# Frames of quasi-geodesics, visibility and geodesic loops

Filippo Bracci  F. Bracci: Dipartimento Di Matematica
Università di Roma “Tor Vergata”
Via Della Ricerca Scientifica 1, 00133
Roma, Italy<EMAIL_ADDRESS>Dedicated to Prof. Simeon Reich, on the
occasion of his 75th birthday

(Date: August 28, 2024)

###### Abstract.

In this paper we give a characterization in terms of “quasi-geodesics frames’
of visibility and existence of geodesic loops for bounded domains in
$\mathbb{C}^{d}$ which are Kobayashi complete hyperbolic and Gromov
hyperbolic.

###### Key words and phrases:

Gromov hyperbolic spaces; Kobayashi hyperbolic spaces; visibility; extension
of holomorphic maps

###### 2010 Mathematics Subject Classification:

32F45

Partially supported by PRIN 2017 Real and Complex Manifolds: Topology,
Geometry and holomorphic dynamics, Ref: 2017JZ2SW5, by GNSAGA of INdAM and by
the MUR Excellence Department Project MatMod@TOV CUP:E83C23000330006 awarded
to the Department of Mathematics, University of Rome Tor Vergata

###### Contents

  1. 1 Introduction
  2. 2 Notations and preliminaries
  3. 3 Visible domains and frame of quasi-geodesics
  4. 4 Geodesic loops in visible domains

##  1\. Introduction

The concept of visibility in several complex variables for bounded and
unbounded domains in $\mathbb{C}^{d}$ has been recently introduced and turned
out to be a key notion to study continuous extension of biholomorphisms,
estimates for the Kobayashi distance and iteration theory (we refer the reader
to, e.g., [11, 6, 12, 2, 3, 4, 14, 18, 19, 20]).

In this note we consider $\Omega\subset\mathbb{C}^{d}$ a bounded domain (with
no assumption on the regularity of $\partial\Omega$) and we assume that its
Kobayashi distance $K_{\Omega}$ is complete (see, e.g., [17] for definition
and properties of the Kobayashi distance).

The domain $\Omega$ is visible if, roughly speaking, the geodesics which
converge to different points in the boundary bend inside the domain.

Due to the completeness of $K_{\Omega}$ and by Arzelà-Ascoli’s theorem, a non-
compactly divergent sequence of geodesics rays or lines admits a subsequence
converging uniformly on compacta to a geodesics ray or line.

One of the basic observations of the present paper is that, if $\Omega$ is
visibile then the convergence is actually uniform (not just on compacta)—see
Lemma 3.1.

Moreover, visibile domains have a natural family of geodesics which exhibit
certain peculiarities (they form a kind of “frame”). Motivated by this, we
give the following definition:

######  Definition 1.1.

Let $\Omega\subset\subset\mathbb{C}^{d}$ be a domain such that
$(\Omega,K_{\Omega})$ is complete hyperbolic and let $A\geq 1$, $B\geq 0$. A
family $\mathcal{F}$ is a frame of $(A,B)$-quasi-geodesics if

  1. (1)

for every $\gamma\in\mathcal{F}$ there exists $R_{\gamma}\in(0,+\infty]$ such
that, if $R_{\gamma}<+\infty$, $\gamma:[0,R_{\gamma}]\to\Omega$ is a
$(A,B)$-quasi-geodesic segment, while, if $R_{\gamma}=+\infty$,
$\gamma:[0,+\infty)\to\Omega$ is a $(A,B)$-quasi-geodesic ray,

  2. (2)

if $\gamma\in\mathcal{F}$ is a quasi-geodesic ray (i.e.,
$R_{\gamma}=+\infty$), then there exists $p_{\gamma}\in\partial\Omega$ such
that $\lim_{t\to+\infty}\gamma(t)=p_{\gamma}$,

  3. (3)

there exists a compact subset $K\subset\Omega$ such that $\gamma(0)\in K$ for
every $\gamma\in\mathcal{F}$,

  4. (4)

for every sequence $\\{\gamma_{k}\\}\subset\mathcal{F}$ there exist a
subsequence $\\{\gamma_{k_{m}}\\}$ and $\gamma\in\mathcal{F}$ such that
$\\{\gamma_{k_{m}}\\}$ converges uniformly to $\gamma$,

  5. (5)

there exist $\epsilon>0$ and $\delta>0$ such that for every $z\in\Omega$ with
$\hbox{dist}(z,\partial\Omega)<\epsilon$ there exists $\gamma\in\mathcal{F}$
such that $K_{\Omega}(z,\gamma([0,+\infty])<\delta$.

