# Constructing Bowditch boundaries of some relatively hyperbolic groups that
are homeomorphic to the $n$-dimensional Sierpiński carpet

Lucas H. R. de Souza

###### Abstract

In this paper we prove that if some relatively hyperbolic groups have Bowditch
boundary homeomorphic to the $n$-sphere, then they are also relatively
hyperbolic with respect to another set of parabolic subgroups and its Bowditch
boundary is homeomorphic to the $n-1$-dimensional Sierpiński carpet.

††Mathematics Subject Classification (2010). Primary: 20F65, 54D35; Secondary:
57M07, 54E45, 57S30.††Keywords: Blowup, Bowditch boundary, relatively
hyperbolic action, sphere, Sierpiński Carpet.

###### Contents

  1. 1 Preliminaries
    1. 1.1 The perspectivity property
      1. 1.1.1 Artin-Wraith glueings
      2. 1.1.2 Perspective compactifications
    2. 1.2 Blowups
      1. 1.2.1 Convergence actions
      2. 1.2.2 The blowup construction
  2. 2 Blowing up spheres

## Introduction

Tshishiku and Walsh [28] (with a correction in [29]) showed the following
topological characterization of the $1$-dimensional Sierpiński Carpet: if we
take a countable dense subset of the $2$-sphere and, for each point $p$ on
this set, we take a quotient of the closed $2$-disk to the $2$-sphere that
collapses the boundary to the point $p$, then the inverse limit of this family
of maps is homeomorphic to the $1$-dimensional Sierpiński Carpet. Later, we
generalized their result [23] for $n-1$-dimensional Sierpiński Carpets, with
$n\neq 4$. More classical characterizations of the Sierpiński Carpet are given
in [31] and [8] (where the same problem with the dimension appears).

In this paper we use such characterization of the Sierpiński Carpet in order
to show that for some classes of relatively hyperbolic pairs of groups with
their Bowditch boundaries homeomorphic to the $n$-sphere, we can refine their
peripheral structures in a way that their new Bowditch boundaries are
homeomorphic to the $n-1$-dimensional Sierpiński Carpet. Precisely, we prove
the following:

Theorem 2.8 Let $(G,\mathcal{P})$ be a relatively hyperbolic pair with
Bowditch boundary homeomorphic to $S^{n}$ and $\mathcal{Q}$ a proper subset of
$\mathcal{P}$ such that every group in $\mathcal{P}-\mathcal{Q}$ is
hyperbolic. Suppose that one of the following holds:

  1. 1.

$n=1$ or $2$.

  2. 2.

$n=3$ and every group in $\mathcal{P}-\mathcal{Q}$ is virtually torsion-free.

  3. 3.

$n=4$, the topological characterization of the $3$-dimensional Sierpiński
Carpet as an inverse limit of $4$-balls holds, every group
$Q\in\mathcal{P}-\mathcal{Q}$ is torsion-free, it acts properly
discontinuously, cocompactly and by isometries on a simply connected manifold
$M_{Q}$ with negative curvature and the Borel Conjecture holds for $Q$.

  4. 4.

$n\geqslant 5$ and every group in $\mathcal{P}-\mathcal{Q}$ has its hyperbolic
boundary homeomorphic to the $n-1$-sphere.

Then the Bowditch boundary of $G$ with respect to $\mathcal{Q}$ is
homeomorphic to the $n-1$-dimensional Sierpiński carpet.

The methods used here are similar to the one used by Tshishiku and Walsh to
show that the Dahmani boundary of a relatively hyperbolic $(G,\mathcal{P})$ is
homeomorphic to a $1$-dimensional Sierpiński carpet, given that its Bowditch
boundary is homeomorphic to the $2$-sphere (Theorem 1.1 of [28]). In fact, if
the Dahmani boundary is a blowup space (see Section 3.1.2 for the definition),
then their theorem has ours as a consequence, in the case of dimension $2$.

Let $p\in S^{n}$ be a bounded parabolic point. Its stabilizer $H$ acts on
$S^{n}-\\{p\\}$ properly discontinuously and cocompactly and then we are able
to have a glueing between $S^{n}-\\{p\\}$ and the hyperbolic boundary of $H$.
To prove the theorem, it is necessary to show that this glueing is
homeomorphic to the closed $n$-ball (and then we can use the characterization
of the Sierpiński Carpet described above). For different dimensions, we use
different tools to show it.

It is not much clear what really happens in dimension $4$, as it is also not
clear if the characterization of the $3$-dimensional Sierpiński Carpet holds.
Furthermore, as far as I know, the Borel Conjecture is not well understood in
dimension $4$, unlike any other dimension.

As consequences of the theorem above, we have the following:

  1. 1.

Another proof of the well known fact that if $(G,\mathcal{P})$ is a relatively
hyperbolic pair such that $\mathcal{P}$ is not empty and its Bowditch boundary
is homeomorphic to $S^{1}$, then $G$ is virtually free (Corollary 2.9).

  2. 2.

Let $(G,\mathcal{P})$ be a relatively hyperbolic pair with Bowditch boundary
homeomorphic to $S^{2}$ and such that $\mathcal{P}$ is not empty and every
group in $\mathcal{P}$ is hyperbolic. If the Kapovich-Kleiner Conjecture
holds, then $G$ is virtually the fundamental group of a compact hyperbolic
$3$-manifold with non-empty totally geodesic boundary (Corollary 2.10).

The Kapovich-Kleiner Conjecture says that if a hyperbolic group G has its
hyperbolic boundary homeomorphic to the $1$-dimensional Sierpinski carpet,
then $G$ is virtually the fundamental group of a compact hyperbolic
$3$-manifold with non-empty totally geodesic boundary [19].

######  Acknowledgements.

This paper contains part of my PhD thesis. It was written under the
advisorship of Victor Gerasimov, to whom I am grateful. I also would like to
thank Christopher Hruska, for his corrections on the proof of the main
theorem.

##  1 Preliminaries

###  1.1 The perspectivity property

####  1.1.1 Artin-Wraith glueings

This theory is covered in [25]. We need to use it in the following sections
just as a language to define some compactifications of spaces.

######  Definition 1.1.

Let $X$ and $Y$ be topological spaces and an application
$f:Closed(X)\rightarrow Closed(Y)$ such that $\forall A,B\in Closed(X),\
f(A\cup B)=f(A)\cup f(B)$ and $f(\emptyset)=\emptyset$ (we will say that such
map is admissible). We will give a topology for $X\dot{\cup}Y$. Let’s declare
as a closed set $A\subseteq X\dot{\cup}Y$ if $A\cap X\in Closed(X),\ A\cap
Y\in Closed(Y)$ and $f(A\cap X)\subseteq A$. Therefore, let’s denote by
$\tau_{f}$ the set of the complements of this closed sets and
$X+_{f}Y=(X\dot{\cup}Y,\tau_{f})$.

Every space $X$ that is a union of disjoint subspaces $A$ and $B$, where $A$
is open on $X$, can be recovered uniquely as an Artin-Wraith glueing of $A$
and $B$.

######  Definition 1.2.

Let $X+_{f}\\!Y$ and $Z+_{h}\\!W$ be topological spaces and continuous maps
$\psi:X\rightarrow Z$ and $\phi:Y\rightarrow W$. We define
$\psi+\phi:X+_{f}Y\rightarrow Z+_{h}W$ by $(\psi+\phi)(x)=\psi(x)$ if $x\in X$
and $\phi(x)$ if $x\in Y$. If $G$ is a group, $\psi:G\curvearrowright X$ and
$\phi:G\curvearrowright Y$, then we define $\psi+\phi:G\curvearrowright
X+_{f}Y$ by $(\psi+\phi)(g,x)=\psi(g,x)$ if $x\in X$ and $\phi(g,x)$ if $x\in
Y$.

######  Definition 1.3.

Let $X+_{f}W$, $Y$ and $Z$ be topological spaces, $\pi:Y\rightarrow X$ and
$\varpi:Z\rightarrow W$ be two continuous maps. We define the pullback of $f$
with respect to $\pi$ and $\varpi$ by
$f^{\ast}(A)=\varpi^{-1}(f(Cl_{X}\pi(A)))$.

######  Corollary 1.4.

Let $X+_{f}Y$, $X+_{g}Y$ be topological spaces. Then, the map
$id:X+_{f}Y\rightarrow X+_{g}Y$ is continuous if and only if $\forall A\in
Closed(X)$, $f(A)\subseteq g(A)$. $\square$

####  1.1.2 Perspective compactifications

Most of the following are covered in [22].

