&0\mod 2n+1\\\ &2\mod 2^{m-1}p\end{cases}.$

Let $\xi=u^{j}a^{k}$ be an $N^{th}$ root of unity, then the choice of $r$
implies $\xi^{r}=u^{2j}$ and

$\psi(u^{j}Y^{k})=\frac{\xi^{r}}{(1-\xi)^{4}}(-3-3\xi+\xi^{r-1}).$

Thus we may obtain

$\displaystyle C(\Gamma)=\sum_{\xi\neq
1}\frac{-3\xi^{r}-3\xi^{r+1}+\xi^{2r-1}}{(1-\xi)^{4}}$

where $\xi$ runs over $N^{th}$ roots of unity. Thus $C(\Gamma)\neq 0$ using
the exact same arguments in the proof of Proposition 5.1. ∎

##  6\. The groups $P_{48}$ and $P_{120}$

######  Proposition 6.1.

Fix positive integers $p$ such that $gcd(p,48)=1$. Let
$\Gamma\cong\mathbb{Z}/p\mathbb{Z}\times P_{48}$ be the matrix group generated
by $u\cdot id$, $X$ and $Y$, where $u$ is a primitive $p^{th}$ root of unity
and $X$ and $Y$ are described in Proposition 4.3. Then

(6.1) $\displaystyle
C(\Gamma)=\sum_{\gamma\in{\Gamma\backslash\\{id\\}}}\psi(\gamma)\neq 0,$

where $\psi(\gamma)$ is defined in (5.1)

###### Proof.

A careful analysis of the representation of the groups given in Proposition
4.3 shows that the elements $\gamma\in\Gamma$ for which $|\gamma_{11}|=1$ are
precisely the $8p$ elements of the form $\\{u^{j}Y^{k}\\}$ where $0\leq k<8$
and $0\leq j\leq p-1$. We split $C(\Gamma)$ as:

$\displaystyle C(\Gamma)=C_{1}(p)+C_{2}(p),$

where

(6.2) $\displaystyle C_{1}(p)=\sum_{|\gamma_{11}|\neq 1}\psi(\gamma)\hskip
28.45274pt\text{and}\hskip
28.45274ptC_{2}(p)=\sum_{\begin{subarray}{c}\gamma\in\Gamma\backslash\\{id\\}\\\
|\gamma_{11}|=1\end{subarray}}\psi(\gamma).$

Using the two Lemmas below, we observe that for $p>26$ we have

$\displaystyle|C(\Gamma)|$ $\displaystyle>||C_{2}(p)|-|C_{1}(p)||$
$\displaystyle>\bigg{|}-\frac{28}{9}p^{4}+i^{p+1}12p^{3}+\frac{5}{9}p^{2}-\frac{1}{144}\bigg{|}-|4.89\times
10^{4}\cdot p|$ $\displaystyle>0.$

For $1\leq p\leq 26$, we explicitly calculate $C(\Gamma)$ using computer.
Please refer to the Appendix A.

∎

######  Lemma 6.1.

For $p\geq 2$ and $gcd(p,48)=1$, we have

$\displaystyle|C_{1}(p)|\leq 4.89\times 10^{4}\cdot p.$

###### Proof.

By explicit calculation, we note that $(Y^{2}X)^{2}$ is an anti-diagonal
matrix. We first consider the sum over the $8p$ elements of the form
$\gamma=u^{j}Y^{k}(Y^{2}X)^{2}$ where $j\in\\{0,1,\dots,p-1\\}$ and
$k\in\\{0,1,2,\dots,7\\}$. We observe that for these elements
$\gamma_{11}=\operatorname{\mathrm{tr}}\gamma=0$ and hence
$\psi(\gamma)=u^{2j}(1-4(-u^{2j}-1))$. Thus

$\displaystyle\sum_{k=0}^{7}\sum_{j=0}^{p-1}\psi(u^{j}Y^{k}(Y^{2}X)^{2})=\sum_{k=0}^{7}\sum_{j=0}^{p-1}u^{2j}(1-4(-u^{2j}-1))=0.$

Here we use that $gcd(p,48)=1$ and $u$ is a primitive $p^{\text{th}}$ root of
unity.

Recall the constant $c$ from Proposition 4.3. We use Sage to observe that
$|\gamma_{11}|\leq|c|$ for $\gamma\in\Gamma$ such that $|\gamma_{11}|\neq 1$.
We now find an upper bound for the remaining $32p$ terms appearing in
$C_{1}(p)$ for which $\gamma_{11}\neq 0$. Note that for these elements
$\gamma$, we have $\det\gamma=u^{2j}$, $\gamma_{11}=u^{j}\alpha$ and
$\operatorname{\mathrm{tr}}\gamma=u^{j}(\alpha+\bar{\alpha})$ where
$|\alpha|\leq|c|$. We obtain the following bound using triangle inequality:

$\displaystyle|\psi(\gamma)|$
$\displaystyle=\bigg{|}\frac{u^{2j}}{(1-u^{j}\alpha)^{4}}\bigg{(}1-3u^{j}(\alpha+\overline{\alpha})+4\frac{u^{j}(\alpha+\overline{\alpha})-u^{2j}-1}{1-u^{j}\alpha}\bigg{)}\bigg{|}$
$\displaystyle=\bigg{|}\frac{u^{2j}}{(1-u^{j}\alpha)^{4}}\bigg{(}-3-3u^{j}(\alpha+\overline{\alpha})+4\frac{u^{j}\overline{\alpha}-u^{2j}}{1-u^{j}\alpha}\bigg{)}\bigg{|}$
$\displaystyle\leq\frac{7+6|c|}{(1-|c|)^{4}}.$

In the last step, we note that the term
$(u^{j}\overline{\alpha}-u^{2j})/(1-u^{j}\alpha)$ has absolute value 1. Using
$|c|=1/\sqrt{2}$, we obtain:

$\displaystyle|C_{1}(p)|\leq 32p\frac{7+6|c|}{(1-|c|)^{4}}\leq 4.89\times
10^{4}p.$

∎

######  Lemma 6.2.

We have

$\displaystyle C_{2}(p)$
$\displaystyle=-\frac{28}{9}p^{4}+i^{p+1}12p^{3}+\frac{5}{9}p^{2}-\frac{1}{144}.$