It turns out that visibile domains have a frame of geodesics (see Proposition
3.2). One of the main result of the paper is to prove the converse for Gromov
hyperbolic domains:

######  Theorem 1.2.

Let $\Omega\subset\subset\mathbb{C}^{d}$ be a domain such that
$(\Omega,K_{\Omega})$ is complete hyperbolic and Gromov hyperbolic. Assume
that $\partial\Omega$ does not contain nontrivial analytic discs. Then
$\Omega$ is visible if and only if there exists a frame of $(A,B)$-quasi-
geodesics for some $A\geq 1,B\geq 0$.

The hypothesis on the non-existence of nontrivial analytic discs on the
boundary is technical. It is presently not known whether a visible complete
hyperbolic and Gromov hyperbolic domain might have nontrivial analytic discs
on the boundary (in [4, Theorem 1.2] it is shown that for a visibile complete
hyperbolic and Gromov hyperbolic domain with no geodesic loops–see below for
the definition—and with $C^{0}$-boundary there can not be nontrivial analytic
discs on the boundary).

The domain $\Omega$ is a Gromov model domain provided the identity map extends
as a homeomorphism from the Gromov compactification of $\Omega$ to the
Euclidean closure $\overline{\Omega}$ (see [6]).

Due to the equivalence between Gromov’s topology and Carathéodory’s prime end
topology (see, e.g., [5, Ch. 4]) in dimension one, Gromov model domains play
essentially the rôle of simply connected Jordan domains, while, visible
domains play the rôle of simply connected domains with locally connected
boundary with respect to the extension to the boundary of biholomorphisms (see
[6] for a careful explanation of this fact).

The following are examples of Gromov model domains: $C^{2}$-bounded strongly
pseudoconvex domains [1], Gromov hyperbolic (bounded or unbounded) convex
domain [7, 8, 9], smooth bounded D’Angelo finite type convex domains [21, 22],
smooth bounded D’Angelo finite type pseudoconvex domains in $\mathbb{C}^{2}$
[16], bounded Gromov hyperbolic Lipschitz $\mathbb{C}$-convex domains [12].

It turns out (see [6, 12]) that a bounded Kobayashi complete hyperbolic and
Gromov hyperbolic domain $\Omega$ is a Gromov model if and only if $\Omega$ is
visible and has no geodesic loops. Where, a geodesic loop for
$\overline{\Omega}$ is a geodesic line $\gamma:(-\infty,+\infty)\to\Omega$
such that the cluster set of $\gamma$ at $+\infty$ coincides with the cluster
set of $\gamma$ at $-\infty$.

In this direction, we first show (see Proposition 4.3) that if $\Omega$ is
visible and there exist $p\in\partial\Omega$ and $U$ an open neighborhood of
$p$ such that $U\cap\Omega$ has at least two connected components having $p$
on their boundary then there exists a geodesic loop for $\overline{\Omega}$.

The second main result of the paper is a characterization of (non)existence of
geodesic loops in terms of frames of quasi-geodesics.

######  Definition 1.3.

Let $\Omega\subset\subset\mathbb{C}^{d}$ be a domain such that
$(\Omega,K_{\Omega})$ is complete hyperbolic and let $A\geq 1$, $B\geq 0$. A
frame of $(A,B)$-quasi-geodesics $\mathcal{F}$ is looping if there exist two
quasi-geodesic rays $\gamma,\eta\in\mathcal{F}$ such that
$\lim_{t\to+\infty}\gamma(t)=\lim_{t\to+\infty}\eta(t)$ and which stay at
infinite distance each other. We say that the quasi-geodesic frame
$\mathcal{F}$ is non-looping otherwise.