######  Definition 1.5.

Let $G$ be a group, $X$ and $Y$ Hausdorff topological spaces with $X$ locally
compact and $Y$ compact, $L:G\curvearrowright G$ the left multiplication
action, $\varphi:G\curvearrowright X$ a properly discontinuous cocompact
action, $\psi:G\curvearrowright Y$ an action by homeomorphisms and $K\subseteq
X$ a compact such that $\varphi(G,K)=X$. Define $\Pi_{K}:Closed(X)\rightarrow
Closed(G)$ as $\Pi_{K}(S)=\\{g\in G:\varphi(g,K)\cap S\neq\emptyset\\}$ and
$\Lambda_{K}:Closed(G)\rightarrow Closed(X)$ as $\Lambda_{K}(F)=\varphi(F,K)$.

######  Definition 1.6.

Let $G$ be a group, $X$ a Hausdorff locally compact space and
$\varphi:G\curvearrowright X$ a properly discontinuous and cocompact action.
We say that a Hausdorff compactification $\bar{X}$ of $X$ is perspective if
$\varphi$ extends continuously to an action on $\bar{X}$ and $\forall
u\in\mathcal{U},\ \forall K\subseteq X$ compact, $\\#\\{g\in
G:\varphi(g,K)\notin Small(u)\\}<\aleph_{0}$, where $\mathcal{U}$ is the only
uniform structure compatible with the topology of $\bar{X}$ (given by Theorem
1, $\S 4$, Chapter II of [3]). If $\bar{X}$ is metrizable, this condition is
equivalent to that $\forall\varepsilon>0$, $\forall K\subseteq X$ compact,
$\\#\\{g\in G:diam\ \varphi(g,K)>\varepsilon\\}<\aleph_{0}$ for any choice of
a metric compatible with the topology of $\bar{X}$.

We denote by $EPers_{0}(\varphi)$ the category whose objects are perspective
compactifications of $X$ and morphisms are continuous equivariant maps that
are the identity when restricted to $X$. If $G=X$ and the action is the left
multiplication action then we use the notation $EPers_{0}(G)$ for such
category.

######  Proposition 1.7.

(Theorem 3.2 of [25]) Let $K\subseteq X$ be a fundamental domain of $\varphi$.
The functor $\Pi:EPers(G)\rightarrow EPers(\varphi)$ that sends
$G+_{\partial}Y$ to $X+_{\partial_{\Pi_{K}}}Y,\ L+\psi:G\curvearrowright
G+_{\partial}Y$ to $\varphi+\psi:G\curvearrowright X+_{\partial_{\Pi_{K}}}Y$
and $id+\phi:G+_{\partial_{1}}Y_{1}\rightarrow G+_{\partial_{2}}Y_{2}$ to
$id+\phi:X+_{(\partial_{1})_{\Pi_{K}}}Y_{1}\rightarrow
X+_{(\partial_{2})_{\Pi_{K}}}Y_{2}$, is a isomorphism of categories.

Furthermore, its inverse is the functor $\Lambda:EPers(\varphi)\rightarrow
EPers(G)$ that sends $X+_{f}Y$ to $G+_{f_{\Lambda_{K}}}Y,\
\varphi+\psi:G\curvearrowright X+_{f}Y$ to $id+\psi:G\curvearrowright
G+_{f_{\Lambda_{K}}}Y$ and $id+\phi:X+_{f_{1}}Y_{1}\rightarrow
X+_{f_{2}}Y_{2}$ to $id+\phi:G+_{(f_{1})_{\Lambda_{K}}}Y_{1}\rightarrow
G+_{(f_{2})_{\Lambda_{K}}}Y_{2}$.

Since $\Pi$ and $\Lambda$ do not depend of the choice of the fundamental
domain (Propositions 3.18 and 3.20 of [25]), we denote $\partial_{\Pi_{K}}$ by
$\partial_{\Pi}$ and $f_{\Lambda_{K}}$ by $f_{\Lambda}$.

######  Proposition 1.8.

Let $G$ be a group and $\varphi_{i}:G\curvearrowright X_{i}$ properly
discontinuous and cocompact actions, $i=\\{1,2\\}$, where $X_{1}$ and $X_{2}$
are Hausdorff locally compact metric ANR spaces. If
$\Pi_{i}:EPers_{0}(G)\rightarrow EPers_{0}(\varphi_{i})$ and
$\Lambda_{i}:EPers_{0}(\varphi_{i})\rightarrow EPers_{0}(G)$ are the functors
induced by $\varphi_{i}$, then $\Pi_{2}\circ\Lambda_{1}$ and
$\Pi_{1}\circ\Lambda_{2}$ preserves $E\mathcal{Z}$-structures. $\square$

######  Remark.

Actually, this proposition says more information. It says that, in this case,
this correspondence agrees with Guilbault and Moran’s correspondence [16].

###  1.2 Blowups

####  1.2.1 Convergence actions

######  Definition 1.9.

Let $G$ be a group, $X$ a topological space and $\varphi:G\curvearrowright X$
an action by homeomorphisms. We say that $\varphi$ is properly discontinuous
if for every compact set $K\subseteq X$, the set $\\{g\in G:\varphi(g,K)\cap
K\neq\emptyset\\}$ is finite. We say that $\varphi$ is a covering action if
$\forall x\in X$, there exists an open neighbourhood $U$ of $x$ such that
$\forall g\in G-\\{1\\}$, $\varphi(g,U)\cap U=\emptyset$. We say that
$\varphi$ is cocompact if the quotient space $X/\varphi$ is compact.

######  Remark.

Suppose that $X$ is locally compact and Hausdorff. It is well known that if
$\varphi$ is a covering action, then the projection map $X\rightarrow
X/\varphi$ is a covering map, if $\varphi$ is properly discontinuous and free
(in special if $G$ is torsion-free), then $\varphi$ is a covering action and
if $\varphi$ is properly discontinuous then $X/\varphi$ is Hausdorff.

######  Definition 1.10.

Let $\varphi:G\curvearrowright X$ be an action by homeomorphisms. A point
$p\in X$ is bounded parabolic if the action $\varphi|_{Stab_{\varphi}p\times
X-\\{p\\}}:Stab_{\varphi}p\curvearrowright X-\\{p\\}$ is properly
discontinuous and cocompact.

######  Definition 1.11.

Let $G$ be a group, $X$ a Hausdorff compact space and
$\varphi:G\curvearrowright X$ an action by homeomorphisms. We say that
$\varphi$ has the convergence property if for every wandering net $\Phi$ (i.e.
a net such that two elements are always distinct) has a subnet $\Phi^{\prime}$
such that there exists $a,b\in X$ such that $\Phi^{\prime}|_{X-\\{b\\}}$
converges uniformly to $a$. We say that $\Phi^{\prime}$ is a collapsing net
with attracting point $a$ and repelling point $b$.

######  Remark.

Regardless the definition above allow us to consider these actions, we are not
considering $\varphi$ as a convergence action if the set $X$ has cardinality
$2$, unless that $G$ is virtually cyclic, $X=Ends(G)$ and $\varphi$ is the
action induced by the left multiplication action $G\curvearrowright G$.

######  Definition 1.12.

Let $G$ be a group acting properly discontinuously on a locally compact
Hausdorff space $X$ and acting on a Hausdorff compact space $Y$ with the
convergence property. The attractor-sum compactification of $X$ is the unique
compactification of $X$ with $Y$ as its remainder and such that the action of
$G$ on it (that extends both actions) has the convergence property. We denote
such compactification by $X+_{f_{c}}Y$ (where $c$ means that the action still
has the convergence property).

######  Remark.

The existence and uniqueness of the attractor-sum compactification is due to
Gerasimov (Proposition 8.3.1 of [13]).

######  Definition 1.13.

Let $\varphi:G\curvearrowright X$ be a convergence action and $p\in X$. We say
that $p$ is a conical point if there is an infinite set $K\subseteq G$ such
that $\forall q\neq p$, $Cl_{X}(\\{(\varphi(g,p),\varphi(g,q)):g\in
K\\})\cap\Delta X=\emptyset$.

######  Definition 1.14.

Let $G$ be a group, $X$ a metrizable compact space and
$\varphi:G\curvearrowright X$ an action by homeomorphisms. We say that
$\varphi$ is hyperbolic if it has the convergence property and every point of
$X$ is conical. In this case, we say that $G$ is hyperbolic and $X$ is its
hyperbolic boundary. Then we denote it by $X=\partial_{\infty}(G)$.