###### Proof.

We observe that

$\psi(u^{j}Y^{k})=\frac{u^{2j}}{(1-u^{j}a^{k})^{4}}\big{(}-3-3u^{j}a^{k}+u^{j}a^{-k}\big{)}.$

Note that $u^{j}a^{k}$ runs over all $N:=8p$ roots of unity. Let $0\leq r\leq
N-1$ be the unique remainder modulo $N$ determined by Chinese remainder
theorem satisfying $r\equiv 0\mod 8$ and $r\equiv 2\mod p.$ The solutions to
this congruence equation are precisely

$r=\begin{cases}6p+2&\text{when }p\equiv 1\mod 4\\\ 2p+2&\text{when }p\equiv
3\mod 4\end{cases}.$

Let $\xi=u^{j}a^{k}$, then the choice of $r$ implies $\xi^{r}=u^{2j}$ and

$\psi(u^{j}Y^{k})=\frac{\xi^{r}}{(1-\xi)^{4}}(-3-3\xi+\xi^{r-1}).$

Thus we may obtain

$\displaystyle C_{2}(p)=\sum_{\xi\neq
1}\frac{-3\xi^{r}-3\xi^{r+1}+\xi^{2r-1}}{(1-\xi)^{4}},$

where $\xi$ runs over $N^{th}$ roots of unity.

In Lemma 5.2, we obtained an explicit polynomial expression in (5.2) for the
sum

$F(N,r)=\sum_{\xi\neq 1}\frac{\xi^{r}}{(1-\xi)^{4}},$

where $1\leq r\leq N$.

When $p\equiv 1\mod 4$, we substitute $N=8p$ and $r\equiv 6p+2\mod p$ to
obtain

$\displaystyle C_{2}(p)$ $\displaystyle=-3F(8p,6p+2)-3F(8p,6p+3)+F(8p,4p+3)$
$\displaystyle=-\frac{28}{9}p^{4}-12p^{3}+\frac{5}{9}p^{2}-\frac{1}{144}.$

In the last term, we used $2r-1\equiv 4p+3\mod N$.

When $p\equiv 3\mod 4$, we substitute $r\equiv 2p+2\mod p$ to obtain

$\displaystyle C_{2}(p)$
$\displaystyle=-\frac{28}{9}p^{4}+12p^{3}+\frac{5}{9}p^{2}-\frac{1}{144}.$

Notice that this expression has opposite sign for the term $12p^{3}$ compared
to previous case. ∎

######  Proposition 6.2.

Fix positive integers $p$ such that $gcd(p,120)=1$. Let
$\Gamma\cong\mathbb{Z}/p\mathbb{Z}\times P_{120}$ be the matrix group
generated by $u\cdot id$, $X$ and $Y$, where $u$ is a primitive $p^{th}$ root
of unity and $X$ and $Y$ are described in Proposition 4.3. Then

(6.3) $\displaystyle
C(\Gamma)=\sum_{\gamma\in{\Gamma\backslash\\{id\\}}}\psi(\gamma)\neq 0,$

where $\psi(\gamma)$ is defined in (5.1).

###### Proof.

A careful analysis of the representation of the groups given in Proposition
4.3 shows that the elements $\gamma\in\Gamma$ for which $|\gamma_{11}|=1$ are
precisely the $10p$ elements of the form $\\{u^{j}Y^{k}\\}$ where $0\leq k<10$
and $0\leq j\leq p-1$. We split $C(\Gamma)$ as:

$\displaystyle C(\Gamma)=C_{1}(p)+C_{2}(p),$

where

$\displaystyle C_{1}(p)=\sum_{|\gamma_{11}|\neq 1}\psi(\gamma)\hskip
28.45274pt\text{and}\hskip
28.45274ptC_{2}(p)=\sum_{\begin{subarray}{c}\gamma\in\Gamma\backslash\\{id\\}\\\
|\gamma_{11}|=1\end{subarray}}\psi(\gamma).$

Using the two Lemmas below, we observe that for $p>46$ we have

$\displaystyle|C(\Gamma)|>||C_{2}(p)|-|C_{1}(p)||>0.$

For $1\leq p\leq 46$, we explicitly calculate $C(\Gamma)$ using computer,
please refer to the Appendix A.

∎

######  Lemma 6.3.

For $p\geq 2$ and $gcd(p,120)=1$, we have

$\displaystyle|C_{1}(p)|\leq 2.68\times 10^{6}\cdot p.$

###### Proof.

We use Sage to observe that $|\gamma_{11}|\leq|c|$ for $\gamma\in\Gamma$ such
that $|\gamma_{11}|\neq 1$. The constant $c$ is described in Proposition 4.3.
We now find an upper bound for the $110p$ terms appearing in $C_{1}(p)$ for
which $|\gamma_{11}|\neq 1$. The bound on $\psi(\gamma)$ for such $\gamma$ is
obtained in exactly the same way as in the proof of Lemma 6.1. In particular,
we have

$\displaystyle|C_{1}(p)|\leq 110p\frac{7+6|c|}{(1-|c|)^{4}}\leq 2.68\times
10^{6}p.$

Here we used that $|c|=(2\sin(\pi/5))^{-1}$. ∎

######  Lemma 6.4.

We have

$\displaystyle
C_{2}(p)=\begin{cases}-\frac{265}{9}\,p^{4}-20\,p^{3}+\frac{25}{18}\,p^{2}-\frac{1}{144}&p\equiv
1\mod 5\\\
\frac{575}{9}\,p^{4}+20\,p^{3}-\frac{35}{18}\,p^{2}-\frac{1}{144}&p\equiv
2\mod 5\\\
\frac{455}{9}\,p^{4}-8\,p^{3}-\frac{11}{18}\,p^{2}-\frac{1}{144}&p\equiv 3\mod
5\\\
-\frac{265}{9}\,p^{4}-20\,p^{3}+\frac{25}{18}\,p^{2}-\frac{1}{144}&p\equiv
4\mod 5\end{cases}.$

###### Proof.

We observe that

$\psi(u^{j}Y^{k})=\frac{u^{2j}}{(1-u^{j}a^{k})^{4}}\big{(}-3-3u^{j}a^{k}+u^{j}a^{-k}\big{)}.$

