# A black lens in bubble of nothing

Shinya Tomizawa<EMAIL_ADDRESS>Ryotaku Suzuki  sryotaku@toyota-
ti.ac.jp Mathematical Physics Laboratory, Toyota Technological Institute
Hisakata 2-12-1, Nagoya 468-8511, Japan

###### Abstract

Applying the inverse scattering method to static and bi-axisymmetric Einstein
equations, we construct a non-rotating black lens inside bubble of nothing
whose horizon is topologically lens space $L(n,1)=S^{3}/{\mathbb{Z}}_{n}$.
Using this solution, we discuss whether a static black lens can be in
equilibrium by the force balance between the expansion and gravitational
attraction.

###### pacs:

04.50.+h 04.70.Bw

††preprint: TTI-MATHPHYS-15

##  I Introduction

The studies on higher-dimensional black hole solutions to Einstein’s equations
have played important roles in the microscopic derivation of black hole
entropy Strominger:1996sh  and in the fundamental research on the scenario of
large extra dimensions Argyres:1998qn  through black hole production in an
accelerator. The recent developments in solution-generation techniques enable
us to find various exact solutions of higher-dimensional black holes, our
understanding of them is still not enough. For example, according to the
topology theorem for five-dimensional black holes Galloway:2005mf ; Cai:2001su
; Hollands:2007aj ; Hollands:2010qy , the topology of the spatial cross
section of the event horizon must be either sphere $S^{3}$, ring $S^{1}\times
S^{2}$ or lens spaces $L(p,q)$, if the spacetime with commuting two rotational
Killing vector fields and a timelike Killing vector field is asymptotically
flat. As for the first two topologies, the exact solutions to vacuum
Einstein’s equations Tangherlini:1963bw ; Myers:1986un ; Emparan:2001wn ;
Pomeransky:2006bd  have already been found. For lens space topology, however,
it has been difficult to find a regular vacuum solution since the resultant
solutions have naked singularities.

The inverse scattering method (ISM) is perhaps one of the most powerful tools
to obtain exact solutions of Einstein equations with $(D-2)$ Killing
isometries, where $D$ is a spacetime dimension. In particular, combined with
the concept of the rod structure Harmark:2004rm , this method has succeeded to
derive five-dimensional vacuum black hole solutions. The first example of the
construction of black hole solutions by the ISM is the re-derivation of the
five-dimensional Myers-Perry black hole solution Pomeransky:2005sj . Next, the
black ring with $S^{2}$-rotation was re-derived Tomizawa:2005wv  by the ISM
(this was first derived in Refs. Mishima:2005id ; Figueras:2005zp ), but it
turned out that the generation of the black ring with $S^{1}$ rotation has a
certain problem on how to choose the seed, since an easy choice of the seed
always results in the generation of a singular solution. The suitable seed to
derive the black ring with $S^{1}$-rotation was first considered in
Iguchi:2006rd ; Tomizawa:2006vp . Subsequently, the more general black ring
solution with both $S^{1}$ and $S^{2}$ rotations was constructed by Pomerasnky
and Sen’kov Pomeransky:2006bd .

Using the ISM, some authors attempted to construct asymptotically flat black
lens solutions to the five-dimensional vacuum Einstein equations. First,
Evslin Evslin:2008gx  attempted to construct a static black lens with the lens
space topology of $L(n^{2}+1,1)$ but found that curvature singularities cannot
be eliminated, whereas both conical and orbifold singularities can be removed.
Moreover, by using the ISM, Chen and Teo Chen:2008fa  constructed black hole
solutions with the horizon topology of $L(n,1)=S^{3}/{\mathbb{Z}_{n}}$ but
these solutions must have either conical singularities or naked curvature
singularities. Thus, the major obstacle in constructing a black lens solution
is always the existence of naked singularities. However, the sudden
breakthrough has come from supersymmetric solutions. Kunduri and Lucietti
Kunduri:2014kja  succeeded in the derivation of the first regular
supersymmetric solution of an asymptotically flat black lens with the horizon
topology of $L(2,1)=S^{3}/{\mathbb{Z}}_{2}$. This solution was further
generalized to the more general supersymmetric black lens with the horizon
topology $L(n,1)=S^{2}/{\mathbb{Z}}_{n}\ (n\geq 3)$ Tomizawa:2016kjh ;
Breunholder:2017ubu . Getting a useful clue from the work of Kundhuri and
Lucietti, Ref Tomizawa:2019acu  attempted to construct the vacuum solution of
the black lens with $L(2,1)$ without singularities but the solution has
unavoidable closed timelike curves (CTCs). Thereafter, Ref. Lucietti:2020phh
discuss nonexistence of vacuum black lenses.

Supersymmetric black lenses carry the mass, electric charge (saturating BPS
bound), two angular momenta and magnetic fluxes Kunduri:2014kja ;
Tomizawa:2006vp . As is discussed in Ref. Tomizawa:2006vp , there exists no
limit such that all the magnetic fluxes vanish. Therefore, as for the
supersymmetric solutions, the existence of the magnetic fluxes seem to play an
essential role in supporting the horizon of the black lens. In general,
however, it is not clear whether such magnetic fluxes necessarily need to
construct a black lens. Recently, a different type of solutions within a class
of generalized Weyl solutions, static black hole binaries and black rings in
expanding bubbles of nothing, was studied in Ref. Astorino:2022fge , although
so far equilibrium configuration of black holes in bubble had been studied in
the context of Kaluza-Klein theory Elvang:2002br ; Tomizawa:2007mz ;
Iguchi:2007xs . As is well-known, an asymptotically flat, static black ring
cannot be in equilibrium since the horizon collapse due to the self-
gravitational force. However, the black ring in Astorino:2022fge  is allowed
to be in static equilibrium by the balance between the expanding force of a
bubble and the gravitational force, so it has no conical singularities. This
solution leads a simple, interesting question to us. Is a non-rotating black
lens in bubble of nothing allowed to be in equilibrium? To study such a
solution may enable us to know what (except for magnetic fluxes) is needed to
obtain a regular black lens. Thus, the end of this paper is to investigate
whether expanding bubble of nothing admits the existence of a black lens in
equilibrium. In this paper, to derive such a solution, we apply the ISM to the
five-dimensional vacuum Einstein equations with staticity and bi-axisymmetry,
and construct a $1$-soliton solution by considering a static black ring inside
of bubble in Astorino:2022fge  as a seed solution. Note that our procedure in
the ISM is entirely the same as the work of Chen-Teo where the seed solution
is chosen as a static black ring, namely, the only different point is the seed
solution.

We organize the remaining part of this paper as follows. In the following
section II, under the assumptions of staticity and bi-axisymmetry, we present
a vacuum solution of a non-rotating black lens with the horizon topology
$L(n,1)$ in bubble of nothing as a $1$-soliton solution in five dimensions by
using the ISM. In section III, we impose the boundary conditions under such
that the spacetime has none of curvature, conical, and orbifold singularities
on the axis and horizon. In section IV, we further impose no closed timelike
curves (CTCs) in the domain of communication. In section V, we discuss whether
the non-rotating black lens in bubble of nothing indeed exist. In section VI,
we confirm the limit of our solution to the Chen-Teo static solution. In the
final section VII, we devote ourselves to the summary and discussion on our
results.

##  II  Static black ring in bubble of nothing as a seed solution

In general, the metric for a stationary and bi-axisymmetric spacetime can be
written in the canonical coordinates as

$\displaystyle
ds^{2}=g_{ab}dx^{a}dx^{b}+f(d\rho^{2}+dz^{2}),\quad(a,b=t,\phi,\psi),$ (1)

where $g_{ab}$ and $f$ depend on only $(\rho,z)$. The following constraint
condition must be satisfied

$\displaystyle{\rm det}(g_{ab})=-\rho^{2}.$ (2)

According to the procedure of Chen and Teo Chen:2008fa , we construct the
static black lens in bubble of nothing by the ISM, where the static black ring
as the seed solution is replaced with the black ring in bubble of nothing
Astorino:2022fge  (see Fig. 1 on the rod structure.).

Figure 1: Rod structure of the black ring inside bubble of nothing.

Therefore, let us start with the exact solution of the black ring in bubble of
nothing to the five-dimensional vacuum Einstein equations, whose metric is
given by

$\displaystyle G_{0}$ $\displaystyle=$ $\displaystyle{\rm diag\
}\left(-\rho^{2}\frac{\mu_{1}}{\mu_{0}\mu_{2}},\frac{\mu_{0}\mu_{2}}{\mu_{1}\mu_{3}},\mu_{3}\right),$
(3) $\displaystyle f_{0}$ $\displaystyle=$ $\displaystyle
C_{f}\frac{\mu_{3}W_{01}^{2}W_{03}W_{12}^{2}W_{23}}{W_{02}^{2}W_{13}W_{00}W_{11}W_{22}W_{33}},$
(4)

where for $i,j=0,1,2,3$,

$\displaystyle\mu_{i}$ $\displaystyle:=$
$\displaystyle\sqrt{\rho^{2}+(z-z_{i})^{2}}-(z-z_{i}),$ (5)
$\displaystyle\bar{\mu}_{i}$ $\displaystyle:=$
$\displaystyle-\frac{\rho^{2}}{\mu_{i}},$ (6) $\displaystyle W_{ij}$
$\displaystyle:=$ $\displaystyle\rho^{2}+\mu_{i}\mu_{j}.$ (7)

First, let us remove a trivial anti-soliton with the BZ vector $(0,0,1)$ at
$z=z_{3}$:

$\displaystyle g_{0}$ $\displaystyle=$ $\displaystyle{\rm diag\
}\left(1,1,-\frac{\bar{\mu}_{3}^{2}}{\rho^{2}}\right)G_{0}={\rm diag\
}\left(1,1,-\frac{\rho^{2}}{\mu_{3}^{2}}\right)G_{0},$ (8)

In turn, let us add back a non-trivial anti-soliton with $(0,-a,1)$ to this,
and then we can obtain a new $1$-soliton solution, a solution of a static
black lens in bubble of nothing,