Then we prove:

######  Theorem 1.4.

Let $\Omega\subset\subset\mathbb{C}^{d}$ be a domain such that
$(\Omega,K_{\Omega})$ is complete hyperbolic and Gromov hyperbolic. Assume
$\partial\Omega$ does not contain nontrivial analytic discs and that $\Omega$
is visible. Then $\Omega$ has no geodesic loops if and only if it has a non-
looping frame of $(A,B)$-quasi-geodesics for some $A\geq 1$ and $B\geq 0$.

As a consequence of Theorem 1.2, Theorem 1.4 and the previous discussion we
have:

######  Theorem 1.5.

Let $\Omega\subset\subset\mathbb{C}^{d}$ be a domain such that
$(\Omega,K_{\Omega})$ is complete hyperbolic and Gromov hyperbolic. Assume
$\partial\Omega$ does not contain nontrivial analytic discs. Then $\Omega$ is
a Gromov model domain if and only if it has a non-looping frame of
$(A,B)$-quasi-geodesics for some $A\geq 1$ and $B\geq 0$.

The definition of frames of (non-looping) quasi-geodesics might seem at a
first sight rather technical and useless. However, unravelling the proofs in
[7, 8, 9, 10, 21, 22, 23, 16] where several kind of (Gromov hyperbolic)
bounded domains are proved to be Gromov models, one sees that actually the
main work was exactly to construct non-looping quasi-geodesic frames in the
sense of our definition. As another example, we sketch an argument for
constructing a frame of non-looping quasi geodesics in strongly pseudoconvex
domains, taking for granted that a $C^{2}$ bounded strongly pseudoconvex
domain $\Omega$ is Kobayashi hyperbolic and Gromov hyperbolic. Indeed, one can
construct easily a frame of non-looping quasi-geodesics as follows. If
$p\in\partial\Omega$ and $U_{p}$ is an open neighborhood of $p$ such that
$\Omega\cap U_{p}$ is biholomorphic to a strongly convex domain $V_{p}$, then
one can consider the family of all real segments in $V_{p}$ steaming from a
fixed point in $V_{p}$. Arguing as in [7, 8] one can see that this family is a
frame of non-looping quasi-geodesics in $V_{p}$. Hence, its preimage
$\mathcal{F}_{p}$ is a frame of non-looping quasi-geodesics in $\Omega\cap
U_{p}$. Taking $\mathcal{F}$ to be the union of all $\mathcal{F}_{p}$ and
using the localization of the Kobayashi distance at the boundary, one can show
that $\mathcal{F}$ is a frame of non-looping quasi-geodesics for $\Omega$ and
hence, by Theorem 1.5, $\Omega$ is a Gromov model domain (a result well known,
as remarked above, after [1]).

The paper is organized as follows. In Section 2 we recall some definitions and
state some preliminary results we need in the proofs. In Section 3 we prove
Theorem 1.2 and in Section 4 we prove Theorem 1.4.

##  2\. Notations and preliminaries

In this section we let $\Omega\subset\mathbb{C}^{d}$ be a bounded domain. We
denote by $k_{\Omega}$ the infinitesimal Kobayashi pseudometric of $\Omega$
and by $K_{\Omega}$ the Kobayashi distance of $\Omega$. We refer the reader to
[17] for definitions and properties.

If $\Omega\subset\subset\mathbb{C}^{d}$ is complete hyperbolic, that is,
$(\Omega,K_{\Omega})$ is a complete metric space, it follows from the Hopf-
Rinow theorem that $(\Omega,K_{\Omega})$ is geodesic and thus, every couple of
points in $\Omega$ can be joined by a geodesic (i.e. length minimizing curve)
for $K_{\Omega}$. If $p,q\in\Omega$, we denote by $[p,q]_{\Omega}$ any
geodesic joining $p$ and $q$.