Note that $u^{j}a^{k}$ runs over all $N:=10p$ roots of unity. Let $0\leq r\leq
N-1$ be the unique remainder modulo $N$ determined by Chinese remainder
theorem satisfying $r\equiv 0\mod 10$ and $r\equiv 2\mod p.$ The explicit
value of $r$ is given below:

$r=\begin{cases}8p+2&\text{when }p\equiv 1\mod 5\\\ 4p+2&\text{when }p\equiv
2\mod 5\\\ 6p+2&\text{when }p\equiv 3\mod 5\\\ 2p+2&\text{when }p\equiv 4\mod
5\end{cases}.$

Let $\xi=u^{j}a^{k}$, then the choice of $r$ implies $\xi^{r}=u^{2j}$ and

$\psi(u^{j}Y^{k})=\frac{\xi^{r}}{(1-\xi)^{4}}(-3-3\xi+\xi^{r-1}).$

Thus we may obtain

$\displaystyle C_{2}(p)=\sum_{\xi\neq
1}\frac{-3\xi^{r}-3\xi^{r+1}+\xi^{2r-1}}{(1-\xi)^{4}},$

where $\xi$ runs over $N^{th}$ roots of unity.

In Lemma 5.2, we obtained an explicit polynomial expression in (5.2) for the
sum

$F(N,r)=\sum_{\xi\neq 1}\frac{\xi^{r}}{(1-\xi)^{4}},$

where $1\leq r\leq N$.

When $p\equiv 1\mod 5$, we substitute $N=10p$ and $r\equiv 8p+2\mod p$ to
obtain

$\displaystyle C_{2}(p)$
$\displaystyle=-3F(10p,8p+2)-3F(10p,8p+3)+F(10p,6p+3)$
$\displaystyle=-\frac{265}{9}\,p^{4}-20\,p^{3}+\frac{25}{18}\,p^{2}-\frac{1}{144}.$

In the last term, we used $2r-1\equiv 6p+3\mod N$.

The remaining cases are dealt exactly in the same way. ∎

##  7\. The groups $P^{\prime}_{8.3^{m}}$

######  Proposition 7.1.

Fix positive integer $p$ such that $gcd(p,8\cdot 3^{m})=1$. Let
$\Gamma\cong\mathbb{Z}/p\mathbb{Z}\times P^{\prime}_{8.3^{m}}$ be the matrix
group generated by $u\cdot id$, $X$, $Y$ and $Z$, where $u$ is a primitive
$p^{th}$ root of unity and $X,\ Y,\ Z$ are described in Proposition 4.4. Then

$\displaystyle
C(\Gamma)=\sum_{\gamma\in{\Gamma\backslash\\{id\\}}}\psi(\gamma)\neq 0,$

where $\psi(\gamma)$ is defined in (5.1)

###### Proof.

First a careful analysis from the representation of the groups given in
Section 4.1, shows that elements of $\Gamma$ can be uniquely written as
$u^{k}\cdot Z^{i_{1}}\cdot X^{i_{2}}\cdot Y^{i_{3}}$ where
$k\in\\{1,2,\dots,p\\},i_{1}\in\\{1,2,\dots,3^{m}\\},i_{2}\in\\{1,2\\}$ and
$i_{3}\in\\{1,2,3,4\\}$. Then looking at the representation of the group
$P^{\prime}_{8.3^{m}}$, we note that for $\gamma\in\Gamma$ such that
$|\gamma_{11}|=1$, we must have $i_{1}\in 3\mathbb{N}$ and $i_{2}=2$. Then
just like last proposition, we can split $C(\Gamma)$ as:

$\displaystyle C(\Gamma)=C_{1}(p,m)+C_{2}(p,m),$

where

$\displaystyle C_{1}(p,m)=\sum_{|\gamma_{11}|\neq 1}\psi(\gamma)\hskip
28.45274pt\text{and}\hskip
28.45274ptC_{2}(p,m)=\sum_{\begin{subarray}{c}\gamma\in\Gamma\backslash\\{id\\}\\\
|\gamma_{11}|=1\end{subarray}}\psi(\gamma)$

Using the two Lemmas below, we observe that for $\tilde{p}=p\cdot 3^{m-1}\geq
24$, we have

$\displaystyle|C(\Gamma)|>||C_{2}(p,m)|-|C_{1}(p,m)||>0.$

For $1\leq\tilde{p}\leq 23$, we explicitly calculate $C(\Gamma)$ using
computer. Please refer to the Appendix A.

∎

######  Lemma 7.1.

For $m,p\in\mathbb{N}$ and $gcd(p,8\cdot 3^{m})=1$, we have

$\displaystyle|C_{1}(p,m)|\leq 80(135\sqrt{2}+191)\cdot 3^{m-1}\cdot p$

###### Proof.

Recall that any element $\gamma\in\Gamma$ can be represented as
$\gamma=u^{k}Z^{i_{1}}X^{i_{2}}Y^{i_{3}}$. Using the explicit description of
matrices $X$, $Y$ and $Z$, we observe that if $|\gamma_{11}|\neq 1$, then
$|\gamma_{11}|\leq\frac{1}{\sqrt{2}}$. For $\gamma\in\Gamma$, we have
$\det\gamma=u^{2j}$, $\gamma_{11}=u^{j}\alpha$ and
$\operatorname{\mathrm{tr}}\gamma=u^{j}(\alpha+\bar{\alpha})$ where
$|\alpha|\leq 1/\sqrt{2}$. Using the bound in the proof of Lemma 6.1, we have

$\displaystyle|\psi(\gamma)|\leq\frac{7+6|\alpha|}{(1-|\alpha|)^{4}}\leq\frac{7+3\sqrt{2}}{(1-1/\sqrt{2})^{4}}.$

Since there are $8p3^{m}-4p3^{m-1}=20p3^{m-1}$ elements $\gamma$ such that
$|\gamma_{11}|\neq 1$, we obtain

$\displaystyle|C_{1}(p,m)|\leq 20\cdot
3^{m-1}p\frac{7+3\sqrt{2}}{(1-1/\sqrt{2})^{4}}=80(135\sqrt{2}+191)\cdot
3^{m-1}\cdot p.$

∎

######  Lemma 7.2.

We have

$C_{2}(p,m)=\sum_{|\gamma_{11}|=1}\psi(\gamma)=\frac{20}{9}\tilde{p}^{4}-\frac{7}{9}\tilde{p}^{2}-\frac{1}{144},$

where $\tilde{p}=p\cdot 3^{m-1}$.