$\displaystyle g_{1}$ $\displaystyle=$
$\displaystyle-\rho^{2}\frac{\mu_{1}}{\mu_{0}\mu_{2}}dt^{2}+\frac{\mu_{0}\mu_{2}(\mu_{1}W_{03}^{2}W_{23}^{2}+4a^{2}z_{3}^{2}\mu_{0}\mu_{2}\mu_{3}^{2}W_{13}^{2})}{\mu_{1}\mu_{3}(\mu_{1}W_{03}^{2}W_{23}^{2}-4a^{2}z_{3}^{2}\rho^{2}\mu_{0}\mu_{2}W_{13}^{2})}d\phi^{2}$
(9)
$\displaystyle-2az_{3}\frac{2\mu_{0}\mu_{2}W_{03}W_{13}W_{23}W_{33}}{\mu_{3}(\mu_{1}W_{03}^{2}W_{23}^{2}-4a^{2}z_{3}^{2}\rho^{2}\mu_{0}\mu_{2}W_{13}^{2})}d\phi
d\psi$
$\displaystyle+\frac{\mu_{1}\mu_{3}^{2}W_{03}^{2}W_{23}^{2}+4a^{2}z_{3}^{2}\rho^{4}\mu_{0}\mu_{2}W_{13}^{2}}{\mu_{3}(\mu_{1}W_{03}^{2}W_{23}^{2}-4a^{2}z_{3}^{2}\rho^{2}\mu_{0}\mu_{2}W_{13}^{2})}d\psi^{2},$

and

$\displaystyle f_{1}$ $\displaystyle=$ $\displaystyle
f_{0}\frac{\mu_{1}W_{03}^{2}W_{23}^{2}-4a^{2}z_{3}^{2}\rho^{2}\mu_{0}\mu_{2}W_{13}^{2}}{\mu_{1}W_{03}^{2}W_{23}^{2}}.$
(10)

It is easy to confirm that in the limit of $a\to 0$, this solution coincides
with the static black ring in bubble of nothing Astorino:2022fge . One should
note that $t,\psi$ is dimensionless and $\phi$ has the dimension of length. In
the following section, after an appropriate coordinate transformation, we will
impose the periodicity of $\phi,\psi$ so that conical singularities do not
exist on symmetry of axises.

##  III Boundary conditions on the rods

Figure 2: Rod structure of the black lens inside bubble of nothing.

In order to impose the appropriate boundary conditions so that the solution
has the rod structure in Fig. 2 and neither conical singularities nor orbifold
singularities, let us introduce new parameters $b:=z_{3}a$ and new coordinates
$(\phi^{\prime},\psi^{\prime})$ defined by

$\displaystyle\frac{\partial}{\partial\phi^{\prime}}=\sqrt{\frac{z_{30}}{z_{30}-2b^{2}}}\left(\frac{\partial}{\partial\phi}+\frac{b}{z_{30}}\frac{\partial}{\partial\psi}\right),\quad\frac{\partial}{\partial\psi^{\prime}}=\sqrt{\frac{z_{30}}{z_{30}-2b^{2}}}\left(\frac{\partial}{\partial\psi}+2b\frac{\partial}{\partial\phi}\right),$
(11)

and then the angular components are written as

$\displaystyle g_{\phi^{\prime}\phi^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{z_{30}}{z_{30}-2b^{2}}\left(g_{\phi\phi}+\frac{b^{2}}{z_{30}^{2}}g_{\psi\psi}+\frac{2b}{z_{30}}g_{\phi\psi}\right),$
(12) $\displaystyle g_{\psi^{\prime}\psi^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{z_{30}}{z_{30}-2b^{2}}(g_{\psi\psi}+4b^{2}g_{\phi\phi}+4bg_{\phi\psi}),$
(13) $\displaystyle g_{\phi^{\prime}\psi^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{z_{30}}{z_{30}-2b^{2}}\left[\left(1+\frac{2b^{2}}{z_{30}}\right)g_{\phi\psi}+2bg_{\phi\phi}+\frac{b}{z_{30}}g_{\psi\psi}\right],$
(14)

where it should be noted that the constraint condition (2) is preserved.

Focusing on the two-dimensional space $\Sigma=\\{(\rho,z)|\
\rho>0,-\infty<z<\infty\\}$, let us study the rod structure of the obtained
solution. The rod structure Emparan:2001wk ; Harmark:2004rm  enables us to
understand stationary and axisymmetric solutions (more precisely, solutions
with $(D-2)$ commuting Killing vectors) easily in a diagrammatic may. The
$z$-axis ($\rho=0$) of the metric, which corresponds to a fixed point set of a
certain Killing isometry, is decomposed into five parts, $I_{-}=\\{(\rho,z)\
|\ \rho=0,\ z<z_{0}\\}$, $I_{0}=\\{(\rho,z)\ |\ \rho=0,\ z_{0}<z<z_{1}\\}$,
$I_{1}=\\{(\rho,z)\ |\ \rho=0,\ z_{1}<z<z_{2}\\}$, $I_{2}=\\{(\rho,z)\ |\
\rho=0,\ z_{2}<z<z_{3}\\}$, $I_{+}=\\{(\rho,z)\ |\ \rho=0,\ z_{3}<z\\}$. Thus,
the boundary $\partial\Sigma$ of $\Sigma$ are composed of $I_{\pm},I_{i}\
(i=0,\ldots,2)$ and the asymptotic region $I_{\infty}=\\{(\rho,z)|\
\sqrt{\rho^{2}+z^{2}}\to\infty\ $ with $z/\sqrt{\rho^{2}+z^{2}}$ finite$\\}$.

Now, we impose the conditions on each rod so that the solution has the same
rod structure as Fig. 2 and has no conical singularities.

  * (1)

$I_{3}=\\{(\rho,z)\ |\ \rho=0,\ z>z_{3}\\}$:
The Killing vector $v_{3}:=(0,0,1)=\partial/\partial\psi^{\prime}$ vanishes.
The condition for the absence of conical singularities on $I_{3}$ is given by

$\displaystyle\lim_{\rho\to
0}\sqrt{\frac{\rho^{2}f_{1}}{g_{\psi^{\prime}\psi^{\prime}}}}=\frac{\Delta\psi^{\prime}}{2\pi}\Longleftrightarrow
C_{f}\frac{z_{30}-2b^{2}}{z_{30}}=\left(\frac{\Delta\psi^{\prime}}{2\pi}\right)^{2}$
(15)

for $z\in(z_{3},\infty)$. Hence, if we choose the periodicity of
$\psi^{\prime}$ as $\Delta\psi^{\prime}=2\pi$, the condition can be satisfied
on $I_{3}$ by setting

$\displaystyle C_{f}=\frac{z_{30}}{z_{30}-2b^{2}}.$ (16)

  * (2)

$I_{0}=\\{(\rho,z)\ |\ z_{0}<z<z_{1},\rho=0\\}$:
The Killing vector $v_{01}:=(0,1,0)=\partial/\partial\phi^{\prime}$ vanishes.
The condition for the absence of conical singularities on $I_{0}$ is given by

$\displaystyle\lim_{\rho\to
0}\sqrt{\frac{\rho^{2}f_{1}}{g_{\phi^{\prime}\phi^{\prime}}}}=\frac{\Delta\phi^{\prime}}{2\pi}\Longleftrightarrow\frac{2C_{f}z_{10}^{2}(z_{30}-2b^{2})}{z_{20}^{2}}=\left(\frac{\Delta\phi^{\prime}}{2\pi}\right)^{2}.$
(17)

  * (3)

$I_{2}=\\{(\rho,z)\ |\ z_{2}<z<z_{3},\rho=0\\}$:
The Killing vector
$v_{23}:=(0,1,\frac{bz_{21}}{z_{30}z_{32}-2b^{2}z_{31}})=\partial/\partial\tilde{\phi}^{\prime}$
vanishes. The conical-free condition is given by

$\displaystyle\lim_{\rho\to
0}\sqrt{\frac{\rho^{2}f_{1}}{g_{1ab}v_{23}^{a}v_{23}^{b}}}=\frac{\Delta\tilde{\phi}^{\prime}}{2\pi}\Longleftrightarrow\frac{2C_{f}(z_{30}z_{32}-2b^{2}z_{31})^{2}}{z_{32}z_{31}(z_{30}-2b^{2})}=\left(\frac{\Delta\tilde{\phi}^{\prime}}{2\pi}\right)^{2}.$
(18)

Since $\phi^{\prime}$ and $\tilde{\phi}^{\prime}$ have the scale of length, it
is useful to introduce angular coordinates $\varphi$ and $\tilde{\varphi}$
with $2\pi$ periodicity by $\varphi:=L\phi^{\prime}$ and
$\tilde{\varphi}:=L\tilde{\phi}^{\prime}$. Then, together with Eq. (16), the
conditions (17) and (18) can be written as,

$\displaystyle(\ref{eq:conical_01})$ $\displaystyle\Longleftrightarrow$
$\displaystyle\frac{2z_{10}^{2}z_{30}}{z_{20}^{2}}=L^{2}\left(\frac{\Delta\varphi}{2\pi}\right)^{2}$
(19) $\displaystyle(\ref{eq:conical_23})$ $\displaystyle\Longleftrightarrow$
$\displaystyle\frac{2z_{30}(z_{30}z_{32}-2b^{2}z_{31})^{2}}{z_{32}z_{31}(z_{30}-2b^{2})^{2}}=L^{2}\left(\frac{\Delta\tilde{\varphi}}{2\pi}\right)^{2}.$
(20)

Moreover, we impose the boundary condition on the parameters $(z_{i},b)$ so
that the spatial topology of the horizon is lens space
$L(n;1)=S^{3}/{\mathbb{Z}}_{n}$ ($n\in{\mathbb{N}}$). From the mathematical
discussion in Ref. Hollands:2007aj , this condition is represented by

$\displaystyle{\rm
det}(\bar{v}_{01},\bar{v}_{23})=n\Longleftrightarrow\frac{Lbz_{21}}{z_{30}z_{32}-2b^{2}z_{31}}=n,$
(21)

where $(\bar{v}_{01},\bar{v}_{23})=L(v_{01},v_{23})$.

(4) $I_{-}=\\{(\rho,z)\ |\ z<z_{0},\rho=0\\}$: The Killing vector
$v_{-}:=(1,0,0)=\partial/\partial t$ vanishes. This semi-infinite rod
corresponds to an accelerating horizon.

(5) $I_{1}=\\{(\rho,z)\ |\ z_{1}<z<z_{2},\rho=0\\}$: The Killing vector
$v_{1}:=(1,0,0)=\partial/\partial t$ vanishes. This finite rod corresponds to
an event horizon.