######  Remark.

Such space $X$, if it exists, is unique, up to unique equivariant
homeomorphism (Lemma 5.1 of [5]).

An equivalent definition is that the minimal convergence action on $X$ induces
a cocompact action on the space of distinct triples of $X$ (Theorem 8.1 of
[4]). The classical definitions of hyperbolic group and hyperbolic boundary
are also equivalent to that, i.e. $X$ is the hyperbolic boundary of a
$\delta$-hyperbolic proper geodesic metric space, where $G$ acts properly
discontinuously cocompactly and by isometries (Proposition 1.13 of [5] and
Theorem 0.1 of [4]. In this present text we need to use just the one stated in
the definition above.

######  Definition 1.15.

Let $G$ be a group, $X$ a Hausdorff compact space and
$\varphi:G\curvearrowright X$ a minimal action by homeomorphisms. We say that
$\varphi$ is relatively hyperbolic if it has the convergence property and its
limit set has only conical and bounded parabolic points. If $\mathcal{P}$ is a
representative set of conjugation classes of stabilizers of bounded parabolic
points of $X$, then we say that $G$ is relatively hyperbolic with respect to
$\mathcal{P}$ (or equivalently that $(G,\mathcal{P})$ is a relatively
hyperbolic pair). We say that $X$ is is the Bowditch boundary of the pair
$(G,\mathcal{P})$ and we denote it by $X=\partial_{B}(G,\mathcal{P})$.

######  Remark.

If $(G,\mathcal{P})$ is a relatively hyperbolic pair, then its Bowditch
boundary is uniquely defined, up to unique equivariant homeomorphism
(Corollary 6.1(e) of [14]).

An equivalent definition is that the minimal convergence action on $X$ induces
a cocompact action on the space of distinct pairs (1C of [30] and Main Theorem
of [12]).

####  1.2.2 The blowup construction

Next we show the general idea of the construction of the parabolic blowup. The
details are covered in [24]. However, this general idea and Propositions 1.16
and 1.17 are enough information for the subsequent section of this text.

Let $G$ be a group, $Z$ a compact Hausdorff space, $\varphi:G\curvearrowright
Z$ an action by homeomorphisms, $P\subseteq Z$ the set of bounded parabolic
points of $\varphi$, $P^{\prime}\subseteq P$ a representative set of orbits,
$\mathcal{C}=\\{C_{p}\\}_{p\in P^{\prime}}$ a family of compact Hausdorff
spaces, $H=\\{H_{p}\\}_{p\in P^{\prime}}$, with $H_{p}\subseteq G$ minimal
sets such that $1\in H_{p}$ and $Orb_{\varphi|_{H_{p}\times
Z}}p=Orb_{\varphi}p$ and $\\{Stab_{\varphi}p+_{\partial_{p}}C_{p}\\}_{p\in
P^{\prime}}$ a family of spaces with the equivariant perspective property with
actions $\eta=\\{\eta_{p}\\}_{p\in P^{\prime}}$, such that the action
$L_{p}+\eta_{p}:Stab_{\varphi}p\curvearrowright
Stab_{\varphi}p+_{\partial_{p}}C_{p}$ is by homeomorphisms (where $L_{p}$ is
the left multiplication action).

We define for $p\in P^{\prime}$ and $q\in Orb_{\varphi}p,\ C_{q}=C_{p}$ and
define $h_{p,q}$, for $q\in Orb_{\varphi}p$, the only element of $H_{p}$ such
that $\varphi(h_{p,q},p)=q$. Take, for $p\in P^{\prime}$ and $q\in
Orb_{\varphi}p$, the compact space $Stab_{\varphi}p+_{\partial_{q}}C_{q}$,
where $\partial_{q}=(\partial_{p})_{\ast}$, is the pullback for
$\tau_{h_{p,q}^{-1}}:Stab_{\varphi}q\rightarrow Stab_{\varphi}p$, the
conjugation map $\tau_{h_{p,q}^{-1}}(x)=h_{p,q}^{-1}xh_{p,q}$, and
$id_{C_{p}}$.

Since $P$ is the set of bounded parabolic points, we have that $\forall p\in
P$, $\varphi|_{Stab_{\varphi}p\times(Z-\\{p\\})}$ is proper and cocompact, so
we are able to take the space
$X_{p}=(Z-\\{p\\})+_{(\partial_{p})_{\Pi}}C_{p}$. We have also that the
quotient map $\pi_{p}:X_{p}\rightarrow Z$, such that $\pi_{p}|_{Z-\\{p\\}}$ is
the identity map and $\pi_{p}(C_{p})=p$, is continuous.

Let $X=\lim\limits_{\longleftarrow}(\\{X_{p}\\}_{p\in P},\\{\pi_{p}\\}_{p\in
P})$. The space $X$, together with the induced action of $G$ on it, is called
parabolic blowup of $(Z,\varphi)$ by $(\mathcal{C},\eta)$.

The next proposition characterizes blowup spaces:

######  Proposition 1.16.

(Proposition of [24]) Let $X$ and $Z$ be Hausdorff compact topological spaces,
$\psi:G\curvearrowright X$ and $\varphi:G\curvearrowright Z$ actions by
homeomorphisms, $P$ the set of bounded parabolic points of $Z$ and a
continuous surjective equivariant map $\pi:X\rightarrow Z$ such that $\forall
x\in Z-P$, $\\#\pi^{-1}(x)=1$. For $p\in P$, take
$X=W_{p}+_{\delta_{p}}\pi^{-1}(p)$, where $W_{p}=X-\pi^{-1}(p)$ and
$\delta_{p}$ is the only admissible map that gives the original topology. Let
also decompose $\psi|_{Stab_{\varphi}p\times X}$ as
$\psi_{p}^{1}+\psi_{p}^{2}$, with respect to the decomposition of the space.
Those are equivalents:

  1. 1.

The map $\pi$ is topologically quasiconvex and $\forall p\in P$, $X\in
EPers(\psi_{p}^{1})$.

  2. 2.

There exists an equivariant homeomorphism $T$ to a parabolic blowup $Y$ that
commutes the diagram:

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{T}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{\prime}}$$\textstyle{Z}$

Where $\pi^{\prime}$ is the projection map.

In this case the map $\pi$ is called a blowup map.

If the action $\psi$ have the convergence property, then the conditions in (1)
are not necessary, i.e. every surjective equivariant continuous map $\pi$ such
that $\forall x\in Z-P$, $\\#\pi^{-1}(x)=1$ is a blowup map (Theorem 3.1 of
[24]). In the special case that $\psi$ have the convergence property and
$\varphi$ is relatively hyperbolic, we have that every surjective equivariant
continuous map is a blowup map (Proposition 3.2 of [24]), since every limit
point that is not bounded parabolic is conical and preimage of a conical point
is a single point (Proposition 7.5.2 of [13]).

If the action $\varphi$ is relatively hyperbolic, then we have more
properties:

######  Proposition 1.17.

Let $G$ be a finitely generated group, $X$ and $Z$ Hausdorff compact spaces
$\psi:G\curvearrowright X$ and $\varphi:G\curvearrowright Z$ actions by
homeomorphisms, $P\subseteq Z$ be the set of bounded parabolic points and
$\pi:X\rightarrow Z$ be a blowup map. If $\varphi$ is minimal and relatively
hyperbolic, then we have:

  1. 1.

(Theorem 3.9 of [24]) If $\forall p\in P$, the action of $Stab_{\varphi}\ p$
on $\pi^{-1}(p)$ have the convergence property, then $\psi$ has the
convergence property.

  2. 2.

(Corollary 3.15 of [24]) If $\forall p\in P$, the action of $Stab_{\varphi}\
p$ on $\pi^{-1}(p)$ is minimal and hyperbolic, then $\psi$ is hyperbolic (and
then $X\cong\partial_{\infty}G$).

  3. 3.

(Corollary 3.17 of [24]) If $\forall p\in P$, the action of $Stab_{\varphi}\
p$ on $\pi^{-1}(p)$ is minimal and relatively hyperbolic with respect to the
parabolic subgroups $\mathcal{P}_{p}$, then $\psi$ is relatively hyperbolic
with respect to the set of parabolic subgroups
$\mathcal{P}=\bigcup\\{\mathcal{P}_{p}:p\in P\\}$ (and then
$X\cong\partial_{B}(G,\mathcal{P})$).

######  Remark.

Theorem 3.9 of [24] is stated for geometric compactification instead of
minimal relatively hyperbolic action, which is more general.