###### Proof.

Note that $|\gamma_{11}|=1$ if and only if
$\gamma=u^{j}Z^{3k}X^{2}Y^{\ell}=-u^{j}Z^{3k}Y^{\ell}=u^{j}Z^{3k}Y^{\ell+2}$.
We observe that $Z^{3}=\beta^{3}\cdot id$, where $\beta$ is primitive
$3^{m}$th root of unity. Thus

$\psi(u^{j}Z^{3k}Y^{\ell})=\psi(u^{j}\beta^{3k}Y^{\ell}).$

Since $u^{j}\beta^{3k}$ runs over $\tilde{p}=p3^{m-1}$ roots of unity, we may
rewrite

$\psi(w^{j}Y^{k})\frac{w^{2j}}{(1-w^{j}a^{k})^{4}}\big{(}-3-3w^{j}a^{k}+w^{j}a^{-k}\big{)}.$

Note that $w^{j}a^{k}$ runs over all $N:=4\tilde{p}$ roots of unity. Let
$0\leq r\leq N-1$ be the unique remainder modulo $N$ determined by Chinese
remainder theorem satisfying $r\equiv 0\mod 4$ and $r\equiv 2\mod\tilde{p}.$
The solutions to this congruence equation is precisely $r=2\tilde{p}+2$.

Let $\xi=w^{j}a^{k}$, then the choice of $r$ implies $\xi^{r}=w^{2j}$ and

$\psi(w^{j}Y^{k})=\frac{\xi^{r}}{(1-\xi)^{4}}(-3-3\xi+\xi^{r-1}).$

Thus we may obtain

$\displaystyle C_{2}(p,m)=\sum_{\xi\neq
1}\frac{-3\xi^{r}-3\xi^{r+1}+\xi^{2r-1}}{(1-\xi)^{4}}$

where $\xi$ runs over $N^{th}$ roots of unity.

Using the formula (5.2) for $F(N,r):=\sum_{\xi\neq
1}\frac{\xi^{r}}{(1-\xi)^{4}}$ where $1\leq r\leq N$, we obtain

$\displaystyle C_{2}(p,m)$
$\displaystyle=-3F(4\tilde{p},2\tilde{p}+2)-3F(4\tilde{p},2\tilde{p}+3)+F(4\tilde{p},3)$
$\displaystyle=\frac{20}{9}\tilde{p}^{4}-\frac{7}{9}\tilde{p}^{2}-\frac{1}{144}.$

∎

## Appendix A

Here we give tables for the explicit coefficient $C(\Gamma)$ for the groups
$\Gamma$ mentioned previously in Propositions 6.1, 6.2 and 7.1. The values of
$C(\Gamma)$ below are obtained numerically using MATLAB and Sage.

###  A.1. $\Gamma=P_{48}\times\mathbb{Z}/p$:

Here we give the table of $C(\Gamma)$ for $1\leq p\leq 26$, with
$gcd(p,48)=1$.

Values of $p$ |  Values of $C(\Gamma)$
---|---
1 |  -5968.348737366
5 |  -17682.290187324
7 |  -5491.7044210122
11 |  -29599.5039863668
13 |  -116823.112487798
17 |  -318950.767090781
19 |  -322934.584891557
13 |  -724318.562730527
25 |  -1402436.345364025

###  A.2. $\Gamma=P_{120}\times\mathbb{Z}/p$:

Here we give the table of $C(\Gamma)$ for $1\leq p\leq 46$, with
$gcd(p,120)=1$.

Values of $p$ |  Values of $C(\Gamma)$
---|---
1 |  2833.2070524
7 |  179826.7366816
11 |  -792274.9699035
13 |  1807438.9684806
17 |  5745222.85120843
19 |  -3699419.76217178
23 |  17634781.2649327
29 |  -20336547.9959923
31 |  -27809085.4970539
37 |  120775875.7114752
41 |  -84581711.0963804
43 |  216829661.9498101

###  A.3. $\Gamma=P^{\prime}_{8.3^{k}}\times\mathbb{Z}/p$:

Here we give the table of $C(\Gamma)$ for $1\leq\tilde{p}\leq 23$, where
$\tilde{p}=p\cdot 3^{m-1}$ and $gcd(p,8.3^{m})=1$.

Values of $m$ |  Values of $p$ |  Values of $C(\Gamma)$
---|---|---
1 |  1 |  -484.8486298220
1 |  5 |  5030.1210024879
1 |  7 |  3676.7952801659
1 |  11 |  31610.0383198645
1 |  13 |  63761.5753483247
1 |  17 |  185455.4886479516
1 |  19 |  289290.8713510202
1 |  23 |  621453.3057542715
2 |  1 |  -1097.0507252631
2 |  5 |  112137.4453024487
2 |  7 |  431848.4432457592
3 |  1 |  16148.9146758443