##  IV CTCs

We require absence of CTCs on $\Sigma\cup\partial\Sigma$. The necessary and
sufficient conditions to ensure that CTCs do not exist on
$\Sigma\cup\partial\Sigma$ is such that $g_{\phi\phi}$ and $g_{\psi\psi}$ (or
$g_{\phi^{\prime}\phi^{\prime}}$ and $g_{\psi^{\prime}\psi^{\prime}}$) become
nonnegative in the region: The condition for the absence of CTCs is given by

$\displaystyle\mu_{1}W_{03}^{2}W_{23}^{2}-4b^{2}\rho^{2}\mu_{0}\mu_{2}W_{13}^{2}>0,$
(22)

which imposes an upper bound for $b^{2}$ at each point

$\displaystyle
b^{2}<U(\rho,z):=\frac{\mu_{1}W_{03}^{2}W_{23}^{2}}{4\rho^{2}\mu_{0}\mu_{2}W_{13}^{2}}.$
(23)

Therefore, if the minimum $U_{min}$ of $U(\rho,z)$ exists on
$\Sigma\cup\partial\Sigma$ and $b^{2}<U_{min}$ holds, CTCs do not exist in the
region. To prove this, we show that the function $U(\rho,z)$ has a minimum at
$(\rho,z)=(0,z_{3})$ on the rod $I_{\pm},I_{i}\ (i=0,\ldots,2)$.

It is not difficult to show that the function $U(\rho,z)$ has a minimum not on
$\Sigma$ but on $\partial\Sigma$ . To see this, one should note that the norm
of the gradient can be written as

$\displaystyle(\partial_{\rho}U)^{2}+(\partial_{z}U)^{2}=\frac{\mu_{1}^{2}W_{03}^{4}W_{23}^{4}}{4\rho^{6}\mu_{0}^{2}\mu_{2}^{2}W_{00}W_{11}W_{22}W_{13}^{4}}$
$\displaystyle\qquad\times\left((\mu_{0}-\mu_{1}+\mu_{2})^{2}\rho^{4}+2\mu_{0}\mu_{2}(2\mu_{0}\mu_{2}-(\mu_{0}+\mu_{2})\mu_{1}+\mu_{1}^{2})\rho^{2}+\mu_{0}^{2}\mu_{1}^{2}\mu_{2}^{2}\right),$
(24)

where the first line is always positive for $\rho>0$, and the second line is
also positive since

$\displaystyle(\mu_{0}-\mu_{1}+\mu_{2})^{2}\rho^{4}+2\mu_{0}\mu_{2}(\mu_{1}^{2}-(\mu_{0}+\mu_{2})\mu_{1}+2\mu_{0}\mu_{2})\rho^{2}+\mu_{0}^{2}\mu_{1}^{2}\mu_{2}^{2}$
$\displaystyle=\left(\left(\mu_{0}-\mu_{1}+\mu_{2}\right)\rho^{2}+\frac{\mu_{0}\mu_{2}\left(\mu_{1}^{2}-(\mu_{0}+\mu_{2})\mu_{1}+2\mu_{0}\mu_{2}\right)}{\mu_{0}-\mu_{1}+\mu_{2}}\right)^{2}+\frac{4\mu_{0}^{3}\mu_{2}^{3}\left(\mu_{0}-\mu_{1}\right)\left(\mu_{1}-\mu_{2}\right)}{\left(\mu_{0}-\mu_{1}+\mu_{2}\right){}^{2}},$
(25)

where the positivity of the last term can be shown from

$\displaystyle(\mu_{0}-\mu_{1})(\mu_{1}-\mu_{2})$
$\displaystyle=\frac{z_{10}z_{21}(\mu_{1}+\mu_{0})(\mu_{2}+\mu_{1})}{(\sqrt{\rho^{2}+(z-z_{1})^{2}}+\sqrt{\rho^{2}+(z-z_{0})^{2}})(\sqrt{\rho^{2}+(z-z_{2})^{2}}+\sqrt{\rho^{2}+(z-z_{1})^{2}})}>0.$
(26)

Therefore, the gradient of a smooth function $U(\rho,z)$ cannot be zero on
$\Sigma$, which means that $U(\rho,z)$ must have a minimum not on $\Sigma$ but
on $\partial\Sigma$. Hence, in what follows, let us consider a minimum of
$U(\rho,z)$ on $\partial\Sigma$, which corresponds to $I_{i}\
(i=\pm,0,\ldots,2),I_{\infty}$.

First, let us consider the minimum of $U(0,z)$ on the rod $\rho=0$, i.e., on
$I_{i}\ (i=\pm,0,\ldots,2)$. On $I_{+}$, we have

$\displaystyle U(0,z)=\frac{(z-z_{0})(z-z_{2})}{2(z-z_{1})},\quad
U_{,z}(0,z)=\frac{(z-z_{1})^{2}+z_{10}z_{21}}{2(z-z_{1})^{2}}>0,$ (27)

and hence the monotonically increasing function $U(0,z)$ on $I_{+}$ has a
minimum at $z=z_{3}$, which is written as

$\displaystyle U(0,z_{3})=\frac{z_{30}z_{32}}{2z_{31}}.$ (28)

On the other hand, since on $I_{2}$

$\displaystyle
U(0,z)=\frac{(z-z_{1})z_{30}^{2}z_{32}^{2}}{2(z-z_{0})(z-z_{2})z_{31}^{2}},\quad
U_{,z}(0,z)=-\frac{[(z-z_{1})^{2}+z_{10}z_{21}]z_{30}^{2}z_{32}^{2}}{2(z-z_{0})^{2}(z-z_{2})^{2}z_{31}^{2}}<0,$
(29)

the monotonically decreasing function $U(0,z)$ on $I_{2}$ has a minimum at
$z=z_{3}$, and hence

$\displaystyle U(0,z)\geq U(0,z_{3}).$ (30)

Moreover, observing on $I_{0}$,

$\displaystyle U(0,z)$ $\displaystyle=$
$\displaystyle\frac{(z-z_{2})z_{30}^{2}}{2(z-z_{0})(z-z_{1})},$ (31)
$\displaystyle U_{,z}(0,z)$ $\displaystyle=$
$\displaystyle-\frac{[(z-z_{2})^{2}-z_{20}z_{21}]z_{30}^{2}}{2(z-z_{0})^{2}(z-z_{1})^{2}}\begin{cases}<0&(z_{0}<z<z_{2}-\sqrt{z_{20}z_{21}})\\\
>0&(z_{2}-\sqrt{z_{20}z_{21}}<z<z_{1}),\end{cases}$ (32)

we find that the function $U(0,z)$ on $I_{0}$ has a local minimum at
$z=z_{*}:=z_{2}-\sqrt{z_{20}z_{21}}$, and hence

$\displaystyle U(0,z)\geq
U(0,z_{*})=\frac{z_{30}^{2}}{2(z_{20}+z_{21}-2\sqrt{z_{20}z_{21}})},$ (33)

where we note that the ratio of these minima on $I_{0},I_{2},I_{+}$ is
computed as

$\displaystyle\frac{U(0,z_{*})}{U(0,z_{3})}=\left(1+\frac{z_{21}}{z_{32}}\right)\left(1+\frac{z_{32}}{z_{20}}\right)\left(1-\sqrt{\frac{z_{21}}{z_{20}}}\right)^{-2}>1.$
(34)

Furthermore, near $I_{-}$ and $I_{1}$, the function $U(\rho,z)$ behaves as,
respectively,

$\displaystyle
U(\rho,z)\simeq\frac{2(z_{0}-z)(z_{2}-z)(z_{3}-z)^{2}}{(z_{1}-z)\rho^{2}},\quad
U(\rho,z)\simeq\frac{2(z-z_{1})(z_{2}-z)(z_{3}-z)^{2}z_{30}^{2}}{(z-z_{0})z_{31}^{2}\rho^{2}},$
(35)

which implies $U(0,z)=\infty$ on $I_{-}$ and $I_{1}$. To summarize, we have
shown that the minimum of $U(0,z)$ on $I_{i}\ (i=\pm,0,\ldots,2)$ is given by
Eq. (28).

Next, let us consider the function $U(\rho,z)$ in the asymptotic region,
namely, on $I_{\infty}$. We can show that in the asymptotic region,
$U(\rho,z)$ behaves as

$\displaystyle
U(\rho,z)\simeq\frac{1}{1+\frac{z}{\sqrt{\rho^{2}+z^{2}}}}\sqrt{\rho^{2}+z^{2}}.$
(36)

Hence, the function $U(\rho,z)$ diverges on $I_{\infty}$, so it cannot have a
minimum on $I_{\infty}$.

Thus, we conclude that Eq. (28) is also a minimum on
$\Sigma\cup\partial\Sigma$ as well as on $\partial\Sigma$, and hence the
necessary and sufficient condition for the absence of CTCs is given by

$\displaystyle b^{2}<\frac{z_{30}z_{32}}{2z_{31}}.$ (37)

##  V On existence of solutions

From the discussion in Sec III and IV, we have shown that the absence of
conical/orbifold singularities, the black lens condition require

$\displaystyle\frac{2z_{10}^{2}z_{30}}{z_{20}^{2}}=L^{2},$ (38)
$\displaystyle\frac{2z_{30}(z_{32}z_{30}-2b^{2}z_{31})^{2}}{z_{32}z_{31}(z_{30}-2b^{2})^{2}}=L^{2},$
(39) $\displaystyle\frac{Lbz_{21}}{z_{30}z_{32}-2b^{2}z_{31}}=n,$ (40)

Now, to confirm whether there is really a parameter range such that all these
conditions can be satisfied, we reparameterize the rod interval
$z_{i,i-1}:=z_{i}-z_{i-1}\ (i=1,2,3)$ and the redefined BZ parameter $b$ as
follows

$\displaystyle z_{10}=\ell,\quad z_{21}:=x\,\ell,\quad z_{32}=y\,\ell,\quad
L=\sqrt{\ell}\,\hat{L},\quad b=\sqrt{\ell}\,\hat{b},$ (41)

where $\ell$ fixes the size of the bubble on $I_{0}$, $x$ and $y$ are the size
of the horizon and the distance between the horizon and the center which is
often called nut. All the dimensionless parameters but $n$,
$x,y,\ell,\hat{L},\hat{b}$, are assumed to be positive. The condition for
avoiding CTCs is now given by

$\displaystyle b^{2}-\frac{y(1+x+y)}{2(x+y)}<0.$ (42)

From Eq. (40), we have

$\displaystyle\hat{L}=\frac{n(y^{2}-2\tilde{b}^{2}(x+y)+y(1+x))}{\hat{b}\,x}.$
(43)