A geodesic $\gamma:[a,b]\to\Omega$, $-\infty<a<b<+\infty$ is called a geodesic
segment. A geodesic $\gamma:[a,+\infty)\to\Omega$, $a\in\mathbb{R}$, is a
geodesic ray and a geodesic $\gamma:(-\infty,+\infty)\to\Omega$ is a geodesic
line.

A geodesic triangle $T$ is the union of 3 geodesic segments (called sides)
$T=[x,y]_{\Omega}\cup[y,z]_{\Omega}\cup[z,x]_{\Omega}$ joining 3 points $x,\
y,z\in X$.

The complete metric space $(\Omega,K_{\Omega})$ is Gromov hyperbolic if there
exists $\delta>0$ (the Gromov constant of $\Omega$) such that every geodesic
triangle $T$ is $\delta$-thin, that is, every point on a side of $T$ has
distance from the union of the other two sides less than or equal to $\delta$
(see, e.g., [13, 15] for details and further properties).

For an absolutely continuous curve $\gamma:[a,b]\rightarrow\Omega$, we denote
by $l_{k_{\Omega}}(\gamma;[s,t])$ the length of the curve $\gamma$ on $[s,t]$,
$a\leq s<t\leq b$, that is

$l_{k_{\Omega}}(\gamma;[s,t]):=\int_{s}^{t}k_{\Omega}(\gamma(\tau);\gamma^{\prime}(\tau))d\tau.$

Let $A>1$ and $B>0$. An absolutely continuous curve
$\gamma:[a,b]\rightarrow\Omega$ is a $(A,B)$-quasi-geodesic if for every
$a\leq s<t\leq b$, we have :

$l_{k_{\Omega}}(\gamma;[s,t])\leq AK_{\Omega}(\gamma(s),\gamma(t))+B.$

An $(A,B)$-quasi-geodesic ray (or line) is an absolutely continuous curve
whose restriction to any compact interval in the domain of definition is an
$(A,B)$-quasi-geodesic. A quasi-geodesic is just any $(A,B)$-quasi-geodesic
segment/ray/line for some $A\geq 1$ and $B\geq 0$.

One of the main feature of Gromov hyperbolic spaces is the so-called Geodesic
Stability Theorem, which says that every $(A,B)$-quasi-geodesic is shadowed by
a geodesic at a distance which depends only on $A,B$ and the Gromov constant
of the space. In this paper we do not need it directly, but we will use
instead a straightforward consequence for “quasi-geodesic rectangles” (see,
e.g., [22, Observation 4.4]):

######  Lemma 2.1.

Let $\Omega\subset\subset\mathbb{C}^{d}$ be a Kobayashi complete hyperbolic
domain such that $(\Omega,K_{\Omega})$ is Gromov hyperbolic. Let
$a,b,c,d\in\Omega$. Let $A\geq 1$ and $B\geq 0$. If $\Gamma_{1}$ is a
$(A,B)$-quasi-geodesic joining $a$ with $b$, $\Gamma_{2}$ is a $(A,B)$-quasi-
geodesic joining $b$ with $c$, $\Gamma_{3}$ is a $(A,B)$-quasi-geodesic
joining $c$ with $d$, $\Gamma_{4}$ is a $(A,B)$-quasi-geodesic joining $a$
with $d$ (that is, $\Gamma_{1}\cup\Gamma_{2}\cup\Gamma_{3}\cup\Gamma_{4}$ is a
$(A,B)$-quasi-geodesic rectangle with sides $\Gamma_{1},\Gamma_{2},\Gamma_{3}$
and $\Gamma_{4}$) then there exists $N>0$ (which depends only on $A,B$ and the
Gromov constant of $\Omega$) such that every point of one side is contained in
the $N$-tubular neighborhood (with respect to $K_{\Omega}$) of the union of
the other three.

We now turn to the precise definition of visibility.