## References

  * [Ber33] Stefan Bergman, _Über die Kernfunktion eines Bereiches und ihr Verhalten am Rande. I_ , J. Reine Angew. Math. 169 (1933), 1–42. MR 1581372
  * [Ber35] by same author, _Über die Kernfunktion eines Bereiches und ihr Verhalten am Rande. II_ , J. Reine Angew. Math. 172 (1935), 89–128. MR 1581442
  * [Che79] Shiu-Yuen Cheng, _Open problems_ , Conference on Nonlinear Problems in Geometry held in Katata, Sep 3-8 (1979), 2, Tohoku Uniersity, Department of Math, Sendai.
  * [CY80] Shiu Yuen Cheng and Shing Tung Yau, _On the existence of a complete Kähler metric on noncompact complex manifolds and the regularity of Fefferman’s equation_ , Comm. Pure Appl. Math. 33 (1980), no. 4, 507–544. MR 575736
  * [EXX22] Peter Ebenfelt, Ming Xiao, and Hang Xu, _On the Classification of Normal Stein Spaces and Finite Ball Quotients With Bergman–Einstein Metrics_ , Int. Math. Res. Not. IMRN (2022), no. 19, 15240–15270. MR 4490953
  * [Fef76a] C. Fefferman, _Correction to: “Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains”_ , Ann. of Math. (2) 104 (1976), no. 2, 393–394. MR 407321
  * [Fef76b] Charles L. Fefferman, _Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains_ , Ann. of Math. (2) 103 (1976), no. 2, 395–416. MR 407320
  * [FW97] Siqi Fu and Bun Wong, _On strictly pseudoconvex domains with Kähler-Einstein Bergman metrics_ , Math. Res. Lett. 4 (1997), no. 5, 697–703. MR 1484700
  * [HL20] Xiaojun Huang and Xiaoshan Li, _Bergman-Einstein metric on a Stein space with a strongly pseudoconvex boundary_ , 2020, arXiv: 2008.03645.
  * [HX20] Xiaojun Huang and Ming Xiao, _A uniformization theorem for Stein spaces_ , Complex Anal. Synerg. 6 (2020), no. 2, Paper No. 6, 5. MR 4098639
  * [HX21] Xiaojun Huang and Ming Xiao, _Bergman–Einstein metrics, a generalization of Kerner’s theorem and Stein spaces with spherical boundaries_ , Journal für die reine und angewandte Mathematik (Crelles Journal) 2021 (2021), no. 770, 183–203.
  * [Kob59] Shoshichi Kobayashi, _Geometry of bounded domains_ , Trans. Amer. Math. Soc. 92 (1959), 267–290. MR 112162
  * [Kra13] Steven G. Krantz, _Geometric analysis of the Bergman kernel and metric_ , Graduate Texts in Mathematics, vol. 268, Springer, New York, 2013. MR 3114665
  * [Lem95] László Lempert, _Algebraic approximations in analytic geometry_ , Invent. Math. 121 (1995), no. 2, 335–353. MR 1346210
  * [Mil57] John Milnor, _Groups which act on $S^{n}$ without fixed points_, Amer. J. Math. 79 (1957), 623–630. MR 90056
  * [NS06] Stefan Yu Nemirovski and Ruslan G Shafikov, _Conjectures of Cheng and Ramadanov_ , Russian Mathematical Surveys 61 (2006), no. 4, 780–782.
  * [Rud08] Walter Rudin, _Function theory in the unit ball of $\mathbb{C}^{n}$_, Classics in Mathematics, Springer-Verlag, Berlin, 2008, Reprint of the 1980 edition. MR 2446682
  * [Wol11] Joseph A. Wolf, _Spaces of constant curvature_ , sixth ed., AMS Chelsea Publishing, Providence, RI, 2011. MR 2742530
  * [Won77] B. Wong, _Characterization of the unit ball in $\mathbb{C}^{n}$ by its automorphism group_, Inventiones mathematicae 41 (1977), 253–258.
  * [Yau82] Shing-Tung Yau, _Problem Section, Seminar on Differential geometry_ , pp. 669–706, Ann. of Math. Studies, vol. 102, Princeton University Press, 1982.


#  StonyBook: A System and Resource for Large-Scale Analysis of Novels

Charuta Pethe, Allen Kim, Rajesh Prabhakar, Tanzir Pial, Steven Skiena
Department of Computer Science,
Stony Brook University, NY, USA
<EMAIL_ADDRESS>
###### Abstract

Books have historically been the primary mechanism through which narratives
are transmitted. We have developed a collection of resources for the large-
scale analysis of novels, including: (1) an open source end-to-end NLP
analysis pipeline for the annotation of novels into a standard XML format, (2)
a collection of 49,207 distinct cleaned and annotated novels, and (3) a
database with an associated web interface for the large-scale aggregate
analysis of these literary works. We describe the major functionalities
provided in the annotation system along with their utilities. We present
samples of analysis artifacts from our website, such as visualizations of
character occurrences and interactions, similar books, representative
vocabulary, part of speech statistics, and readability metrics. We also
describe the use of the annotated format in qualitative and quantitative
analysis across large corpora of novels. 111Please check out the demo video at
https://youtu.be/kCu-zKFvvQE.

##  1 Introduction

Books have historically been the primary mechanism through which knowledge and
narratives are transmitted through time and space. Books (particularly novels)
are a uniquely interesting class of texts that provide a view of the inner
lives of people, what they think and how they work: knowledge that cannot be
readily gleaned from other sources. Building intelligent systems requires that
machines understand people on some fundamental level – their goals, dreams,
and thought processes. Computational analysis of novels provides a possible
approach to getting this understanding into intelligent computer systems, with
wide-spread implications for common-sense reasoning and human factors
engineering.

But the analysis of longform texts has generally been understudied by the
natural language processing (NLP) research community. Modern neural network
techniques for NLP generally focus on short texts (such as paragraphs or
social media posts) because model capacity limitations preclude the effective
understanding of full longform texts. We believe that improved data and
computational resources will help open this field to a broader research
community.

The field of digital literary studies has spurred advances in computational
linguistics as well as digital humanities. However, the lack of standard
annotation systems and exploratory interfaces for the large-scale
computational analysis of books has hindered the efforts of these communities.
This paper describes our book analysis system
StonyBook222https://github.com/sbu-dsl/stonybook and associated web resource
333https://www.stonybook.org that aims to bridge this gap.

Figure 1: StonyBook: System overview

Our main contributions in this paper are:

  * •

An end-to-end system (in the form of a Python library) for the NLP-driven
annotation of novels in a standard XML specification format, supporting inputs
for different book file formats (Project Gutenberg text files, HathiTrust
page-wise text files) with extensive text-cleaning efforts. We describe the
capability of our pipeline, including named entity recognition, quote
extraction/attribution, and character identification.

  * •

A data resource of 17,289 annotated books from the Project Gutenberg corpus
and 31,918 other distinct annotated books from the HathiTrust Digital Library
corpus. Beyond just annotating texts, we have performed extensive volume
deduplication and the assembly of metadata including authorship year and
current publication status.

  * •

An exploratory web interface that provides statistics and insights about
books, authors, and subjects, and where they lie in context of other books in
the corpora. As examples of the power of this resource, we present interesting
observations on character frequency distributions and part of speech
distributions.