Eliminating $\hat{L}$ from Eqs. (38) and (39) in terms of Eq.(43), we obtain

$\displaystyle
0=2\hat{b}^{2}(x+y+1)\left\\{2n^{2}(x+1)^{2}y^{2}+2n^{2}(x+1)^{2}xy+x^{2}\right\\}$
$\displaystyle\qquad-4\hat{b}^{4}n^{2}(x+1)^{2}(x+y)^{2}-n^{2}(x+1)^{2}y^{2}(x+y+1)^{2},$
(44)

and

$\displaystyle 0=\left\\{y(x+y+1)-2\hat{b}^{2}(x+y)\right\\}^{2}$
$\displaystyle\times\left[4\hat{b}^{4}n^{2}y(x+y)-2\hat{b}^{2}(x+y+1)\left\\{2n^{2}y(x+y)+x^{2}\right\\}+n^{2}y(x+y)(x+y+1)^{2}\right].$
(45)

First, we consider the conical singularity free condition on $I_{2}$ in Eq.
(45), which admits three branches for $\hat{b}^{2}$

$\displaystyle\hat{b}^{2}$ $\displaystyle=$
$\displaystyle\hat{b}_{\pm}^{2}:=\frac{(x+y+1)\left(x^{2}+2n^{2}(x+y)y\pm
x\sqrt{x^{2}+4n^{2}xy+4n^{2}y^{2}}\right)}{4n^{2}(x+y)y},$ (46)
$\displaystyle\hat{b}^{2}$ $\displaystyle=$
$\displaystyle\hat{b}_{0}^{2}:=\frac{y(x+y+1)}{2(x+y)}.$ (47)

From Eq. (46), we can show

$\displaystyle\hat{b}_{\pm}^{2}-\frac{y(1+x+y)}{2(x+y)}=\frac{x(x+y+1)\left(x+2n^{2}y\pm\sqrt{x^{2}+4n^{2}xy+4n^{2}y^{2}}\right)}{4n^{2}(x+y)y}>0,$
(48)

where we note

$\displaystyle(x+2n^{2}y)^{2}-(x^{2}+4n^{2}xy+4n^{2}y^{2})=4n^{2}(n^{2}-1)y^{2}>0.$
(49)

Therefore, this shows that the nonexistence condition of CTCs, (42), cannot be
satisfied for any $x>0$ and $y>0$. On the other hand, substituting Eq. (47)
into Eq. (43), we can show

$\displaystyle\hat{L}=0,$ (50)

which cannot satisfy $\hat{L}>0$. Hence, the solution without the conical
singularity on $I_{2}$ cannot avoid CTCs.

Next, we consider the conical singularity free condition on $I_{0}$ in Eq.
(44), from which we can obtain two branches for $\hat{b}^{2}$,

$\displaystyle\hat{b}^{2}=\tilde{b}^{2}_{\pm}:=\frac{(x+y+1)\left\\{x^{2}+2n^{2}(x+1)^{2}(x+y)y\pm
x\sqrt{x^{2}+4n^{2}(x+1)^{2}(x+y)y}\right\\}}{4n^{2}(x+1)^{2}(x+y)^{2}},$ (51)

which leads to

$\displaystyle\tilde{b}_{\pm}^{2}-\frac{y(1+x+y)}{2(x+y)}=\frac{x(x+y+1)\left(x\pm\sqrt{x^{2}+4n^{2}(x+1)^{2}(x+y)y}\right)}{4n^{2}(x+1)^{2}(x+y)^{2}}.$
(52)

From these, we find that only the branch $\tilde{b}_{-}^{2}$ can satisfy the
nonexistence condition of CTCs (42).

In summary, if one imposes the absence of conical singularities on the whole
symmetry of axis $I_{0},I_{2},I_{+}$, the presence of CTCs cannot be avoided
around the center $(\rho,z)=(0,z_{3})$. However, if one imposes it only on
$I_{0}$ and  $I_{+}$, one can obtain the solutions without CTCs, in which the
horizon admits the lens space topology $L(n;1)$ for $n\geq 1$.

##  VI Consistency with Chen-Teo static black lens

Here, we confirm that our solution coincides with the asymptotically flat,
static black lens solution by Chen-Teo Chen:2008fa  in a certain scaling
limit, for which the following variables are used,

$\displaystyle z_{0}=-\lambda^{2},\quad z_{1}=-\lambda c\kappa^{2},\quad
z_{2}=\lambda c\kappa^{2},\quad z_{3}=\lambda\kappa^{2},\quad
L=\sqrt{2}\lambda\bar{L},\quad a=\frac{\bar{a}}{\sqrt{2}\kappa^{2}}.$ (53)

With the rescaled coordinates,

$\displaystyle\rho\to\lambda\bar{\rho},\quad z\to\lambda\bar{z},$ (54)

and the rescaled parameters,

$\displaystyle\bar{z}_{0}:=z_{1}/\lambda=-c\kappa^{2},\quad\bar{z}_{1}:=z_{2}/\lambda=c\kappa^{2},\quad\bar{z}_{1}:=z_{3}/\lambda=\kappa^{2}$
(55)

the limit $\lambda\to\infty$ pushes $z_{0}$ away to $-\infty$, and one can see
that the rod structure in $(\bar{\rho},\bar{z})$ recovers that of the static
black lens in Ref. Chen:2008fa . In the limit $\lambda\to\infty$, Eq. (38)
requires

$\displaystyle\bar{L}=1.$ (56)

and then, Eqs. (39) and (40) leads, respectively, to the nonexistence
condition of the conical singularities on
$\bar{z}\in(\bar{z}_{2},\bar{z}_{3})$ and the condition of horizon topology
$L(n;1)$ in Ref. Chen:2008fa ,

$\displaystyle\frac{(1-c-\bar{a}^{2}(1+c))^{2}}{(1-\bar{a}^{2})^{2}(1-c^{2})}=1,\quad\frac{2\bar{a}c}{1-c-\bar{a}^{2}(1+c)}=n,$
(57)

where $\bar{a}$ corresponds to $a$ in Ref. Chen:2008fa . Moreover, the limit
of the nonexistence condition for CTCs (37) can be written as

$\displaystyle\bar{a}^{2}<\frac{c-1}{c+1},$ (58)

which corresponds to the parameter region, “Region I”, in Ref. Chen:2008fa .
Here, it should be noted that the solution in “Region II” in Chen:2008fa ,
which admits naked singularities and CTCs, are excluded from our solution by
the nonexistence condition (37) of CTCs.

##  VII Summary and Discussions

In this paper, using the ISM for static and bi-axisymmetric Einstein
equations, we have constructed the non-rotating black lens inside a bubble of
nothing whose horizon is topologically lens space
$L(n,1)=S^{3}/{\mathbb{Z}}_{n}$. Our work is entirely the parallel to the work
of Chen-Teo Chen:2008fa , where the static black ring as a seed solution is
replaced with a static black ring in bubble of nothing Astorino:2022fge .
Using this solution, we have investigated whether a static black lens can be
in equilibrium by the force balance between the expansion and gravitational
attraction. If we require the absence of CTCs in the domain of outer
communication, the non-rotating black lens must have conical singularities
between the horizon and the center. It has been shown, however, that for black
lens, the existence of expanding bubble does not exclude conical singularities
and hence the two forces, the force of the bubble expansion and gravitational
attraction, cannot be in static equilibrium unlike for the black ring.

Refs. Chen:2008fa ; Tomizawa:2019acu  have argued whether a rotating black
lens can be in equilibrium by the balance between the gravitational force
(attraction) and the centrifugal force (repulsive force), and concluded that
it cannot be in equilibrium. The generalization of this rotating black lens
solution to one in expanding bubble may be an interesting issue, since whether
such a black lens without conical singularities exists depends on the balance
among the gravitational force, the centrifugal force and the expansion of the
bubble. Such a rotating black lens solution is expected to be two or more than
soliton solution. This deserves our future work.

###### Acknowledgements.

This work is supported by Toyota Technological Institute Fund for Research
Promotion A. RS was supported by JSPS KAKENHI Grant Number JP18K13541. ST was
supported by JSPS KAKENHI Grant Number 21K03560.

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complex hyperbolic lattices and (relative) strict hyperbolization]A note on complex hyperbolic lattices and strict hyperbolization

Kejia Zhu
Department of Mathematics
University of California at Riverside

[2020]20F65, 22E40 (primary), 32J27, 32J05, 	57N65 (secondary)

We study the connection between the fundamental groups of complex hyperbolic manifolds and those of spaces arising from the (relative) strict hyperbolization process due to Charney–Davis and Davis–Januszkiewicz–Weinberger. Viewing a non-uniform lattice \(\Gamma\) in \(\text{PU}(n,1)\) as a relatively hyperbolic group with respect to its cusp subgroups in the usual way, we show that when $n\geq 2$, \(\Gamma\) is not isomorphic to
any relatively hyperbolic group arising from the relative strict hyperbolization process, via work of Lafont–Ruffoni. We also prove that a uniform lattice in $\text{PU}(n,1)$ is not the fundamental group of a Charney-Davis strict hyperbolization when $n\geq 2$, assuming the initial complex satisfies some mild conditions.

§ INTRODUCTION

The strict hyperbolization introduced by Charney–Davis [5] gives rich examples of closed aspherical manifolds with hyperbolic fundamental groups. More generally, their hyperbolization procedure turns a finite simplicial complex $K$ into a strictly negatively curved piecewise hyperbolic finite cell complex $\mathcal H(K)$. There is a relative version, due to Charney–Davis [5] and Davis–Januszkiewicz–Weinberger [7], known as the relative strict hyperbolization [5] that constructs compact aspherical manifolds with prescribed boundary as follows. If $K$ is a finite simplicial complex with subcomplex $L$, we may associate to \((K,L)\) a topological space \(\mathcal{R}=\mathcal{R}(K,L)\) with boundary. It is known that the boundary of \(\mathcal{R}\) may be identified with \(L\) up to subdivision, and with this identification, the components of \(L\) \(\pi_1\)-inject \(\mathcal{R}\). Let $\cP_\mathcal{R}$ be a set of representatives of the conjugacy classes of the fundamental groups of the components of $L$, viewed as subgroups of $\pi_1(\mathcal R)$. Then the group $\pi_1(\mathcal R)$ is known to be relatively hyperbolic with respect to $\mathcal P_\mathcal{R}$. See [10] and [1] for details. It is generally unknown whether the spaces arising from the relative strict hyperbolization process are complex hyperbolic. In this note, we show the following result:

The process of strict hyperbolization of a compact homogeneous $n$-dimensional simplicial complex without boundary and the process of relative strict hyperbolization of a finite simplicial complex with subcomplex cannot produce complexes with fundamental groups isomorphic to complex hyperbolic lattices of (complex) dimension $n\ge2$, in particular, the process of (relative) strict hyperbolization above cannot produce complex hyperbolic manifolds of (complex) dimension at least $2$.