Corollary 3.17 of [24] has as immediate consequence a Drutur and Sapir Theorem
(Corollary 1.14 of [10]) which says that if $G$ is relatively hyperbolic with
respect to $\mathcal{P}$ and every group H in $\mathcal{P}$ is relatively
hyperbolic with respect to $\mathcal{P}_{H}$, then $G$ is relatively
hyperbolic with respect to $\bigcup_{H\in\mathcal{P}}\mathcal{P}_{H}$.

##  2 Blowing up spheres

######  Proposition 2.1.

(Tshishiku-Walsh [29] for $n=2$, de S. [23] for $n\neq 2,4$) Let $P$ be a
countable dense subset of the sphere $S^{n}$, with $n\neq 4$. Let, for every
$p\in P$, $\pi_{p}:D^{n}\rightarrow S^{n}$ be the map that collapses the
boundary of the $n$-ball to the point $p$. Then the limit space
$\lim\limits_{\longleftarrow}\\{D^{n},\pi_{p}\\}_{p\in P}$ is homeomorphic to
the $n-1$-dimensional Sierpiński carpet.

######  Remark.

It is not clear if this proposition holds for $n=4$.

######  Proposition 2.2.

Let $G$ be a group that has an action by homeomorphisms on $S^{n}$. Let
$P\subseteq S^{n}$ be the set of bounded parabolic points,
$P^{\prime}\subseteq P$ be a set of representatives and $\pi:X\rightarrow
S^{n}$ a blowup map. Let $Q^{\prime}\subseteq P^{\prime}$,
$Q=Orb_{\varphi}Q^{\prime}\subseteq P$ and suppose that $Q$ is countable and
dense on $S^{n}$. If $n=4$, then we also suppose that the last proposition
holds for dimension $4$. If $\forall p\in Q^{\prime}$, $X_{p}$ is the closed
ball $D^{n}$ with the boundary equal to the preimage of $p$ and $\forall p\in
P^{\prime}-Q^{\prime}$, $X_{p}$ is homeomorphic to $S^{n}$ and the map to
$S^{n}$ is a homeomorphism, then $X$ is homeomorphic to the Sierpiński carpet
of dimension $n-1$.

###### Proof.

It is immediate from the last proposition. ∎

Tshishiku and Walsh showed that if $(G,\mathcal{P})$ is a relatively
hyperbolic pair with its Bowditch boundary homeomorphic to the sphere $S^{2}$
and $X$ then the Dahmani boundary $\partial_{D}(G,\mathcal{P})$ is the
$1$-dimensional Sierpiński carpet (Theorem 1.1 of [28]). Our methods used next
are similar to theirs.

What we do next is to establish the necessary condition to prove the Theorem
2.8.

######  Lemma 2.3.

Let $G$ be a group, $H$ a finite index subgroup of $G$, $X$ a proper
$\delta$-hyperbolic space, $\varphi:G\curvearrowright X$ a properly
discontinuous cocompact action and $\varphi^{\prime}$ the restriction of
$\varphi$ to $H$. If $\varphi^{\prime}$ acts by isometries, then the induced
functor $\Pi$ sends the hyperbolic compactification of $G$ to the hyperbolic
compactification of $X$.

######  Remark.

$G$ is hyperbolic since $H$ is hyperbolic and it has a finite index.

###### Proof.

The functor $\Pi$ sends the hyperbolic compactification
$G+_{\partial_{\infty}}\partial_{\infty}(G)$ to a space
$W=X+_{f}\partial_{\infty}(G)$, and the action of $G$ on both has the
convergence property (by the construction of $\Pi$ and Proposition 8.3.1 of
[13]). It implies that the action of $H$ on $W$ has the convergence property.
But $\partial_{\infty}(G)=\partial_{\infty}(H)$ (since $H$ has finite index
over $G$), which implies that $W=X+_{f}\partial_{\infty}(G)$ is the only
compactification of $X$ such that the action of $H$ on it extends
$\varphi^{\prime}$ and it has the convergence property and the restriction to
the action of $H$ on the boundary is hyperbolic (see Definitions 1.12 and
1.14). However, the hyperbolic compactification of $X$ has also this property,
since $H$ acts on $X$ properly discontinuously, cocompactly and by isometries.
So $W$ is the hyperbolic compactification of $X$. ∎

######  Definition 2.4.

Let $M$ be a compact manifold without boundary. We say that $M$ is aspherical
if $\pi_{n}(M,p)$ is trivial for every $n>1$. We say that $M$ is topologically
rigid if for every compact manifold without boundary $N$, every homotopic
equivalence $M\rightarrow N$ is homotopic to a homeomorphism.

######  Conjecture 2.5.

Let $G$ be a group. We say that the Borel Conjecture holds for $G$ if every
compact aspherical manifold without boundary such that $\pi_{1}(M,p)=G$ is
topologically rigid.

######  Remark.

See more about the conjecture in [1].

######  Proposition 2.6.

(Ferry, [11]) If $X$ is an AR space and $Z\subseteq X$ is a $\mathcal{Z}$-set
such that $Z$ is homeomorphic to $S^{n-1}$ and $X-Z$ is homeomorphic to
$\mathbb{R}^{n}$, with $n\geqslant 5$, then $X$ is homeomorphic to the closed
$n$-ball.

######  Remark.

It is stated in page 479 of [2].

######  Proposition 2.7.

Let $G$ be a group $\varphi:G\curvearrowright S^{n}$ an action by
homeomorphisms, with $n\geqslant 5$, $P$ the set of bounded parabolic points,
$P^{\prime}\subseteq P$ a set of representatives, $Q^{\prime}\subseteq
P^{\prime}$, $Q=Orb_{\varphi}Q^{\prime}\subseteq P$, and
$\eta=\\{\eta_{p}\\}_{p\in P^{\prime}}$ a family of actions with
$L_{p}+\eta_{p}:Stab_{\varphi}p\curvearrowright
Stab_{\varphi}p+_{\delta_{p}}C_{p}$ such that
$Stab_{\varphi}p+_{\delta_{p}}C_{p}$ has the perspectivity property (where
$L_{p}$ is the left multiplication action), $C_{p}$ is a single point if
$p\notin Q^{\prime}$ and $C_{p}$ is homeomorphic to $S^{n-1}$ otherwise.
Suppose that $Q$ is countable and dense on $S^{n}$. If, for every $p\in
Q^{\prime}$, there exists a space $E_{p}+_{f_{p}}C_{p}$ that forms an
$E\mathcal{Z}$-structure for $Stab_{\varphi}p$ that it is in correspondence
with $Stab_{\varphi}p+_{\delta_{p}}C_{p}$ by the functor induced by the action
of $Stab_{\varphi}p$, then the blowup space $X$ with respect to $\eta$ is
homeomorphic to a $n-1$-dimensional Sierpiński carpet.

###### Proof.

Let $p\in P^{\prime}-Q^{\prime}$. Let’s name the actions
$\varphi_{p}=\varphi|_{Stab_{\varphi}p\times(S^{n}-\\{p\\})}$ and
$\psi_{p}+\eta_{p}:Stab_{\varphi}p\curvearrowright E_{p}+_{f_{p}}C_{p}$ (the
action given by the $E\mathcal{Z}$-structure). If
$\Pi_{p}:EPers(Stab_{\varphi}p)\rightarrow EPers(\varphi_{p})$ and
$\Lambda^{\prime}_{p}:EPers(\psi_{p})\rightarrow EPers(Stab_{\varphi}p)$ are
the functors induced by $\varphi_{p}$ and $\psi_{p}$, respectively, then
$\Pi_{p}\circ\Lambda^{\prime}_{p}$ sends $E\mathcal{Z}$-structures to
$E\mathcal{Z}$-structures (Proposition 1.8). So the space
$X_{p}=\Pi_{p}(Stab_{\varphi}p+_{\delta_{p}}C_{p})=(S^{n}-\\{p\\}))+_{f^{\prime}_{p}}C_{p}$,
where $f^{\prime}_{p}$ is some admissible map, form an
$E\mathcal{Z}$-structure of $Stab_{\varphi}p$ with $\mathcal{Z}$-boundary the
subspace $C_{p}$. But $S^{n}-\\{p\\}$ is homeomorphic to $\mathbb{R}^{n}$ and
$C_{p}$ is homeomorphic to $S^{n-1}$, which implies, by Ferry’s Theorem, that
$X_{p}$ is homeomorphic to the closed $n$-ball. Thus, it follows from
Proposition 2.2 that $X$ is a $n-1$-dimensional Sierpiński carpet. ∎

Then we have:

######  Theorem 2.8.

Let $(G,\mathcal{P})$ be a relatively hyperbolic pair with Bowditch boundary
homeomorphic to $S^{n}$ and $\mathcal{Q}$ a proper subset of $\mathcal{P}$
such that every group in $\mathcal{P}-\mathcal{Q}$ is hyperbolic. Suppose that
one of the following holds:

  1. 1.