Figure 1 shows an overview of the system and the contributions, described in
detail in the following sections. Due to copyright restrictions, we cannot
directly distribute annotated texts from the HathiTrust dataset. However, the
list of unique IDs for books from both corpora (Project Gutenberg and
HathiTrust) along with analysis artifacts have been made available publicly on
the website, facilitating a wide range of analyses.

##  2 Related Work

GutenTag Brooke et al. (2015) is a tool that provides functionality to build
subcorpora from the Project Gutenberg corpus, and a general tagging framework
in order to facilitate interaction between the digital humanities and
computational linguistics communities. The Gutenberg module of NLTK Bird and
Loper (2004) provides a limited set of functionalities for a few books in the
corpus.

The BookNLP pipeline Bamman et al. (2014) generates NLP-driven book analyses
in multiple formats, with tokenization, part-of-speech tagging, dependency
tagging, character identification, and quote attribution. This pipeline uses
CoreNLP Manning et al. (2014) for syntax parsing. BookNLP defines a set of
initial characters and creates a set of allowable variants for each one,
similar to Davis et al. (2003) and Elson et al. (2010), after which each
character is greedily assigned to the most recently linked entity of which it
is a variant.

Groza and Corde (2015) use an ontology-based approach to identify characters,
that is specific to folktales. Vala et al. (2015) present a graph-based
approach to cluster variants of character name mentions, involving heuristics
to decide whether to merge two vertices. The CHARLES tool Vala et al. (2016)
provides a collaborative interface for manual annotation of characters. Dekker
et al. (2019) compare four different approaches for character identification,
by evaluating against a manually annotated dataset. Jahan and Finlayson (2019)
propose a refined definition of character, and an approach that uses an
animacy detector followed by a supervised classification model to label which
of the animated entities qualify as characters. Egbert and Mahlberg (2020)
focus on analysis of register variation in fiction.

##  3 System Components

In this section, we describe the primary data sets we employ, and the analysis
phases we use to clean and process these into annotated texts. This work is
focused on the effort and discipline with which we unify disparate corpora,
tools, and metadata to create a substantial resource for research on novels.

###  3.1 Datasets

Our primary resources are two well-known collections of scanned texts:

  * •

Project Gutenberg – We identified 17,289 books by 4,425 authors from the
English language section of the Project Gutenberg corpus, tagged with the term
"fiction" as subject. Previous research, such as Rae et al. (2019), utilized
Project Gutenberg extensively and introduced PG-19, a book-level language
modeling benchmark dataset derived from it. The StonyBook pipeline can aid
researchers working with Project Gutenberg books.

  * •

HathiTrust Digital Library – We identified a subset of 31,918 books from
14,496 authors from the HathiTrust Digital Library  dataset, distinct from the
Project Gutenberg dataset. We obtained this subset by filtering from a set of
English fiction books Underwood (2016).

We followed Kim et al. (2021) for removing duplicate copies of books from both
the datasets based on title, author, and content similarity.

###  3.2 Annotation Format

After each of the phases described in the subsequent sections, we store the
text along with its annotations in XML format, adding or removing tags as
required after each phase. At the end of the pipeline, we have an XML file
conforming to a standard specification, consistent across corpora, simplifying
our analysis. The final annotated file for each book includes information
about POS, NER, dependency, and character counts. An example can be found in
Figure 2.

To process these XML files, we provide utility functions developed using lxml
Behnel et al. (2005) that we use to generate our analysis. This will be
especially useful to those who prefer to store the annotations in a different
format (e.g. collection of CSV files as done in BookNLP).

###  3.3 Preprocessing

To annotate Gutenberg headers and footers, we employ an improved version of
the cleanup module from the Gutenberg Python
package444https://pypi.org/project/Gutenberg/. To annotate front and back
matter in books from the HathiTrust corpus, we use a neural classification
model McConnaughey et al. (2017).

###  3.4 Section Header Identification

We use an exhaustive rule-based search Pethe et al. (2020) to identify and tag
section headers. This covers section headers including key words like
"Chapter", "Book", etc. We also factor in the chapter numbering to ensure
consistency in our tagging.

###  3.5 Syntax Parsing

We obtain the following annotations, as described in Figure 1: tokenization,
sentence splitting, part-of-speech tagging, lemmatization, NER, dependency
parsing, coreference resolution, and quote extraction and attribution. We
provide the option to use either module for annotation: CoreNLP through Stanza
Qi et al. (2020) or spaCy with neuralcoref Honnibal and Montani (2017).

Figure 2: Processed XML including character information (gcc, fpcc, spcc stand
for gendered, first-person and second-person coreference counts respectively)

###  3.6 Character Identification

We perform the following steps to canonically identify the characters in a
book:

#### Candidate augmentation:

We expand the ner PERSON tags to include any prefixed honorifics, and tag all
occurrences of the entity as candidates.

#### Gender inference:

We infer the character’s gender using three methods: dictionary lookup of the
honorific, pronoun coreference majority vote, and an external data source
555Gender Guesser (https://pypi.org/project/gender-guesser/).

#### Name mention clustering:

For single name mentions, we map to the closest occurrence of a full name with
either the same first name or last name, in the order of preference for gender
inferred from honorific, coreferences, and name. We then find unique
combinations of first/last names and gender. We sort these by frequency, and
keep only those characters with more than 2 total mentions, to obtain the
final character list.

Figure 2 shows a sample template of the processed XML after character
identification. This includes information about character names, genders,
occurrences in the text, and types of coreferences and their counts.

##  4 Web Interface

The results and metadata of analysis of each of almost 50,000 novels appears
on our website https://www.stonybook.org. Our website features detailed
analysis of each book, described in this section. Additionally, we provide an
author-wise and subject-wise view of all these novels. For easy perusal of our
website, we provide a list of links to analyses of popular books. This list of
books was procured using ratings and popularity data from GoodReads
666https://www.goodreads.com/ and Amazon 777https://www.amazon.com/.

###  4.1 Characters

Unifying and aggregating character references into distinct character personas
enables us to measure the importance and interactions between characters, and
how this evolves over the length of the book.

Figure 3: Popular characters and their occurrences in ‘Oliver Twist’ (vertical
dashed lines denote chapter breaks).

Figure 3 shows a part of the exploratory web interface for the book ‘Oliver
Twist’ by Charles Dickens. It shows where each of the top 10 most frequent
characters occur throughout the narrative. This gives us an idea of the
positional importance of characters, which has potential applications in
generating character representations, and book-level QA involving questions
about specific characters.