Recall, by definition, a complex hyperbolic lattice is a discrete subgroup of $\text{PU}(n,1)$ with finite quotient measure, and a complex hyperbolic manifold is the quotient space of the complex hyperbolic $n$-space $\Hy_{\mathbb C}^n$ by a torsion-free lattice in the $\text{PU}(n,1)$, so the proof of Theorem<ref> follows quickly from the Theorem <ref> and Theorem <ref>.

The proof for the relative case (Theorem <ref>) depends on the work of Lafont–Ruffoni. Lafont–Ruffoni proved the following Theorem <ref> regarding the properties that the relative hyperbolic group  \((\pi_1(\mathcal{R}), \mathcal{P}_\mathcal{R})\) would satisfy: Given a relatively hyperbolic group $(G,\mathcal{P})$ acting isometrically on a CAT(0) cube complex $X$, we define a property below that $(G,\mathcal{P})$ arises as \((\pi_1(\mathcal{R}), \mathcal{P}_\mathcal{R})\) would satisfy:

(1) $X/G$ is compact.

(2) Each $p\in \mathcal P$ acts elliptically on $X$.

(3) For each cube $C$ of $X$, $\text{Stab}_G(C)$ is either maximal parabolic, or else is full relatively quasi-convex, hyperbolic, and virtually compact special.

The relative hyperbolic group \((\pi_1(\mathcal{R}), \mathcal{P}_\mathcal{R})\) admits an action on a CAT(0) cube complex \(X\) isometrically satisfying the Property <ref>

Recall that if $(\Gamma,\cP)$ is a relatively hyperbolic group pair and $\Gamma_0\leq \Gamma$ has finite index then $(\Gamma_0,\cP_0)$ is also a relatively hyperbolic group pair, where $\cP_0$ is the set of representatives of the conjugacy classes of
\begin{equation}\label{eqn1}\{P^g\cap\Gamma_0|g\in \Gamma, P\in \mathcal P\}.\end{equation}
Suppose $(\Gamma,\cP)$ has Property <ref>, then it follows immediately that $(\Gamma_0,\cP_0)$ has Property <ref>. We will thus freely pass to finite index subgroups in the proof.

Note Property <ref> closely resembles those of a relative geometric action, introduced by Einstein and Groves ([8]). The difference lies in $(3)$, where relatively geometric actions require a cell stabilizer to be either finite or conjugate to a finite index subgroup of a peripheral subgroup. In [3], Bregman, Groves, and the author proved that \((\Gamma,\mathcal{P})\) does not admit a relative geometric action on a CAT(0) cube complex when \(\Gamma\) is a non-uniform lattice in \(\text{PU}(n,1)\), and \(\cP\) is a set of conjugacy class representatives for the cusp subgroups of \(\Gamma\). Recall we are interested in the chance that relatively strict hyperbolized objects from Lafont–Ruffoni can be complex hyperbolic, it is thus natural to consider whether such pairs can arise from the relative strict hyperbolization considered by Lafont–Ruffoni. The following theorem gives a negative answer.

When $n\ge2$, a non-uniform lattice of $\PU(n,1)$ as relatively hyperbolic groups (relative to the cusp subgroups), does not satisfy Property <ref>, in particular, it does not arise as $(\pi_1(\mathcal R(K, L)),\mathcal P_\mathcal{R})$.

Now given a uniform lattice $\Gamma$ in \(\text{PU}(n,1)\), by Selberg’s lemma, we can always pass it to a torsion-free finite index subgroup. Then it acts freely properly on the complex hyperbolic $n$-space $\HC^n$ and the quotient space is a compact Kähler manifold with negative curvature. Recall a Kähler group is the fundamental group of a compact Kähler manifold. In particular, $\Gamma$ is a Kähler group. Also recall a cubulated group is a group acting geometrically on a CAT(0) cube complex. Delzant–Gromov in [6][Corollary, p.52] showed that any Kähler group that admits a full system of convex cuts is commensurable to a surface group, where a convex cut is a codimension one quasi-convex subgroup. Recall a virtually special hyperbolic group is cubulated hyperbolic (see [19]-Lemma 7.14), in particular, it admits a full system of convex cuts,
we apply the result of Delzant–Gromov to the following result of Lafont and Ruffoni <cit.> to get the non-relative version of the Theorem <ref> (see
Theorem <ref>):

Let $n > 0$ and $K$ an \(n\)-dimensional simplicial complex which is compact, homogeneous, and without boundary. If $\mathcal H(K)$ is the Charney–Davis strict hyperbolization of $K$, then $\pi_1(\mathcal H(K))$ is Gromov hyperbolic and virtually compact special.

If \(K\) is a compact homogeneous $n$-dimensional simplicial complex without boundary, then \(\pi_1(\mathcal{H}(K))\) is not commensurable to a Kähler group if its cohomology dimension is strictly greater than $2$. In particular,  \(\pi_1(\mathcal{H}(K))\) is not isomorphic to a uniform lattice in $\PU(n,1)$ for $n\ge2$.

By Lafont and Ruffoni <cit.>, \(\pi_1(\mathcal{H}(K))\) is
Gromov hyperbolic and virtually special. Suppose \(\pi_1(\mathcal{H}(K))\) is commensurable to a Kähler group, then it has a finite index subgroup, say $G$, which is Kähler, Gromov hyperbolic and virtually special (recall a finite index subgroup of a Kähler is still Kähler). Now by [6][Corollary, p.52], $G$, thus \(\pi_1(\mathcal{H}(K))\), is commensurable to a surface group, whose cohomological dimension is $2$, so the cohomological dimension of \(\pi_1(\mathcal{H}(K))\) can not be strictly greater than $2$. In particular, \(\pi_1(\mathcal{H}(K))\) is not isomorphic to a uniform lattice $\Gamma<\PU(n,1)$ for $n\ge2$ because otherwise by Selberg's lemma, $\Gamma$ has a finite index torsion-free subgroup, which is Kähler and with cohomological dimension $2n\ge 4$, therefore \(\pi_1(\mathcal{H}(K))\) is commensurable to a Kähler group of cohomological dimension $2n\ge 4$, contradiction.

Belegradek in <cit.> showed that if $K$ is a finite simplicial complex, then $\mathcal H(K)$ is not homotopy equivalent to a compact  Kähler manifold of complex dimension $n\ge2$. The theorem <ref> gives a stronger result in the sense of fundamental groups by assuming $K$ satisfies some mild conditions. If the relative strict hyperbolization $\mathcal{R}(K,L)$ is an $n$-manifold, Belegradek in <cit.> proved that the interior of $\mathcal{R}(K,L)$ is not homeomorphic to an open set of a compact Kähler manifold of complex dimension $m\ge2$. The theorem <ref> directly deals with the fundamental group of the whole $\mathcal{R}(K,L)$ without assuming $\mathcal{R}(K,L)$ to be a manifold.

Outline: The remainder of this note is devoted to the proof of <ref>. In Section <ref>, we first introduce the Dehn filling and related definitions. Then we apply some techniques from Groves and Manning to prove a key technical result, i.e. the Theorem <ref>. In Section <ref>, we first review the Borel–Serre and Toroidal compactifications and construct three compact Eilenberg–MacLane spaces by filling cusps, whose fundamental groups are sitting in a factorization. In Section <ref>, we prove Theorem <ref> by applying the result of Delzant–Gromov to the factorization above to deduce a contraction if we assume $(\Gamma,\cP)$ satisfies Property <ref>.

Acknowledgments: The author would like to thank his advisor, Daniel Groves, for answering his questions. He also would like to thank Corey Bregman, Matthew Durham, Shaver Phagan, and Lorenzo Ruffoni for their helpful comments. In particular, the author would like to thank Corey Bregman for permitting the author to employ some methods and techniques from our joint work in an unpublished version of [3] which are not used in the final version with Daniel Groves.

§ RELATIVELY HYPERBOLIC DEHN FILLING

Dehn filling is a well-known procedure for changing hyperbolic structures in $3$-manifold topology (<cit.>). Mimicking the effect of the classical Dehn filling of a manifold on its fundamental group, Osin [17] and Groves and Manning [11] developed a generalized notion of Dehn filling, which is entirely group-theoretic and a-priori devoid of any geometric content. We now recall some basic definitions associated with this generalized notion of a Dehn filling of a group pair $(G,\mathcal{P})$, which are frequently used in the techniques developed by Groves and Manning [11]:

Given a group pair $(G,\mathcal P)$, where $\mathcal P = \{P_1, ..., P_m\}$ and we choose a normal subgroup $N_i$ for each peripheral group $P_i$. Define $\mathcal{N}:=\{N_i\}$. Let $K$ be the normal closure generated by the union $\bigcup N_i$.

The Dehn filling of $(G,\mathcal P)$ with respect to $\mathcal{N}$ is the pair $(\overline{G},\overline{\mathcal P})$ from the quotient $G\to G/K=:\overline{G}$, where $\bar{\mathcal P}$ is the set of images of $\mathcal P$ under this quotient. The elements of $\mathcal N$ are called the filling kernels. When we want to specify the filling kernels we will write $G(N_1,\ldots, N_m)$ for the quotient $\overline{G}$. The filling is peripherally finite if each normal subgroup $N_i$ has finite index in $P_i$. Given a subgroup $Q<G$, we call $G(N_1,\ldots,N_m)$ a $Q$-filling if for all $g \in G$, and $P_i\in \cP$,  $|Q \cap P_i^g|=\infty$ implies $N_i^g\subseteq Q$. If $\mc{Q}=\{Q_1,\ldots, Q_l\}$ is a family of subgroups, then $G(N_1,\ldots, N_m)$ is a $\mc{Q}$–filling if it is a $Q$–filling for every $Q\in \mc{Q}$. We say that a property $\mathcal X$ holds for all sufficiently long Dehn fillings of $(G,\mathcal P)$ if there is a finite subset $X\subset G\setminus\{1\}$ so that whenever $N_i \cap B =\emptyset$ for all $i$, the corresponding Dehn filling $G(N_1,..., N_n)$ has property $\mathcal X$.

A subgroup $G_1$ of $G$ is full if whenever $P$ is a parabolic subgroup of $G$ such that $|G_1\cap  P|$ is infinite, we have $G_1\cap P$ is a finite index subgroup of $P$.