$n=1$ or $2$.

  2. 2.

$n=3$ and every group in $\mathcal{P}-\mathcal{Q}$ is virtually torsion-free.

  3. 3.

$n=4$, the topological characterization of the $3$-dimensional Sierpiński
Carpet as an inverse limit of $4$-balls holds, every group
$Q\in\mathcal{P}-\mathcal{Q}$ is torsion-free, it acts properly
discontinuously, cocompactly and by isometries on a simply connected manifold
$M_{Q}$ with negatively bounded curvature and the Borel Conjecture holds for
$Q$.

  4. 4.

$n\geqslant 5$ and every group in $\mathcal{P}-\mathcal{Q}$ has hyperbolic
boundary homeomorphic to $S^{n-1}$.

Then $\partial_{B}(G,\mathcal{Q})$ is a $n-1$-dimensional Sierpiński carpet.

######  Remark.

$G$ is relatively hyperbolic with respect to $\mathcal{Q}$ by a theorem of
Drutu and Sapir (Corollary 1.14 of [10]).

If $n=1$, then every group $P\in\mathcal{P}$, stabilizer of a bounded
parabolic point $p$, acts properly discontinuously and cocompactly on
$\partial_{B}(G,\mathcal{P})-\\{p\\}$, which is homeomorphic to $\mathbb{R}$.
Then $P$ is two-ended, which implies that it is hyperbolic.

If the Dahmani boundary is equivalent with the parabolic blowup with a
Bowditch boundary as a base, then this result for $n=2$ is a special case of
Theorem 1.1 of [28].

###### Proof.

Let $X$ be the blowup space of $\partial_{B}(G,\mathcal{P})$ with fibers the
hyperbolic boundary of the groups in $\mathcal{P}-\mathcal{Q}$ and a single
point for each group in $\mathcal{Q}$. We have that
$X=\partial_{B}(G,\mathcal{Q})$, by Proposition 1.17.

Suppose $n=1$

Let $Q\in\mathcal{P}-\mathcal{Q}$ be the stabilizer of a bounded parabolic
point $q$. Since $X_{q}$ is a compactification of
$\partial_{B}(G,\mathcal{P})-\\{q\\}$ with $\partial_{\infty}(Q)$ as boundary,
$\partial_{B}(G,\mathcal{P})-\\{q\\}$ is homeomorphic to $\mathbb{R}$,
$\\#\partial_{\infty}(Q)=2$ and the only compactification of $\mathbb{R}$ with
two points is homeomorphic to the closed $1$-ball, it follows that $X_{q}$ is
homeomorphic to the $1$-ball. Thus, by Proposition 2.2 the space
$\partial_{B}(G,\mathcal{Q})$ is a $0$-dimensional Sierpiński carpet (i.e. a
Cantor set).

Suppose $n=2$.

Let $Q\in\mathcal{P}-\mathcal{Q}$ be the stabilizer of a bounded parabolic
point $q$. The action of $Q$ on $\partial_{B}(G,\mathcal{P})-\\{q\\}$ is
properly discontinuous, which implies that it has a finite kernel $K$. The
quotient $R=Q/K$ has an induced action on
$\partial_{B}(G,\mathcal{P})-\\{q\\}$ that is faithful. By Theorem 1.3 of
[17], there are finite index subgroups $Q^{\prime}<Q$ and $R^{\prime}<R$ such
that the restriction of the quotient map is an isomorphism between
$Q^{\prime}$ and $R^{\prime}$. Let
$\varphi_{q}:Q^{\prime}\curvearrowright\partial_{B}(G,\mathcal{P})-\\{q\\}$ be
the restriction of the action of $G$ on $\partial_{B}(G,\mathcal{P})$. By the
construction of $Q^{\prime}$, we have that $\varphi_{q}$ is a faithful action
(and it is cocompact, since $Q^{\prime}$ has finite index over $Q$).

By Theorem 0.3 of [9] all elements of $\mathcal{P}$ are virtual surface
groups. By taking a finite index subgroup, we can suppose that $Q^{\prime}$ is
a surface group (which is torsion-free). So
$(\partial_{B}(G,\mathcal{P})-\\{q\\})/\varphi_{q}$ is a compact $2$-manifold,
with fundamental group isomorphic to $Q^{\prime}$. Since $Q^{\prime}$ is
hyperbolic, then it is the fundamental group of a hyperbolic surface. By the
classification of compact surfaces and existence of Riemannian metrics of
constant curvature on compact surfaces, there exists a Riemannian metric on
$\partial_{B}(G,\mathcal{P})-\\{q\\}$ with constant negative curvature such
that the action $\varphi_{q}$ is by isometries. So
$\partial_{\infty}(Q^{\prime})=\partial_{\infty}(Q)$ is homeomorphic to
$S^{1}$ and the hyperbolic compactification of
$\partial_{B}(G,\mathcal{P})-\\{q\\}$ is a closed $2$-ball. But such
compactification is $X_{q}$ (of the construction of the blowup space) since
the functor induced by the action of $Q$ preserves hyperbolic
compactifications, by the Lemma 2.3.

Suppose $n=3$.

Let $Q\in\mathcal{P}-\mathcal{Q}$ be the stabilizer of a bounded parabolic
point $q$, $Q^{\prime}$ a finite index torsion-free subgroup of $Q$ and
$\varphi_{q}:Q^{\prime}\curvearrowright\partial_{B}(G,\mathcal{P})-\\{q\\}$
the restriction of the action of $G$ on $\partial_{B}(G,\mathcal{P})$. Since
$Q^{\prime}$ is torsion-free, $\varphi_{q}$ is a covering action (and it is
already cocompact).

Since $\partial_{B}(G,\mathcal{P})-\\{q\\}$ is connected and locally connected
and $\varphi_{q}$ is a covering action, there exists, by Proposition 6.3 of
[16], a metric $d$ on the space $\partial_{B}(G,\mathcal{P})-\\{q\\}$ such
that the metric space is proper and geodesic and $\varphi_{q}$ is an action by
isometries. Since $Q^{\prime}$ is hyperbolic, then
$\partial_{B}(G,\mathcal{P})-\\{q\\}$ is quasi-isometric to $Q^{\prime}$ and
then it is also metric hyperbolic.

The quotient space $(\partial_{B}(G,\mathcal{P})-\\{q\\})/\varphi_{q}$ is a
compact $3$-manifold, with fundamental group isomorphic to $Q^{\prime}$ and
irreducible (i.e. every locally flat $2$-sphere in it bounds a closed
$3$-ball), since its universal cover,
$\partial_{B}(G,\mathcal{P})-\\{q\\}\cong\mathbb{R}^{3}$, is irreducible
(Proposition 1.6 of [18] and the Generalized Schoenflies Theorem [7]). So, by
Theorem 4.1 of [2], $\partial_{\infty}(Q^{\prime})=\partial_{\infty}(Q)$ is
homeomorphic to $S^{2}$ and the hyperbolic compactification of
$\partial_{B}(G,\mathcal{P})-\\{q\\}$ is a closed $3$-ball. But such
compactification is $X_{q}$ since the functor induced by the action of $Q$
preserves hyperbolic compactifications (again, by the Lemma 2.3).

Suppose that $n=4$.

Let $Q\in\mathcal{P}-\mathcal{Q}$ be the stabilizer of a bounded parabolic
point $q$ and
$\varphi_{q}:Q\curvearrowright\partial_{B}(G,\mathcal{P})-\\{q\\}$ the
restriction of the action of $G$ on $\partial_{B}(G,\mathcal{P})$. Since $Q$
is torsion-free, $\varphi_{q}$ is a covering action (and it is already
cocompact).