Figure 4: Character interaction network for ‘Oliver Twist’ (Blue nodes
represent male characters, pink nodes represent female characters, node size
is proportional to the number of occurrences of the character).

We also extract a character interaction network from the book, where we
consider two characters to be frequently interacting if they occur within a
fixed window (here we empirically set the window size to 30 tokens, but this
is a modifiable parameter) of each other more than 5 times. We display this
network in the web interface, as shown in Figure 4, along with character
occurrences.

Figure 5: Gender statistics for ‘Oliver Twist’.

We also display the distribution of the percentage of male and female
characters across the entire corpus, and denote where each book lies in the
distribution, as shown in Figure 5.

In addition to these visualizations, we also display a comprehensive list of
characters in the book, with all their aliases, mention counts, and pronoun
coreference counts.

###  4.2 Most Similar Books

Figure 6: Top most similar books for ‘Oliver Twist’.

Book embeddings provide vector representations facilitating queries of the
similarities of different works. We train a doc2vec model Le and Mikolov
(2014) on all the books in both the corpora (Project Gutenberg and HathiTrust)
combined. We train for 10 epochs using a window size of 5, to generate 100
dimensional vectors. We restrict our vocabulary to the top 200,000 words and
exclude words with counts less than 100. We strip stop words and lemmatize
each token. Then for each book, we list the top 10 most similar books from
each of the corpora as shown in Figure 6.

###  4.3 Representative Vocabulary

For each book, we compute the ratio of the normalized frequency of each word,
and compute the ratio of this with the globally normalized word frequency
across all books. We sort words by this ratio to obtain lists of most and
least representative words in the book, filtered to the 10,000 most common
words in the corpus. We also list the words that are most common in the corpus
but are absent in the book, as missing words. Figure 7 shows a part of our web
interface that shows representative vocabulary.

Figure 7: Representative vocabulary for ‘Oliver Twist’.

###  4.4 Part of Speech Statistics

Figure 8: Part of speech distribution graphs for ‘Oliver Twist’.

For each of the following parts of speech (POS), we compute the count and
percentage among all POS tags within the book, as well as the percentile among
all the books: Noun, Adjective, Verb, Adverb, Pronoun, Interjection,
Preposition, and Conjunction. Additionally, as shown in Figure 8, we also
display the distribution of POS percentages across all books in the corpus,
and denote where each particular book lies in the distribution.

###  4.5 Readability Metrics

Figure 9: Readability metrics for ‘Oliver Twist’.

We compute the following readability metrics for each book: Flesch Reading
Ease, Dale Chall Readability Score Dale and Chall (1948), Automated
Readability Index (ARI), Coleman-Liau Index Coleman and Liau (1975), Gunning
Fog Index Kincaid et al. (1975), SMOG Index Mc Laughlin (1969), Spache
Readability Score and Linsear Write Formula Grade level.

##  5 Exploring the Resource

In this section, we present some of the aggregate analyses powered from the
dataset. These should be viewed as explorations and demonstrations more than
definitive treatments of these issues, in the hopes that they inspire further
work.

#### Percentage frequency vs. character rank:

We consider all books in our combined corpus (Project Gutenberg + HathiTrust)
that contain at least nine distinct characters. We rank characters in
decreasing order of frequency, and for each rank, we compute the average
percentage of this rank’s mention across all books. Figure 10 shows the
resulting plot, which reveals that the distribution of the percentage of
character mentions as a function of rank is similar to the Zipf’s law and
Benford’s law distributions Benford (1938).

Figure 10: Mean percentage of the number of mentions of the top 9 most
frequent characters for each character rank (across 45,939 books with at least
9 characters), with Benford’s law and Zipf’s law distributions plotted for
comparison

#### Ratio of top 2 characters’ mentions:

For all the books in the combined corpus, we compute the ratio of number of
mentions of the most frequent character to the second most frequent character.
For a large majority of books, this ratio is less than 10. However, the
remaining books, i.e. the outliers reveal an interesting insight: Books with
unusually high ratios feature a main protagonist who dominates most of the
narrative. A notable example of an outlier is Robinson Crusoe, told to the
children (67.75), understandably because it is a tale of a man castaway alone
on a deserted island.

#### Protagonist’s gender over time:

For books from the HathiTrust dataset, the year of publication is available as
metadata. We sort these books by year of publication and divide them into ten
equal subsets. Figure 11 shows the percentage of books with male protagonists
in each subset, as a function of time. The period from 1851 to 1886 shows a
sharp decline in the percentage of male protagonists. This time period
features many books from prominent female authors, including Margaret
Oliphant, Martha Finley, and May Agnes Fleming.

Figure 11: Percentage of books with male protagonists by year of publication

#### POS Correlations:

Figure 12 shows the pairwise Pearson correlations of eight parts of speech.
Interestingly, the parts of speech form two distinct groups inversely
correlated with each other. Intuitively, noun-adjective and verb-adverb are
pairs with positive correlations. However, these pairs are in different
groups, indicating that books lean towards being either more action-oriented
or description-oriented.

Figure 12: Pairwise correlations of POS tags: The parts of speech form two
distinct groups inversely correlated with each other.

##  6 Conclusion

We present an end-to-end system for NLP-driven annotation of literary works,
along with a standard XML annotation format. Additionally, we also present a
resource of annotated books in this standard format, along with analysis. This
work will bridge the gap between computational analysis and linguistic
exploration, and enable easier processing of books for further downstream
tasks such as social network extraction, character embedding generation,
event/action prediction, knowledge graph embedding, and causality analysis.