Let $\cP_0=\{P_{0,1},\ldots,P_{0,r}\}$ be the collection of peripheral subgroups of \(\overline{\Gamma}_0\) induced by $\cP$ as in Equation<ref>. The following theorem is regarding Property <ref>. The proof of part (1), in particular, the filling, is motivated by the proof of Theorem A4 in the appendix of [10] by Groves and Manning.

Let $(\Gamma,\cP)$ be a relatively hyperbolic group with peripheral subgroups residually finite. If $(\Gamma,\mathcal P)$ admits an action on a CAT(0) cube complex $X$ satisfying Property <ref>. Then there is a finite index subgroup $\Gamma_0\leq \Gamma$ and a Dehn filling $\psi:\Gamma_0\rightarrow\overline{\Gamma}_0$ satisfying:

* $\overline{\Gamma}_0$ is hyperbolic and virtually special

* $K:=\ker\psi$ is the normal closure of the subgroup generated by the peripheral subgroups in $\mathcal{P}_0$, and \(K\cong \underset{g\in \overline \Gamma_0}{\text{\Large$\ast$}}(P_{0,1}*\cdots*P_{0,r})\), where the inner free product is over $\overline \Gamma_0$-orbits of elements of $\cP_0$.

For (1), the strategy is to construct a sufficiently long peripherally finite $\mathcal Q$-filling (see the second paragraph) and then apply the Corollary 6.6 of [12]. Suppose $(\Gamma,\cP)$ satisfies the Property <ref> on a CAT(0) cube complex $X$. Since $X/\Gamma$ is compact, there are only finitely many $\Gamma$-orbits of cubes in $X$. By assumption, the stabilizer of each cube is full relatively quasi-convex (recall maximal parabolic subgroups are naturally full relatively quasi-convex), and conversely, each element of $P_j\in \cP$ stabilizes some cube. Therefore, the hypotheses of Corollary 6.6 of [12] is satisfied by the action of $\Gamma$ on $X$.

Let $T_1,\ldots, T_k$ be representatives of the $\Gamma$-orbits of cubes in $X$, and let $\mathcal{Q}=\{Q_1,\ldots, Q_k\}$, where $Q_i$ is the finite index subgroup of the stabilizer of $T_i$ fixing $T_i$ pointwise. Now we construct a sufficiently long $\mathcal Q$-filling: for each $P_j\in\mathcal{P}$ there are finitely many infinite intersections $P_j\cap Q_i^g$ for $Q_i\in\mathcal{Q}, g\in G$; each such $P_j\cap Q_i^g$ is finite index in $\mathcal{P}$ (recall each $Q_i$ as a cell stabilizer is full relatively quasi-convex). Let $N$ be the normal core of the intersection of these $P_j\cap Q_i^g$, where the normal core here is the largest normal subgroup of $\Gamma$ that is contained in the intersection. Given any collection of finite index $\{N_j'\unlhd P_j\}$, the collection $\{N_j:=N_j'\cap N\}$ gives a peripherally finite $\mathcal{Q}'$–filling. Since each $P_j$ is residually finite, we are able to choose the $N_j'$ to avoid any finite subset of $G\setminus \{1\}$, so the $\mathcal{Q}'$–filling is also sufficiently long.

Now we apply the Corollary 6.6 of [12] to this sufficiently long $\mathcal Q'$-filling, which asserts that the quotient $\overline X=K\backslash X$ is a CAT(0) cube complex. Since the filling is peripherally finite, the peripheral subgroups of $(\overline{\Gamma},\overline{\cP})$ are finite, thus $\overline{\Gamma}$ is hyperbolic.

Observe that $\bar{X}/\overline{\Gamma}=(X/K)/(\Gamma/K)=X/\Gamma$ is compact, so $\overline{\Gamma}$ acts cocompactly on $\bar{X}=K\backslash X$. Also, note the stabilizers are either finite or quasi-convex and virtually special. So the Theorem D of [12] now asserts that $\overline{\Gamma}$ is virtually special, thus $\overline{\Gamma}$ is residually finite and virtually torsion-free. Let $\overline \Gamma_0$ be a finite index torsion-free normal subgroup of $\overline \Gamma$, in particular, $\overline{\Gamma}_0$ is hyperbolic and virtually special. This proves (1).

Define $\Gamma_0$ to be the preimage of $\overline{\Gamma}_0$ in the filling $K\to \Gamma\to \overline\Gamma=\Gamma(N_1,...,N_n)$, in particular $\Gamma_0$ is a finite index subgroup of $\Gamma$ that is hyperbolic relative to $\mathcal{P}_0$ defined as in Equation<ref>. Also note that $K$ is contained in $\Gamma_0$.

For (2), we first claim that each peripheral subgroup group $P_{0,i}$ of $\Gamma_0$ is conjugate to one of the filling kernels $N_j$: recall by Equation<ref>, each $P_{0,i}=\Gamma_0\cap P_j^g$ for some $P_j\in \cP$ and $g\in \Gamma$. Recall $N_j^g< K\leq \Gamma_0$ and $N_j^g$ is a finite index subgroup of $P_j^g$, so $N_j^g$ is a finite index subgroup of $\Gamma_0\cap P_j^g=P_{0,i}$.  On the other hand, by Theorem 1.1 of [17], we may assume that $P_j/N_j\to \overline{\Gamma}$ is injective, thus $P_j^g/N_j^g\to\overline{\Gamma}$ is also injective. Recall $\overline\Gamma_0<\overline\Gamma$ is torsion-free and $P_j^g/N_j^g$ is finite, so the injection implies that $ P_{0,i}=\Gamma_0\cap P_j^g\subset N_j^g$, it follows that $N_j^g= P_{0,i}$, as desiorange. In particular, $\llangle \bigcup_i P_{0,i}\rrangle=\llangle \bigcup_j N_{j}^g\rrangle$. Therefore,
$$K=\llangle \bigcup_i N_{i}\rrangle=\llangle \bigcup_i N_{i}^g\rrangle=\llangle \bigcup_i P_{0,i}\rrangle.$$
Now by the Theorem 4.8 of [13], for a sufficiently long Dehn filling, we have that
\[K\cong \underset{g\in \overline{\Gamma}}{\text{\Large$\ast$}}(N_1*\cdots*N_m)\]
Under the conjugation action of $\overline{\Gamma}_0$, we can rewrite this as

\begin{align*}K&\cong \underset{g\in \overline{\Gamma}_0}{\text{\Large$\ast$}}(*_{\bar{t}\in \overline{\Gamma}/\overline{\Gamma}_0} (N_1*\cdots*N_m))\\
&\cong \underset{g\in \overline{\Gamma}_0}{\text{\Large$\ast$}}(*_{t\in \Gamma/\Gamma_0} (N_1*\cdots*N_m))\\
&\cong \underset{g\in \overline \Gamma_0}{\text{\Large$\ast$}}(P_{0,1}*\cdots*P_{0,r}),
\end{align*}

where the last isomorphism follows from that each $P_{0,i}$ is a conjugate of some $N_j$ and when $t$ ranges over elements of $\Gamma/\Gamma_0$, the conjugates $N_j$ in the inner free product above range over peripheral subgroups $P_{0,i}$.

§ COMPACTIFICATION OF $M$

It is well-known that a non-uniform lattice $\Gamma$ in $\PU(n,1)$ is virtually torsion-free. So we can assume $\Gamma$ is torsion-free by passing to a finite-index subgroup if necessary. Therefore, the quotient space \(M:=\Gamma\backslash\Hy^n_\C\) is a non-compact hyperbolic complex manifold of finite volume with finitely many cusps.

§.§ The structure of cusps

Here we give a concise description of the cusps in $M$, for more details see <cit.>. Let $\{\mathcal{C}_1,\ldots, \mathcal{C}_r\}$ be the cusps of $M$. By the Margulis lemma, each cusp $\mathcal{C}_i$ of $M$ is homeomorphic to $\Sigma_i \times [0,\infty)$ where $\Sigma_i$ is an $(2n-1)$-dimensional manifold as the quotient of $\mc{H}_{2n-1}(\R)$, the $(2n-1)$-dimensional Heisenberg group, by the peripheral subgroup $P_i$ (see <cit.> for details). Then the corresponding cusp subgroup is isomorphic to $\pi_1(\Sigma_i)$. Each cusp corresponds to a conjugacy class of subgroups fixing a point on the boundary at infinity of \(\Hy^n_\C\). Farb in [9] proved that $\Gamma$ is hyperbolic relative to the collection of its cusp subgroups, which we denote by $\cP$.

There is a fiber sequence

$$S^1\to \Sigma_i\to \mathcal O_i,$$
where $\mathcal O_i$ is a Euclidean orbifold finitely coveorange by $2n-2$-dimensional torus (see <cit.> for a comprehensive explanation). Note in general $\mathcal O_i$ can be singular, thus this is not generally a locally trivial fibration. But in our case, since each $P_i$ is
torsion-free, $\Sigma_i$ is smooth and we freely pass to the torus cover to obtain an actual fiber bundle

\begin{equation}\label{circleseq}S^1\to \Sigma_i\to T_i,
\end{equation}

In the following three subsections, we will construct three different compact Eilenberg–MacLane spaces by compactifying $M$, which will be used in the Theorem <ref>:

§.§ Borel-Serre compactification

Borel and Serre in [4] constructed a compactification $M_0$ of $M$, which works in general for lattices in noncompact symmetric spaces. In our case, i.e. the rank 1 case and $\Gamma$ is torsion-free, the $M_0$ is a smooth manifold with boundary.

If we lift $\{\mathcal{C}_i\}$ to the universal cover $\Hy_{\mathbb C}^n$, the pre-image of $\sqcup_i\mathcal{C}_i$ is a  union of pairwise disjoint horoballs centeorange on the parabolic fixed points in $\partial_\infty\Hy_{\mathbb C}^n$, which is $\Gamma$-equivariant. The $\Gamma$-stabilizer of each parabolic fixed point conjugate to some $P_i$ that acts freely, geometrically on the corresponding horosphere with quotient $\Sigma_i$. The quotient of the complement of the interiors of these horoballs is therefore a compact aspherical manifold $M_0$ with boundary $\partial M_0=\sqcup_i\Sigma_i$. In particular, $M_0$ is an Eilenberg–MacLane space $K(\Gamma,1)$, thus $H_i(M_0)\cong H_i(\Gamma)$. The pair $(M_0,\partial M_0)$ is called the Borel–Serre compactification of $M$. Each peripheral subgroup $P_i$ is associated with a unique boundary component $\Sigma_i$ and vice versa. Thus, $M_0$ is obtained from $M$ by removing a neighborhood of the cusps, in particular, $M$ and $M_0$ are homotopy equivalent and thus $\pi_1(M_0)=\pi_1(M)=\Gamma$.