Let $p,q\in\partial\Omega$, $p\neq q$. We say that the couple $(p,q)$
satisfies the visibility condition with respect to $K_{\Omega}$ if there exist
a neighborhood $V_{p}$ of $p$ and a neighborhood $V_{q}$ of $q$ and a compact
subset $K$ of $\Omega$ such that $V_{p}\cap V_{q}=\emptyset$ and
$[x,y]_{\Omega}\cap K\neq\emptyset$ for every $x\in V_{p}\cap\Omega$, $y\in
V_{q}\cap\Omega$.

We say that $\Omega$ is visible if every couple of points
$p,q\in\partial\Omega$, $p\neq q$, satisfies the visibility condition with
respect to $K_{\Omega}$.

We say that a geodesic line $\gamma:(-\infty,+\infty)\to\Omega$ is a geodesic
loop in $\overline{D}$ if $\gamma$ has the same cluster set $\Gamma$ in
$\overline{\Omega}$ at $+\infty$ and $-\infty$. In such a case we say that
$\Gamma$ is the vertex of the geodesic loop $\gamma$.

By [11, Lemma 3.1], we have:

######  Lemma 2.2.

Suppose $\Omega\subset\mathbb{C}^{d}$ is a bounded complete hyperbolic domain.
If $\Omega$ is visible then every geodesic ray lands, i.e., if
$\gamma:[0,+\infty)\to\Omega$ is a geodesic then there exists
$p\in\partial\Omega$ such that $\lim_{t\to+\infty}\gamma(t)=p$. In particular,
the vertex of every geodesic loop in $\overline{\Omega}$ is a point in
$\partial\Omega$.

In the sequel, we will also need the following lemma (see [6, Lemma A.2])

######  Lemma 2.3 (D’Addezio).

Let $D\subset\mathbb{C}^{d}$ be a complete hyperbolic bounded domain. Assume
that $\partial D$ does not contain non-trivial analytic discs. If
$\\{z_{n}\\},\\{w_{n}\\}\subset D$ are two sequences such that
$\lim_{n\to+\infty}z_{n}=p,\lim_{n\to+\infty}w_{n}=q$ with $p,q\in\partial D$
and

$\sup_{n}K_{D}(z_{n},w_{n})<+\infty,$

then $p=q$.

##  3\. Visible domains and frame of quasi-geodesics

As a matter of notation, if $\\{\gamma_{k}\\}$ is a sequence of curves in
$\mathbb{C}^{d}$ such that for every $k$ the curve $\gamma_{k}$ is defined on
some interval $[0,R_{k}]$, with $R_{k}\in(0,+\infty]$, we say that
$\\{\gamma_{k}\\}$ converges uniformly to a curve $\gamma:[0,a]\to\Omega$
provided that $a=\lim_{k\to+\infty}R_{k}$ and for every $\theta>0$ there
exists $k_{0}\in\mathbb{N}$ such that for every $k\geq k_{0}$ and for every
$t\in[0,R_{k}]$ it holds

$|\gamma_{k}(t)-\gamma(t)|\leq\theta.$

######  Lemma 3.1.

Let $\Omega\subset\subset\mathbb{C}^{d}$ be a domain such that
$(\Omega,K_{\Omega})$ is complete hyperbolic. Suppose $\Omega$ is visible. If
$\\{\gamma_{k}\\}$ is a sequence of geodesic (segment or rays) in $\Omega$
which converges uniformly on compacta to a geodesic ray $\gamma$ then
$\\{\gamma_{k}\\}$ converges uniformly to $\gamma$.

###### Proof.

By hypothesis, for every $k$, either $\gamma_{k}$ is a geodesic segment
defined on $[0,R_{k}]$ for some $R_{k}>0$ or $\gamma_{k}$ is geodesic ray
defined on $[0,+\infty)$.

In case $\gamma_{k}$ is a geodesic ray, since $\Omega$ is visible, by Lemma
2.2 there exists $\gamma_{k}(+\infty):=\lim_{t\to+\infty}\gamma_{k}$, hence,
in this case, in order to unify notations, we can consider $\gamma_{k}$ to be
defined on $[0,+\infty]$ and set $R_{k}=+\infty$.