Since $\varphi_{q}$ is a covering action, properly discontinuous and
cocompact, the quotient space
$(\partial_{B}(G,\mathcal{P})-\\{q\\})/\varphi_{q}$ is a compact $4$-manifold,
with fundamental group isomorphic to $Q$. Its universal cover
$\partial_{B}(G,\mathcal{P})-\\{q\\}\cong\mathbb{R}^{4}$ is contractible,
which implies that $(\partial_{B}(G,\mathcal{P})-\\{q\\})/\varphi_{q}$ is a
$K(Q,1)$ space. We also have that $M_{Q}/Q$ is a compact manifold and a
$K(Q,1)$ space. Since the Borel Conjecture holds for $Q$, the spaces
$(\partial_{B}(G,\mathcal{P})-\\{q\\})/Q$ and $M_{Q}/Q$ are homeomorphic. So
there exists a Riemannian metric on $\partial_{B}(G,\mathcal{P})-\\{q\\}$
which have negatively bounded curvature and the action $\varphi_{q}$ is by
isometries (we can do it in a way that $\partial_{B}(G,\mathcal{P})-\\{q\\}$
and $M_{Q}$ are isometric). Then the hyperbolic compactification of
$\partial_{B}(G,\mathcal{P})-\\{q\\}$ is a closed $4$-ball. But such
compactification is $X_{q}$ since the functor induced by the action of $Q$
preserves hyperbolic compactifications.

In any of those cases it follows from Proposition 2.2 that the space
$\partial_{B}(G,\mathcal{Q})$ is a $n-1$-dimensional Sierpiński carpet.

Suppose $n\geqslant 5$.

Let $Q\in\mathcal{P}-\mathcal{Q}$. Since it is hyperbolic, it has an
$E\mathcal{Z}$-structure given by its Rips complex $P_{d}$, for a big enough
$d>0$, compactified with its hyperbolic boundary (Theorem 1.2 of [2]). This
compactification is in correspondence with
$Q+_{\partial_{\infty}}\partial_{\infty}(Q)$ since the functor induced by the
action of $Q$ preserves hyperbolic compactifications. Then, by the last
proposition, the blowup $X$ is homeomorphic to the $n-1$-dimensional
Sierpiński carpet. But $X=\partial_{B}(G,\mathcal{Q})$. ∎

######  Remark.

Theorem 1.3 of [17] says only that there are finite index subgroups
$Q^{\prime}<Q$ and $R^{\prime}<R$ that are isomorphic. But this isomorphism
comes from the restriction of the quotient map, as seen on its proof. In the
statement of Theorem 1.3 of [17] it is missing the hypothesis of the map being
surjective, which is the case that we needed above.

######  Corollary 2.9.

Let $(G,\mathcal{P})$ be a relatively hyperbolic pair with Bowditch boundary
homeomorphic to $S^{1}$ and such that $\mathcal{P}$ is not empty. Then $G$ is
virtually free.

######  Remark.

This corollary is well known.

###### Proof.

Since every group in $\mathcal{P}$ is hyperbolic, then $G$ is hyperbolic
(Proposition 1.17). It follows from the last theorem that
$\partial_{\infty}(G)$ is a Cantor set. Thus, by 7.A.1 of [15], $G$ is
virtually free. ∎

Examples as the blowups given in Theorem 2.8 appear in every dimension in the
base:

######  Example.

Let $M$ be a compact real hyperbolic $n+1$-manifold with non-empty totally
geodesic boundary and $n\neq 4$. We have that its fundamental group
$G=\pi_{1}(M,p)$ is hyperbolic and $\partial_{\infty}(G)$ is a
$n-1$-dimensional Sierpiński carpet (Theorem 1.1 of [20]). Let
$\\{C_{i}\\}_{i\in\mathbb{N}}$ be the set of the $n-1$-spheres in
$\partial_{\infty}(\pi_{1}(M,p))$ that appears from the boundaries of the
balls removed from $S^{n}$ on the standard definition of the $n-1$-dimensional
Sierpiński carpet and $\mathcal{P}$ a set of conjugation classes of the
stabilizers of the sets $\\{C_{i}\\}_{i\in\mathbb{N}}$. Let $P\in\mathcal{P}$
be the stabilizer of the sphere $C$. We have that $C$ is $P$-invariant,
$\mathcal{P}$ is finite (Proposition 3.4 of [9]), $C$ is the limit set of $P$
(Lemma 3.5 of [9]) and $P$ is a quasi-convex subgroup of $G$ (Proposition 3.7
of [9]) (all proofs work in every dimension). Since the $n-1$-spheres in
$\\{C_{i}\\}_{i\in\mathbb{N}}$ are pairwise disjoint, we have that the family
$\mathcal{P}$ is almost malnormal (i.e. $\forall P,P^{\prime}\in\mathcal{P}$,
$\forall g\in G$, $P=gP^{\prime}g^{-1}$ or $P\cap
gP^{\prime}g^{-1}=\emptyset$). If $g\in G$, the limit set of $gPg^{-1}$ is the
limit set of $gP$ (using the perspectivity property) and the limit set of $gP$
is $gC$, which implies that the limit set of $gPg^{-1}$ is $gC$ and then
$gPg^{-1}=P$ if and only if $g\in P$ (in another words, $P$ is its own
normalizer). Then $(G,\mathcal{P})$ is a relatively hyperbolic pair (Theorem
7.11 of [6]) with Bowditch boundary given by the quotient of
$\partial_{\infty}(G)$ when we collapse all spheres in
$\\{C_{i}\\}_{i\in\mathbb{N}}$ (Main Theorem of [27]), which is a $n$-sphere
(Decomposition Theorem - Theorem 2 of [21]). So this quotient map
$\partial_{\infty}(G)\rightarrow\partial_{B}(G,\mathcal{P})$ is an example of
a blowup map.

In dimension $n=2$, the existence of examples of hyperbolic groups with
hyperbolic boundary homeomorphic to the $1$-dimensional Sierpiński carpet that
are not virtually the fundamental group of a compact hyperbolic $3$-manifold
with non-empty totally geodesic boundary is the negation of the Kapovich-
Kleiner Conjecture [19] and it is still open.

If the Kapovich-Kleiner Conjecture is true, then we have the following
proposition which is similar to the Relative Cannon Conjecture:

######  Corollary 2.10.

Let $(G,\mathcal{P})$ be a relatively hyperbolic pair with Bowditch boundary
homeomorphic to $S^{2}$ and such that $\mathcal{P}$ is not empty and every
group in $\mathcal{P}$ is hyperbolic. If the Kapovich-Kleiner Conjecture
holds, then $G$ is virtually the fundamental group of a compact hyperbolic
$3$-manifold with non-empty totally geodesic boundary. $\square$

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# Signatures of the black hole quantum atmosphere in nonlocal correlations

Adam Z. Kaczmarek<EMAIL_ADDRESS>Dominik Szczȩśniak
Department of Theoretical Physics, Faculty of Science and Technology, Jan
Długosz University in Czȩstochowa, 13/15 Armii Krajowej Ave., 42200
Czȩstochowa, Poland

###### Abstract

Recently, it was suggested that the Hawking radiation may originate not at the
event horizon but in the quantum region outside of it, known as the quantum
atmosphere. The present study attempts to explore this argument further by
assessing its role in shaping quantum correlations near a black hole. Herein,
these are conveniently captured within the geometric measure of nonlocality,
termed as the measurement-induced nonlocality, and found to exhibit signatures
of the atmosphere. In particular, a notable loss of correlations is observed
well outside the event horizon, coinciding with the peak of particles
radiation in the atmosphere region. Still, the correlations are shown to be
always finite therein and to continuously scale with not only the radiation
temperature but also with the horizon size. Hence, some characteristics of the
atmosphere appears to be detectable at the quantum correlations level,
providing novel insight and means to help verify the concept of interest.

††preprint: APS/123-QED

##  I Introduction

In the 1970s, the studies of quantum fields on a curved background have led to
the famous results by Hawking [1]. In his pioneering work, Hawking has shown
that black holes evaporate due to the particle creation. Although providing
groundbreaking insight into the behavior of gravity at the quantum level, the
Hawking radiation also raised some questions. One of the biggest and still
unresolved challenges in this regard is the information paradox, stating that
the evaporating black hole evolves from its initial to final state in
contradiction to the unitary time evolution of quantum mechanics [2]. In the
pursuit to unravel this inconstancy, it is crucial to ask, among other
questions, where exactly the radiation quanta originate? The original
investigations by Hawking and Unruh suggested that it arises from the quantum
excitations at the effective distance ($r$) very near the event horizon
($\Delta r=r-r_{H}\ll r_{H}$, where $r_{H}$ is the Hawking radius) [1, 3].
However, this viewpoint was recently contested by Giddings, advocating for the
notion of the source region ($r_{A}$) well outside the event horizon ($\Delta
r=r_{A}-r_{H}\sim r_{H}$), known as the quantum atmosphere [4].