§.§ Toroidal compactification

There is another natural compactification of $M$ which fills in the cusps with the Euclidean orbifolds called Toroidal compactification. The Toroidal compactification of $M$, is given by filling each fiber in the circle bundle $S^1\to \Sigma_i\to T_i$ to a disk $\mathbb D^2$, resulting a $\mathbb D^2$-bundle over $T_i$:
$$\mathbb D^2\to B_i\to T_i,$$
where $B_i$ is the total space, which admits a deformation retraction to $T_i$. The resulting space is a Kähler manifold $\mathcal{T}(M)$ with boundary $T:=\sqcup T_i$. Moreover, $\partial B_i\cong \Sigma_i$.

Hummel–Schroeder (see [15]) showed that we can always pass to a finite cover of $M$ so that its Toroidal compactification is an aspherical compact Kähler manifold with a smooth boundary $T$. In the language of Dehn-filling, Toroidal compactification (after passing to a finite cover of $M$ if necessary) is given by collapsing the normal subgroup of each cusp subgroup corresponding to the $S^1$ in each circle bundle $S^1\to \Sigma_i\to T_i$. For a more comprehensive description of Toroidal compactification, see <cit.>.

For the following, we will assume that $\mathcal{T}(M)$ is an aspherical compact Kähler manifold with a smooth boundary $T$. In particular, $\mathcal{T}(M)$ is an Eilenberg–MacLane space $K(\mathcal{T}(M),1)$, thus $H_i(\mathcal{T}(M))\cong H_i(\pi_1(\mathcal{T}(M))$. Observe that $M_0\setminus \partial M_0$ is homeomrphic to $M$ and $M$ is homeomorphic to $\mathcal{T}(M)\setminus T$, which induces a map of relative pairs $f: (M_0, \partial M_0)\to (\mathcal{T}(M),T) $ that is a homeomorphism on the interior of $M_0$ and sends each boundary component $\Sigma_i\subset \partial M=\sqcup \Sigma_i$ to the corresponding boundary component $T_i\subset T=\sqcup_i T_i$ by the fibering in Equation<ref>.

The map $f\colon (M_0, \partial M_0)\rightarrow (\mathcal{T}(M),T) $ induces an isomorphism
\[f_*\colon H_{2n}(M_0,\partial M_0)\rightarrow H_{2n}(\mathcal{T}(M),T)\]

Since $\partial B_i\cong \Sigma_i$, removing the interior of $\sqcup_i B_i$ yields a manifold homeomorphic to $M_0$. By Excision theorem, we thus have $H_*(M_0,\partial M_0)\cong H_*(\mc{T}(M),\sqcup_i B_i)$. On the other hand, each $B_i$ is homotopy equivalent to $T_i$, so $H_*(M_0,\partial M_0)\cong H_*(\mc{T}(M),T)$. Restricted to $M_0\cong \mathcal{T}(M)\setminus \text{Int}(\sqcup_i B_i)$, this deformation retraction gives exactly the map $f$ defined above.

§.§ Collapsing the cusps

Let $\Gamma_0\leq \Gamma$ be the finite index subgroup whose existence is guaranteed by Theorem <ref>. Replacing $\Gamma$ with $\Gamma_0$, we may assume that there exists a quotient homomorphism $\varphi\colon \Gamma\rightarrow \overline{\Gamma}$ such that

* $\overline{\Gamma}$ is infinite, torsion-free hyperbolic, and virtually special, in particular, $\overline{\Gamma}$ cubulated hyperbolic.

* Let $\cP=\{P_1,\ldots, P_r\}$ the peripheral subgroups. Then $K=\ker\psi$ is the normal closure of the subgroup of $\Gamma$ generated by elements of $\cP$ and moreover $K$ can be written as a free product

\begin{equation}\label{eqn:freeproduct}K\cong \underset{g\in \overline{\Gamma}}{\text{\Large$\ast$}}(P_{1}*\cdots*P_{r})\end{equation}

Write $\partial M_0=\Sigma_1\sqcup\cdots \sqcup \Sigma_r$, where $\pi_1(\Sigma_i)=P_i$. Since $\Sigma_i$ is a quotient of the Heisenberg group $\mc{H}_{2n-1}(\R)$ by a torsion-free, cocompact subgroup, each $\Sigma_i$ is aspherical.

By construction, $\overline{\Gamma}$ is obtained from $\Gamma$ by killing the peripheral subgroups $P_i$. Now define

$$\mc{C}(M):=M_0\sslash\partial M_0$$

where the notation “$\sslash$" means that we collapse each component of $\partial M_0$ to a point resepctively. Let $p_i\in \mc{C}(M)$ be the image of $\Sigma_i$ and set $S=\{p_1,\ldots, p_r\}$. So the space obtained from $M_0$ by gluing on the cone $C(\Sigma_i)$ for each component $\Sigma_i\subset \partial M_0$ is homeomorphic to $\mathcal{C}(M)$.

[thick, orange](-.2,3.3) arc(-180:180:.345cm and .15cm);
,3.3) to[out=-20,in=200](.3,3.3);
[orange](0,3.29) to[out=20,in=160](.25,3.29);

[thick,orange](1.9,3.3) arc(-180:180:.325cm and .12cm);
,3.3) to[out=-20,in=200](2.42,3.3);
[orange](2.1,3.29) to[out=20,in=160](2.38,3.29);

[thick](-0.1,4) to[out=270,in=90]
(-1,1) to[out=270,in=180](1,0) to[out=0,in=270](3,1) to[out=90,in=270](2.5,4);
[thick](2,4) to[out=270,in=0](1,2.5) to[out=180,in=270](.5,4);
[thick](.2,1.5) to[out=-50,in=240](1.8,1.5);
[thick](.3,1.4) to[out=30,in=140](1.7,1.4);

[thick,blue](-0.1,4) arc(-180:180:.3cm and .15cm);
,4) to[out=-20,in=200](.35,4);
(0.08,3.99) to[out=20,in=160](.33,3.99);

[thick,blue](2,4) arc(-180:180:.25cm and .12cm);
,4) to[out=-20,in=200](2.4,4);
[blue](2.13,3.99) to[out=20,in=160](2.38,3.99);

at ((0.2,4.35)$\scriptstyle \Sigma_1$ ;
at ((2.25,4.325)$\scriptstyle \Sigma_2$ ;

Borel–Serre $M_0$, $\partial M_0=\Sigma_1\bigcup \Sigma_2$

[thick, orange](-.2,3.3) arc(-180:180:.345cm and .15cm);
,3.3) to[out=-20,in=200](.3,3.3);
[orange](0,3.29) to[out=20,in=160](.25,3.29);

[thick,orange](1.9,3.3) arc(-180:180:.325cm and .12cm);
,3.3) to[out=-20,in=200](2.42,3.3);
[orange](2.1,3.29) to[out=20,in=160](2.38,3.29);

[thick](-0.1,4) to[out=270,in=90]
(-1,1) to[out=270,in=180](1,0) to[out=0,in=270](3,1) to[out=90,in=270](2.5,4);
[thick](2,4) to[out=270,in=0](1,2.5) to[out=180,in=270](.5,4);
[thick](.2,1.5) to[out=-50,in=240](1.8,1.5);
[thick](.3,1.4) to[out=30,in=140](1.7,1.4);

[very thick, blue](-0.1,4) to[out=-20,in=200](.5,4);
[blue] (-0.1,4) circle (1pt);
[blue] (.5,4) circle (1pt);

[very thick, blue](2,4) to[out=-20,in=200](2.5,4);
[blue] (2,4) circle (1pt);
[blue] (2.5,4) circle (1pt);

at ((0.2,4.2)$\scriptstyle T_1$ ;
at ((2.25,4.2)$\scriptstyle T_2$ ;

Toroidal $\mathcal{T}(M)$, $T=T_1\bigcup T_2$

[thick, orange](-.2,3.3) arc(-180:180:.345cm and .15cm);
,3.3) to[out=-20,in=200](.3,3.3);
[orange](0,3.29) to[out=20,in=160](.25,3.29);

[thick,orange](1.9,3.3) arc(-180:180:.325cm and .12cm);
,3.3) to[out=-20,in=200](2.42,3.3);
[orange](2.1,3.29) to[out=20,in=160](2.38,3.29);

[thick](.2,4) to[out=240,in=80](-.2,3.3)  to[out=260,in=90](-1,1) to[out=270,in=180](1,0) to[out=0,in=270](3,1) to[out=90,in=275](2.55,3.3) to[out=95,in=300](2.25,4);
[thick](2.25,4) to[out=240,in=85](1.9,3.3) to[out=265,in=0](1,2.5) to[out=180,in=270](.490,3.3) to[out=100,in=290](.2,4);
[thick](.2,1.5) to[out=-50,in=240](1.8,1.5);
[thick](.3,1.4) to[out=30,in=140](1.7,1.4);
[blue] (.2,3.95) circle (1.5pt);
[blue] (2.25,3.95) circle (1.5pt);

at ((0.2,4.2)$\scriptstyle p_1$ ;
at ((2.25,4.2)$\scriptstyle p_2$ ;

Collapsing the cusps $\mathcal{C}(M)$, $S=\{p_1,p_2\}$

Schematic picture of the 3 compactifications of $M$.

The space $\mc{C}(M)$ is the Eilenberg–MacLane space $K(\overline{\Gamma},1)$.