Now, assume by contradiction there exists $\theta>0$ and a sequence
$\\{t_{k}\\}$ such that $t_{k}\in[0,R_{k}]$ and

$|\gamma_{k}(t_{k})-\gamma(t_{k})|\geq\theta,$

where, as before, if $t_{k}=+\infty$ we set
$\gamma(+\infty):=\lim_{t\to+\infty}\gamma(t)$.

Since $\\{\gamma_{k}\\}$ converges uniformly on compacta to $\gamma$, it
follows that either $t_{k}=+\infty$ or $\\{t_{k}\\}$ converges to $+\infty$.
Hence, up to subsequences, we can assume that $\\{\gamma_{k}(t_{k})\\}$
converges to a point $q\in\partial\Omega$ such that

$|q-\gamma(+\infty)|\geq\theta.$

Fix an open neighborhood $U$ of $\gamma(+\infty)$ and an open neighborhood $V$
of $q$ such that $\overline{U}\cap\overline{V}=\emptyset$. By visibility
hypothesis, there exists a compact subset $K\subset\Omega$ such that every
geodesic in $\Omega$ joining a point of $U\cap\Omega$ to a point in
$V\cap\Omega$ has to intersect $K$.

Since $\gamma(t)$ tends to $\gamma(+\infty)$ as $t\to+\infty$, there exists
$R>0$ such that $\gamma(r)\in U\cap\Omega$ for all $r\geq R$. Hence, for every
$r\geq R$, there exists $k_{r}\in\mathbb{N}$ such that $\gamma_{k}(r)\in
U\cap\Omega$ for $k\geq k_{r}$. Up to take $k_{r}$ larger, we can also assume
that $\gamma_{k}(t_{k})\in V\cap\overline{\Omega}$ for $k\geq k_{r}$.

Thus, $\gamma_{k}([r,t_{k}))$ is a geodesic from $U\cap\Omega$ to
$V\cap\Omega$ for $k\geq k_{r}$. Therefore, for every $k\geq k_{r}$ there
exists $s_{k}\in(r,t_{k})$ such that $\gamma_{k}(s_{k})\in K$.

But,

$+\infty>\max_{\xi\in K}K_{\Omega}(z_{0},\xi))\geq
K_{\Omega}(z_{0},\gamma_{k}(s_{k}))=K_{\Omega}(\gamma_{k}(0),\gamma_{k}(s_{k}))=s_{k}>r,$

which gives a contradiction for $r$ large enough. ∎

######  Proposition 3.2.

Let $\Omega\subset\subset\mathbb{C}^{d}$ be a domain such that
$(\Omega,K_{\Omega})$ is complete hyperbolic. If $\Omega$ is visible then
there exists a frame of geodesics.

###### Proof.

Fix $z_{0}\in\Omega$ and let $\mathcal{F}$ be the set of all geodesic steaming
from $z_{0}$, i.e., $\gamma\in\mathcal{F}$ if either there exists
$R_{\gamma}\in(0,+\infty)$ such that $\gamma:[0,R_{\gamma}]\to\Omega$ is a
geodesic segment or $\gamma:[0,+\infty)\to\Omega$ is a geodesic ray, and (in
both cases) $\gamma(0)=z_{0}$. By definition, $\mathcal{F}$ satisfies
condition (1) and (3) of Definition 1.1. Also, since the distance $K_{\Omega}$
is complete, by Hopf-Rinow’s theorem, for every $w\in\Omega$ there exists a
geodesic segment $\gamma$ joining $z_{0}$ and $w$, hence, (5) is trivially
satisfied for any $\epsilon>0$ and $\delta>0$.

Moreover, since $\Omega$ is visible, it follows from Lemma 2.2 that
$\mathcal{F}$ enjoys also (2).

Now, we show that $\mathcal{F}$ satisfies (4). Assume $\\{\gamma_{k}\\}$ is a
sequence of geodesic steaming from $z_{0}$.