## References

  * Bamman et al. (2014) David Bamman, Ted Underwood, and Noah A. Smith. 2014.  A Bayesian mixed effects model of literary character.  In _Proceedings of the 52nd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 370–379, Baltimore, Maryland. Association for Computational Linguistics.
  * Behnel et al. (2005) Stefan Behnel, Martijn Faassen, and Ian Bicking. 2005.  lxml: Xml and html with python.
  * Benford (1938) Frank Benford. 1938.  The law of anomalous numbers.  _Proceedings of the American philosophical society_ , pages 551–572.
  * Bird and Loper (2004) Steven Bird and Edward Loper. 2004.  NLTK: The natural language toolkit.  In _Proceedings of the ACL Interactive Poster and Demonstration Sessions_ , pages 214–217, Barcelona, Spain. Association for Computational Linguistics.
  * Brooke et al. (2015) Julian Brooke, Adam Hammond, and Graeme Hirst. 2015.  GutenTag: an NLP-driven tool for digital humanities research in the Project Gutenberg corpus.  In _Proceedings of the Fourth Workshop on Computational Linguistics for Literature_ , pages 42–47, Denver, Colorado, USA. Association for Computational Linguistics.
  * Coleman and Liau (1975) Meri Coleman and Ta Lin Liau. 1975.  A computer readability formula designed for machine scoring.  _Journal of Applied Psychology_ , 60(2):283.
  * Dale and Chall (1948) Edgar Dale and Jeanne S Chall. 1948.  A formula for predicting readability: Instructions.  _Educational research bulletin_ , pages 37–54.
  * Davis et al. (2003) Peter T Davis, David K Elson, and Judith L Klavans. 2003.  Methods for Precise Named Entity Matching in Digital Collections.  In _2003 Joint Conference on Digital Libraries, 2003. Proceedings._ , pages 125–127. IEEE.
  * Dekker et al. (2019) Niels Dekker, Tobias Kuhn, and Marieke van Erp. 2019.  Evaluating named entity recognition tools for extracting social networks from novels.  _PeerJ Computer Science_ , 5:e189.
  * Egbert and Mahlberg (2020) Jesse Egbert and Michaela Mahlberg. 2020.  Fiction–one register or two?: Speech and narration in novels.  _Register Studies_ , 2(1):72–101.
  * Elson et al. (2010) David Elson, Nicholas Dames, and Kathleen McKeown. 2010.  Extracting social networks from literary fiction.  In _Proceedings of the 48th Annual Meeting of the Association for Computational Linguistics_ , pages 138–147, Uppsala, Sweden. Association for Computational Linguistics.
  * Groza and Corde (2015) Adrian Groza and Lidia Corde. 2015.  Information retrieval in folktales using natural language processing.  In _2015 IEEE International Conference on Intelligent Computer Communication and Processing (ICCP)_ , pages 59–66. IEEE.
  * (13) HathiTrust Digital Library.  https://www.hathitrust.org/.
  * Honnibal and Montani (2017) Matthew Honnibal and Ines Montani. 2017.  spaCy 2: Natural language understanding with Bloom embeddings, convolutional neural networks and incremental parsing.  To appear.
  * Jahan and Finlayson (2019) Labiba Jahan and Mark Finlayson. 2019.  Character identification refined: A proposal.  In _Proceedings of the First Workshop on Narrative Understanding_ , pages 12–18, Minneapolis, Minnesota. Association for Computational Linguistics.
  * Kim et al. (2021) Allen Kim, Charuta Pethe, Naoya Inoue, and Steve Skiena. 2021.  Cleaning dirty books: Post-OCR processing for previously scanned texts.  In _Findings of the Association for Computational Linguistics: EMNLP 2021_ , pages 4217–4226, Punta Cana, Dominican Republic. Association for Computational Linguistics.
  * Kincaid et al. (1975) J Peter Kincaid, Robert P Fishburne Jr, Richard L Rogers, and Brad S Chissom. 1975\.  Derivation of new readability formulas (automated readability index, fog count and flesch reading ease formula) for navy enlisted personnel.  Technical report, Naval Technical Training Command Millington TN Research Branch.
  * Le and Mikolov (2014) Quoc Le and Tomas Mikolov. 2014.  Distributed Representations of Sentences and Documents.  In _International Conference on Machine Learning_ , pages 1188–1196. PMLR.
  * Manning et al. (2014) Christopher Manning, Mihai Surdeanu, John Bauer, Jenny Finkel, Steven Bethard, and David McClosky. 2014.  The Stanford CoreNLP natural language processing toolkit.  In _Proceedings of 52nd Annual Meeting of the Association for Computational Linguistics: System Demonstrations_ , pages 55–60, Baltimore, Maryland. Association for Computational Linguistics.
  * Mc Laughlin (1969) G Harry Mc Laughlin. 1969.  Smog grading-a new readability formula.  _Journal of reading_ , 12(8):639–646.
  * McConnaughey et al. (2017) Lara McConnaughey, Jennifer Dai, and David Bamman. 2017.  The labeled segmentation of printed books.  In _Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing_ , pages 737–747, Copenhagen, Denmark. Association for Computational Linguistics.
  * Pethe et al. (2020) Charuta Pethe, Allen Kim, and Steve Skiena. 2020.  Chapter Captor: Text Segmentation in Novels.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP)_ , pages 8373–8383, Online. Association for Computational Linguistics.
  * Qi et al. (2020) Peng Qi, Yuhao Zhang, Yuhui Zhang, Jason Bolton, and Christopher D. Manning. 2020\.  Stanza: A python natural language processing toolkit for many human languages.  In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics: System Demonstrations_ , pages 101–108, Online. Association for Computational Linguistics.
  * Rae et al. (2019) Jack W Rae, Anna Potapenko, Siddhant M Jayakumar, Chloe Hillier, and Timothy P Lillicrap. 2019.  Compressive transformers for long-range sequence modelling.  _arXiv preprint_.
  * Underwood (2016) Ted Underwood. 2016.  Metadata for English-language Literature in HathiTrust Digital Library Beyond 1923.  https://github.com/tedunderwood/hathimetadata.
  * Vala et al. (2016) Hardik Vala, Stefan Dimitrov, David Jurgens, Andrew Piper, and Derek Ruths. 2016\.  Annotating characters in literary corpora: A scheme, the CHARLES tool, and an annotated novel.  In _Proceedings of the Tenth International Conference on Language Resources and Evaluation (LREC’16)_ , pages 184–189, Portorož, Slovenia. European Language Resources Association (ELRA).
  * Vala et al. (2015) Hardik Vala, David Jurgens, Andrew Piper, and Derek Ruths. 2015.  Mr. bennet, his coachman, and the archbishop walk into a bar but only one of them gets recognized: On the difficulty of detecting characters in literary texts.  In _Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing_ , pages 769–774, Lisbon, Portugal. Association for Computational Linguistics.