Since a neighborhood of each $p_i$ is homeomorphic to the cone on $\Sigma_i$, it follows from van Kampen's theorem that $\pi_1(\mc{C}(M))=\overline{\Gamma}$. It remains to show that $\mc{C}(M)$ is aspherical.
Consider the short exact sequence
\[1\rightarrow K\rightarrow\Gamma\rightarrow\overline{\Gamma}\rightarrow 1.\]
Find the covering space of $M_0$ corresponding $K$, $\pi_K\colon M_K\rightarrow M$. Since $M_0$ is a deformation retraction of $M$ with the universal cover $\HC^n$, so $M_K$ can be seen as a $K(K,1)$. Observe that the universal cover of $\mc{C}(M)$, say $\widetilde{\mc{C}(M)}$, is obtained from $M_K$ by collapsing boundary components of $M_K$ to points separately. Since each cusp subgroup is contained in $K$, $\pi_K$ projects each boundary component of $M_K$ onto some boundary component of $M_0$. By Theorem <ref>-(3), $K\cong\underset{g\in \overline{\Gamma}}{\text{\Large$\ast$}}(P_{1}*\cdots*P_{r})$, also recall each $\Sigma_i$ is a $K(P_i,1)$, it follows that for $j\ge0$,

$$H_j(M_K)\cong H_j(K;\Z)\cong\bigoplus_{g\in \overline{\Gamma}}\left(H_j(P_1;\Z)\oplus\cdots\oplus H_*(P_r;\Z)\right)\cong\bigoplus_{\Sigma\subseteq \partial M_k}H_j(\Sigma).$$

Since $\widetilde{\mc{C}(M)}$ is simply-connected (and connected), by the Hurewicz theorem it is enough to show that $H_i(\widetilde{\mc{C}(M)})=0$ when $i\geq 2$. Let $\overline V$ be a neighborhood of $S$ which is evenly coveorange by the covering map $\pi\colon \widetilde{\mc{C}(M)}\rightarrow \mathcal{C}(M)$. Construct an open cover of $\widetilde{\mc{C}(M)}$, $\{U_i\}_{i=1}^2$, where $U_1:=p^{-1}(\mc{C}(M)\setminus S)$ and $U_2:=p^{-1}(\bar V)$. Note each component of $U_2$ is homeomorphic to the cone on some $\Sigma_i$. We choose $U_1$ and $U_2$ so that $U_1\cap U_2$ is a disjoint union of connected components. Each component is homeomorphic to $\Sigma\times (0,1)$ for some $\Sigma\subseteq \partial M_K$.
It follows that $H_i(U_1\cap U_2)\cong\bigoplus_{\Sigma\subset \partial M_K}H_p(\Sigma) $.

Now apply Mayer–Vietoris to the triad $(\widetilde{\mc{C}(M)},U_1,U_2)$, it gives a long exact sequence:

$$... \to H_{i}(U_1\cap U_2)\to H_i(U_1)\oplus H_i(U_2)\to H_i(\widetilde{\mc{C}(M)})\to H_{i-1}(U_1\cap U_2)\to H_{i-1}(U_1)\oplus H_{i-1}(U_2)\to...$$

Observe that each connected component of $U_2$ as a cone is contractible, thus $H_i(U_2)=0$ for $i>0$. On the other hand,
$H_i(U_1)\cong H_1(M_K)$ for all $i$ since $U_1$ is homeomorphic to $M_K$. Also recall that $H_i(U_1\cap U_2)\cong \bigoplus_{\Sigma\subset \partial M_K}H_i(\Sigma)$, so the long exact sequence becomes

$$...\to \bigoplus_{\Sigma\subseteq \partial M_K}H_{i}(\Sigma)\to H_i(M_K)\to H_i(\widetilde{\mc{C}(M)})\to \bigoplus_{\Sigma\subseteq \partial M_K}H_{i-1}(\Sigma)\to H_{i-1}(M_K)\to...$$

for all $i\geq 2$. In the long exact sequence above, since $H_j(M_K)\cong\bigoplus_{\Sigma\subseteq \partial M_k}H_j(\Sigma)$, it follows that $H_i(\widetilde{\mc{C}(M)})=0$ for all $i \geq 2$.

§ PROOF OF THE THEOREM <REF>

Recall Lemma <ref> says $H_*(\overline{\Gamma};\Z)\cong H_*(\mc{C}(M))$. Observe that $\mc{C}(M)$ is obtained from $M_0$ by collapsing boundary components to points separately. Denote such a quotient map by $q\colon M_0\rightarrow \mc{C}(M)$, which also induces a map of pairs $(M_0,\partial M_0)\to (\mathcal{C}(M),S)$.

The induced map $q_*\colon H_{2n}(M_0,\partial M_0)\rightarrow H_{2n}(\mathcal{C}(M),S)$ is an isomorphism.  In particular, $H_{2n}(\mc{C}(M))$ is isomorphic to $H_{2n}(M_0,\partial M_0)$ and $H_{2n}(\overline \Gamma)\cong\mathbb Z$.

Note $(M_0,\partial M_0)$ and $(\mathcal{C}(M),S)$ are good pairs, it follows from the Theorem 2.13 of [14]), the quotient maps $ M_0\to M_0/\partial M_0$
and $\mc{C}(M)\to \mc{C}(M)/S$ induce the isomorphism
$$H_{2n}(M_0,\partial M_0)\to \tilde H_{2n}(M_0/\partial M_0,\{*\}),$$
where $\{*\}$ is the image of $\partial M_0$, and the isomorphism
$$H_{2n}(\mathcal{C}(M),S)\to\tilde H_{2n}(\mathcal{C}(M)/S,\{*\}),$$
for each $i\ge0$, where now $\{*\}$ represents the common image of $\{c_1,\ldots, c_r\}$ respectively. There is an obvious homeomorphism $ (M_0/\partial M_0,*)\to (\mc{C}(M)/S,*)$, which induces an isomorphism

$$H_{2n}(M_0/\partial M_0,*)\to H_{2n}(\mc{C}(M)/S,*).$$

Therefore, there is a commutative diagram

\[
\xymatrix{H_{2n}(M_0,\partial M_0)\ar[r]^{ q_*}\ar[d]_\cong &H_{2n}(\mc{C}(M),S)\ar[d]^{\cong}\\
H_{2n}(M_0/\partial M_0,*)\ar[r]_\cong & H_{2n}(\mc{C}(M)/S,*)

It follows that the induced map $q_*:H_{2n}(M_0,\partial M_0)\rightarrow H_{2n}(\mc{C}(M))$ is also an isomorphism.
Since the dimension of $\mc{C}(M)$ is $2n\geq 4$, the long exact sequence of the relative homology gives the isomorphism $H_{2n}(\mc{C}(M))\to H_{2n}(\mathcal{C}(M),S)$ induced by inclusion. Therefore,
$$H_{2n}(M_0,\partial M_0)\to H_{2n}(\mc{C}(M))$$
is an isomorphism. By Poincaré-Lefschetz duality, $H_{2n}(M_0,\partial M_0)\cong \Z$. Since the dimension of $\mc{C}(M)$ is $2n\ge 4$, from the long exact sequence of the pair $(\mc{C}(M),S)$, there is an isomorphism induced by inclusion: $H_{2n}(\mc{C}(M))\xrightarrow{\cong}H_{2n}(\mc{C}(M),S)$. By Lemma <ref>, $\mc{C}(M)$ is a $K(\overline\Gamma,1)$, it follows that $H_{2n}(\overline \Gamma)\cong\mathbb Z$.

We are now ready to prove Theorem <ref>.

Suppose $(\Gamma,\cP)$ admits Property <ref> on a CAT(0) cube complex $X$. The strategy is to find a hyperbolic Dehn filling of $\Gamma$ which is cubulated and which factors through $\pi_1(\mathcal{T}(M))$. Define $\mc{D}$ to be the image of $\mc{P}$ under homomorphism on $\pi_1$ induced by $f$, which is a collection of free abelian subgroups of $\pi_1(\mathcal{T}(M))$ such that the pair $(\pi_1(\mathcal{T}(M)),\mc{D})$ is relatively hyperbolic.

Note since $K$ is a normal subgroup containing each peripheral subgroup, and $\mc{T}(M)$ is obtained by collapsing  a subgroup of each peripheral subgroup, we have the desired factorization:

\begin{equation}\label{eqn:KeyDiagram}
\xymatrix{(\Gamma,\cP)\ar[rr]^\varphi\ar[dr]_{f_*}&&(\overline{\Gamma},1)\\&(\pi_1(\mathcal{T}(M)),\mathcal{D})\ar[ur]_\kappa}
\end{equation}

Since $\pi_1(\mathcal{T}(M))$ is Kähler and $\Gamma$ is cubulated hyperbolic, in particular, it admits a full system of convex cuts, by the corollary on page 52 of [6], the homomorphism $\kappa: \pi_1(\mathcal{T}(M))\to \overline{\Gamma}$ must virtually admit a further factorization

\begin{equation}\label{eqn:Surface Factorization}
\xymatrix{(\pi_1(\mathcal{T}(M))_1,\mc{D}_1)\ar[rr]^{\kappa_1}\ar[dr]&&(\overline{\Gamma},1)\\&(\overline{\pi_1(\mathcal{T}(M))},\overline{\mc{D}})\ar[ur]}
\end{equation}

where $\pi_1(\mathcal{T}(M))_1$ is a finite index subgroup of $\pi_1(\mathcal{T}(M))$, $\overline{\pi_1(\mathcal{T}(M))}$ is a surface group, and the image $\overline{\mc{D}}$ of $\mc{D}_1$ (where $\mc{D}_1$ is induced from $\mc{D}$) is a collection of quasi-convex subgroups of $\overline{\pi_1(\mathcal{T}(M))}$.

Since the virtual cohomological dimension of $\overline{\pi_1(\mathcal{T}(M))}$ is 2, but $\dim(\mc{T}(M))=2n\geq 4$, this implies that $(\kappa_1)_*\colon H_{2n}(\pi_1(\mathcal{T}(M))_1,\mc{D}_1)\rightarrow H_{2n}(\overline{\Gamma})$ must be  the zero map. On the other hand, recall $M_0$ is the $K(\Gamma,1)$,$\mathcal{T}(M)$ is the $K(\pi_1(\mathcal{T}(M)),1)$, and $\mathcal C(M)$ is the $K(\overline\Gamma,1)$, so by Lemma <ref>, $\varphi$ induces an isomorphism $\varphi_*\colon H_{2n}(\Gamma,\cP)\to H_{2n}(\overline\Gamma,\{1\})\cong H_{2n}(\overline\Gamma)\cong\Z$, which factors through the isomorphism  $f_*\colon H_{2n}(\Gamma,\cP)\rightarrow H_{2n}(\pi_1(\mathcal{T}(M)),\mc{D})$ by Proposition <ref>.

Therefore $\kappa_*\colon H_{2n}(\pi_1(\mathcal{T}(M)),\mc{D})\rightarrow H_{2n}(\overline{\Gamma},\{1\})$ has to be an isomorphism. In particular, the restriction map $(\kappa_1)_*=({\kappa|_{\pi_1(\mathcal{T}(M))_1}})_*\colon H_{2n}(\pi_1(\mathcal{T}(M))_1,\mc{D}_1)\rightarrow H_{2n}(\overline{\Gamma},\{1\})\cong\Z$ induced from the degree map is non-trivial, which is a contradiction.

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