The above claim by Giddings was initially supported by the estimates of the
effective size of an emitting body, as based on the Stefan-Boltzmann law, and
the simultaneous calculations of the Hawking quanta wavelength [4]. Later on,
the follow-up studies on the stress energy tensor [5, 6] the heuristic
gravitational Schwinger effect argument [5, 7] or the quantum correlations
across the event horizon [8] only backed up this idea. Interestingly, it was
also shown that the firewall phenomenon [9] is somewhat compatible with the
quantum atmosphere [10] and that the location of the latter can be well
estimated based on the thermal behavior of the radiation [11].

This intriguing concept of quantum atmosphere naturally calls for an even
deeper insight, not only in terms of the Hawking radiation origin but also the
resulting alternation of a black hole surrounding. An intuitive step in this
direction is to perform measure of some kind in the background of a black
hole. Due to the intrinsically quantum character of the Hawking radiation, the
inspection of quantum correlations in the vicinity of the event horizon
appears as a suitable approach for this task. In particular, it is expected
here that the distinct structure of Hawking quanta should have profound impact
on a quantum system, allowing to trace its behaviour under the new setting and
to observe potential signatures of the atmosphere.

In the present study, we explore quantum atmosphere from such a new
perspective. This is done by evoking the phenomenon of nonlocality, a
fundamental and potentially frame-independent property of any quantum system
or reality in general [12]. In a nutshell, the setup of two entangled parties
is considered here to be located near a black hole, allowing to capture
thermal characteristics of its atmosphere in analogy to the earlier studies on
quantum systems in the Schwarzschild or the Garfinkle-Horowitz-Strominger
space-times [13, 14]. In this context, the nonlocality itself is quantified
within the so-called measurement-induced nonlocality (MIN), a genuine
correlation measure between parts of the composite system [15]. The MIN is
well-suited for such considerations since it describes correlations in a broad
manner, going beyond the Bell theorem and allowing for nonlocality without
entanglement or the nonlocality without quantumness [16]. This is to say, the
assumed approach allows to conveniently probe the atmosphere region and to
interpret its role in shaping quantum correlations near a black hole. As a
result, the corresponding analysis is expected to provide vital contribution
to the field aimed at comprehending the nature of the Hawking radiation via
the correlation measures [17, 18, 13, 19, 20, 14].

The present study is organized as follows: in section II we present
description of the quantum systems of interest in the Schwarzschild space-
time, next in section III we define our setup and analytically outline the
measurement-induced nonlocality in the presence of the quantum atmosphere,
finally section IV gives our predictions on the thermal evolution of quantum
correlations in the Hartle-Hawking vacuum. To this end, the obtained results
are summarized by some pertinent conclusions.

##  II Dirac fields in the Schwarzschild space-time

In the present analysis, we choose the initial state to be of the fermionic
type. This allows us to be on the same footing with other recent studies on
quantum correlations in the relativistic setting, which frequently consider
Dirac particles [19, 21, 22, 20, 14]. In this regard, it is instructive to
note also that the fermionic modes are more resistant towards the Hawking
radiation than the bosonic ones and have non-zero value even for the radiation
temperature approaching infinity [13, 19, 14].

In the context of the above, the Dirac equation for the curved space-time is
first considered:

$\displaystyle(i\gamma^{a}e^{\mu}_{a}D_{\mu}-m)\psi=0,$ (1)

where $D_{\mu}=\partial_{\mu}-\frac{i}{4}\omega^{ab}_{\mu}\sigma_{ab}$,
$\sigma_{ab}=\frac{i}{2}\\{\gamma_{a},\gamma_{b}\\}$, $e^{\mu}_{a}$ is
vierbein and $\omega^{ab}_{\mu}$ denotes the spin connection. The positive
frequency solutions of Eq. (1) correspond to the regions $I$ and $II$ i.e.
outside and inside the event horizon ($r=r_{h}$), respectively. In order to
obtain a complete basis for the analytic modes with the positive energy, the
Kruskal coordinates are utilized to perform analytical continuation in
accordance to the Damour-Ruffini method [23, 5]. The resulting Dirac fields
are expanded in the appropriate Kruskal basis, as follows:

$\displaystyle\Psi$
$\displaystyle=\sum_{i}d\textbf{k}\frac{1}{\sqrt{2\cosh(\pi\omega_{i}/\kappa)}}$
$\displaystyle\times\Big{[}c^{I}_{\textbf{k}}\zeta^{I+}_{\textbf{k}}+c^{II}_{\textbf{k}}\zeta^{II+}_{\textbf{k}}+d^{I\dagger}_{\textbf{k}}\zeta^{I_{-}}_{\textbf{k}}+d^{II\dagger}_{\textbf{k}}\zeta^{II_{-}}_{\textbf{k}}\Big{]},$
(2)

where $c_{\textbf{k}}$ and $d_{\textbf{k}}^{\dagger}$ are subsequently the
creation and annihilation operators applied to the Kruskal vacuum [17]. Next,
by using the Bogoliubov transformation, it is possible to establish the
relation between operators in a black hole and the Kruskal space-times [24].
In particular, the vacuum and excited states of the black hole coordinates
correspond to the Kruskal two-mode squeezed states as:

$\displaystyle\ket{0_{\textbf{k}}}^{+}=\alpha\ket{0_{\textbf{k}}}_{I}^{+}\ket{0_{-\textbf{k}}}^{-}_{II}+\beta\ket{1_{\textbf{k}}}_{I}^{+}\ket{1_{-\textbf{k}}}^{-}_{II},$
$\displaystyle\ket{1_{\textbf{k}}}^{+}=\ket{1}^{+}_{I}\ket{0_{-\textbf{k}}}_{II}^{-},$
(3)

with the following Bogoliubov coefficients [25]:

$\displaystyle\alpha=\frac{1}{(e^{-\omega_{i}/T}+1)^{1/2}},\;\;\;\beta=\frac{1}{(e^{\omega_{i}/T}+1)^{1/2}},$
(4)

where $T$ denotes Hawking temperature of the emitted radiation. The above
reasoning can next directly link to the Giddings argument [4], by considering
a dimensionally reduced Schwarzschild black hole of line element
$ds^{2}=-f(r)dt^{2}+f(r)^{-1}dr^{2}$.

Figure 1: (A) The measurement-induced nonlocality of the physically accessible
quantum correlations ($\text{MIN}(\rho_{AB_{I}})$) and (B) the normalized
local temperature in the Hartle-Hawking vacuum ($T_{HH}/T_{H}$) as a function
of the normalized distance ($r/r_{H}$) for the selected values of the so-
called Hartle-Hawking constant ($D_{HH}$).

##  III Measurement-induced nonlocality in the Hartle-Hawking vacuum

To encompass how the origin of the Hawking quanta affects quantum systems in
the presence of the quantum atmosphere, the slightly modified version of the
well-established probing scenarios [26, 24, 17] is employed. In details, it is
assumed that Alice and Bob share maximally entangled Bell state of the form:

$\displaystyle\rho_{AB}=\ket{\phi^{+}}\bra{\phi^{+}}\;\;\;\text{for}\;\;\;\ket{\phi^{+}}=\frac{1}{\sqrt{2}}\big{(}\ket{00}+\ket{11}\big{)},$
(5)

while being equipped with the detectors sensitive only to modes $A$ and $B$,
respectively. After that, Alice remains stationary at the asymptotically flat
region of the space-time. On the other hand, Bob, after the initial free-fall,
starts to hover at the distance $r$ from the black hole centre. In this
manner, at some radius $r$ the detector $B$ is expected to experience the
outgoing radiation in the quantum atmosphere region, which is extended beyond
the radius $r_{h}$ of an event horizon. To describe what Bob perceives, the
corresponding vacuum and excited states are being constructed in the Kruskal
frame by using the appropriate Bogoliubov transformation between operators, as
described by Eqs. (3) and (4) [24, 17]. As a consequence, the $\rho_{AB}$
state can be now represented in a new basis, with the Alice and Bob being
under the influence of the radiation quanta. In this regard, the density
matrix $\rho_{AB_{I}B_{II}}$ is obtained by expanding Bob mode into the
Kruskal one ($B\rightarrow B_{I}B_{II}$). Since the region $II$ is physically
inaccessible, it is necessary to perform the trace over the region $II$ i.e.
$\rho_{AB_{I}}=\Tr_{B_{II}}{(\rho_{AB_{I}B_{II}})}$. In this way the density
matrix, describing physically accessible correlations with the $A$ and $B$
confined to the physical region $I$, can be obtained [24, 13, 27]. We remark
that since this is a gedanken setup, the present analysis can be considered to
be complementary to the recent work regarding quantum correlations across the
event horizon [8].

In the outlined framework, the MIN for the bipartite quantum state $\rho$,
shared by the $A$ and $B$ parts, is defined to be [15, 19]:

$\displaystyle\text{MIN}(\rho)=\text{Max}_{\Pi^{A}}\Big{|}\Big{|}\rho-\Pi^{A}(\rho)\Big{|}\Big{|}^{2}.$
(6)

In the case of Eq. (5), the following MIN form for the physically accessible
quantum correlations can be obtained by using the Bloch decomposition [28, 13,
14]:

$\displaystyle\text{MIN}(\rho_{AB_{I}})=\frac{1}{2(1+e^{-\omega/T})}.$ (7)

We note, that since our goal is to characterize possible signatures of the
quantum atmosphere, the local temperature ($T$) should take the Hartle-Hawking
($T_{HH}$) form [11]:

$\displaystyle T_{HH}=T_{H}\sqrt{1-\frac{r_{h}}{r}}$
$\displaystyle\sqrt{1+2\frac{r_{h}}{r}+\big{(}\frac{r_{h}}{r}\big{)}^{2}\Big{(}9+D_{HH}+36\ln(\frac{r_{h}}{r})\Big{)}},$
(8)

with $T_{H}=\frac{1}{4\pi r_{h}}$. In Eq. (8), the $D_{HH}$ is the
undetermined constant of the stress tensor for the Hartle-Hawking vacuum,
termed here as the Hartle-Hawking constant for simplicity. According to [11],
its value cannot be fixed by the Hartle-Hawking boundary conditions. However,
it is known that for $D_{HH}\geq D_{C}\approx 23.03$ the temperature is
positive and decreases after reaching peak at $r_{c}\approx 1.43r_{h}$. Note
that this local temperature is vanishing at the horizon $r_{h}$ and approaches
Hawking temperature at the infinity ($r\rightarrow\infty$). In addition, the
$T_{HH}=0$ result agrees with the lack of the influx/flux of the particle
radiation for the horizon in thermal equilibrium [11]. At the same time,
$T_{HH}$ peaks at the macroscopic distance from the event horizon where the
main excitations occur.

##  IV Thermal evolution of quantum correlations

Figure 2: (A)-(D) The measurement-induced nonlocality of the physically
accessible quantum correlations ($\text{MIN}(\rho_{AB_{I}})$) and (E)-(H) the
normalized local temperature in the Hartle-Hawking vacuum ($T_{HH}/T_{H}$) on
a plane defined by the distance from a black hole ($r$) and the event horizon
size ($r_{H}$). The results are depicted for the selected values of the so-
called Hartle-Hawking constant ($D_{HH}\in\\{23.03,40,60,80\\}$, in order from
left to the right-hand side).

In Fig. 1 (A), the behaviour of the $\text{MIN}(\rho_{AB_{I}})$ measure, as a
function of the normalized distance ($r/r_{H}$), is presented for the selected
values of the $D_{HH}$ constant. Note that only $D_{HH}\geq D_{C}\approx
23.03$ is considered to avoid physically irrelevant solutions, yielding
imaginary or inverse distance-dependent temperature [11]. Although the full
physical meaning of the $D_{HH}$ is still unknown, the obtained results show
that the $\text{MIN}(\rho_{AB_{I}})$ measure behaves qualitatively the same
for each of the considered parameter values. In details, it exhibits
continuous character showing that the quantum correlations of interest
initially decrease but later return to its maximum value at $r>4r_{H}$, as the
distance from the event horizon increases. It is instructive to note that the
$\text{MIN}(\rho_{AB_{I}})$ minimum is located always at $r\approx 1.43r_{H}$,
regardless of the assumed $D_{HH}$ level. The latter is responsible only for
the absolute value of the $\text{MIN}(\rho_{AB_{I}})$ measure and the width of
the correlation loss region. This is to say, the higher the $D_{HH}$ value the
more visible the correlation drop is. Deeper inspection allows relating this
loss to the peak of the Hawking quanta radiation well outside the event
horizon. In Fig. 1 (B), the distance-dependent character of the normalized
local temperature ($T_{HH}/T_{H}$) is presented for convenience. The depicted
results clearly show that the aforementioned peak is indeed located around
$r\approx 1.43r_{H}$, in correspondence to the correlation loss region.
Similarly, it is possible to trace back the destructive role of the $D_{HH}$
parameter on the quantum correlations and directly associate it with the local
temperature (refer to Fig. 1 (B)).

To verify if the above claims are true for different horizon radii, the
$\text{MIN}(\rho_{AB_{I}})$ measure is examined further by analyzing its
dependence on the $r$ and $r_{H}$ distances separately. In Figs. 2 (A)-(D) the
density plots of the $\text{MIN}(\rho_{AB_{I}})$ are given on a plane defined
by these two parameters. Once again several values of the $D_{HH}$ constant
are chosen to enhance some of the effects ($D_{HH}\in\\{23.03,40,60,80\\}$).
Similarly to results presented in Fig. 1 (A), the $\text{MIN}(\rho_{AB_{I}})$
shows minimum just outside the event horizon, which aligns with the previously
observed ratio $r/r_{h}\sim 1.43$. Moreover, the destructive influence of the
$D_{HH}$ parameter is also confirmed. However, the presented results allows to
observe additional aspects of the discussed quantum correlations. When
exploring the role of the $r_{H}$ distance, it can be noticed that the
correlation loss region becomes narrower along with the horizon size decrease.
Simultaneously, the decline onset appears to be shifted slightly toward the
horizon. This effect is best visible for the $D_{HH}=80$ (please refer to Fig.
2 (D)). To this end, the described behavior is obviously strongly related to
the character of the local temperature. The corresponding density plots on the
distance plane for the selected values of the $D_{HH}$ constant are given in
Figs. 2 (E)-(F).

##  V Discussion and conclusions

The conducted analysis reveals that the quantum correlations, as captured
within the MIN measure, allows to observe signatures of the black hole quantum
atmosphere. In general, the obtained results suggest that the quantum
correlations evolve continuously with the distance from the event horizon and
that they are always finite in the Hartle-Hawking vacuum. This is to say, the
whole atmosphere region is accessible at the quantum correlations level and
its characteristics can be recovered within a correlation measure. In
particular, two distinctive effects appear to be detectable in the atmosphere
spectrum:

  * (i)

The notable loss of correlations is observed at the macroscopic distance from
the horizon ($r\approx 1.43r_{H}$). It is important to note that this
signature coincides with the Hawking radiation peak in the quantum atmosphere
[10, 11]. Moreover, the correlation loss region is well-visible even when the
horizon radius is varied, although it is shifted toward a black hole when its
size is decreased. The latter is to some extent similar to the effect related
to the influence of the black hole dimensionality on the atmosphere effective
radius, as described in [29].

  * (ii)

The monotonic increase of quantum correlations is found to follow their
initial degradation, as the distance from a black hole increases. This
increase continues up to the maximum value, which is reached well outside the
event horizon ($r>4r_{H}$). Note that the correlations are observed to
saturate much faster with the distance than the local temperature itself,
meaning that they stop exhibiting distinct characteristics of the atmosphere
earlier. Sill, there is a strong interplay between both of these observables,
suggesting potentially robust probing capabilities of a quantum correlation
measures.

Based on the described observations, it can be argued that the quantum
correlations analysis constitutes promising venue to further discuss the
Hawking radiation nature and to verify the Giddings argument from a new
perspective [4]. This can be done not only in theoretical terms but also has
potential for an experimental realization. Since the discussed phenomena are
difficult or even impossible to examine in a real space-time, the analogue
black holes may be viewed as an example to investigate the atmosphere concept
with a help of quantum detectors [30, 31]. However, it means also that the
consequences of the radiation origin location spans beyond the conventional
black hole physics and profoundly impact behavior of quantum systems near the
even horizon. In other words, the observed effects should be also considered
in terms of the future research on information problems at large scales.
Hence, the presented analysis supports at the same time the importance of the
relativistic quantum information field in approaching problems such as the
information paradox and other black hole phenomena measurable at the quantum
level [25, 19].

To this end, it is instructive to comment on the actual value of the $D_{HH}$
parameter. In principle, the physical meaning of this constant is largely
unknown since its introduction [11]. However, the presented study shows that
it may be possible to determine the $D_{HH}$ parameter based on the quantum
correlation measure. This would simply require to associate the experimentally
observed global correlation minimum (at $r/r_{h}\sim 1.43$) with the unknown
parameter via the relations (7) and (8). Obviously, the main difficulty will
be related to the construction of realistic setup capable of performing such
measures as mentioned above.

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