Note that for every $k$, either $\gamma_{k}$ is a geodesic segment defined on
$[0,R_{k}]$ for some $R_{k}>0$ or $\gamma_{k}$ is geodesic ray defined on
$[0,+\infty)$. However, as remarked above, in this case there exists
$\gamma_{k}(+\infty):=\lim_{t\to+\infty}\gamma_{k}$, hence, we can consider
$\gamma_{k}$ to be defined on $[0,+\infty]$ and set $R_{k}=+\infty$.

By Arzelà-Ascoli’s theorem, it follows that, up to subsequences,
$\\{\gamma_{k}\\}$ converges uniformly on compacta to some
$\gamma\in\mathcal{F}$. By Lemma 3.1 the convergence is actually uniform, and
we are done. ∎

Now we are in a good shape to prove Theorem 1.2:

###### Proof of Theorem 1.2.

One direction follows from Proposition 3.2.

Conversely, Assume that $\Omega$ has a family $\mathcal{F}$ of $(A,B)$-quasi-
geodesics. Hence, there exists a compact subset $K\subset\Omega$ such that
every quasi-geodesic in $\mathcal{F}$ steams from $K$. Assume
$\\{z^{j}_{k}\\}\subset\Omega$ be a sequence converging to
$p_{j}\in\partial\Omega$, $j=1,2$ with $p_{1}\neq p_{2}$.

We argue by contradiction and we assume that
$\\{[z^{1}_{k},z^{2}_{k}]_{\Omega}\\}$ eventually escapes every compact subset
of $\Omega$. By hypothesis, for $k$ large enough, we can find
$\gamma_{k}^{1},\gamma_{k}^{2}\in\mathcal{F}$ and $w_{k}^{j}\in\gamma_{k}^{j}$
such that $K_{\Omega}(z_{k}^{j},w_{k}^{j})<\delta$, $j=1,2$ (here, we a slight
abuse of notation, we denote by $\gamma_{k}^{j}$ also the image of the quasi-
geodesic $\gamma_{k}^{j}$).

By Lemma 2.3, since $\partial\Omega$ does not contain nontrivial analytic
discs, it follows that $\\{w_{k}^{j}\\}$ converges to $p_{j}$, $j=1,2$ for
$k\to+\infty$.

By condition (4) of Definition 1.1, we can assume, up to subsequences, that
$\\{\gamma_{k}^{j}\\}$ converges uniformly to some $\gamma^{j}\in\mathcal{F}$,
$j=1,2$. Hence,

(3.1) $\lim_{t\to+\infty}\gamma^{j}(t)=p_{j},\quad j=1,2.$

We claim now that $\\{[w^{1}_{k},w^{2}_{k}]_{\Omega}\\}$ eventually escapes
every compact subset of $\Omega$, if so does
$\\{[z^{1}_{k},z^{2}_{k}]_{\Omega}\\}$. Indeed, assume by contradiction that
there exists a compact set $Q\subset\subset\Omega$ and
$c_{k}\in[w^{1}_{k},w^{2}_{k}]_{\Omega}$ such that $c_{k}\in Q$ for every $k$.
Since

$[w^{1}_{k},w^{2}_{k}]_{\Omega}\cup[z^{1}_{k},z^{2}_{k}]_{\Omega}\cup[z^{1}_{k},w^{1}_{k}]_{\Omega}\cup[z^{2}_{k},w^{2}_{k}]_{\Omega}$

is a geodesic rectangle, by Lemma 2.1, it follows that there exist $N>0$ and

$t_{k}\in[z^{1}_{k},z^{2}_{k}]_{\Omega}\cup[z^{1}_{k},w^{1}_{k}]_{\Omega}\cup[z^{2}_{k},w^{2}_{k}]_{\Omega}$

such that $K_{\Omega}(t_{k},c_{k})\leq N$. Since for every
$s\in[z^{j}_{k},w^{j}_{k}]_{\Omega}$ we have that $K_{\Omega}(s,z^{j}_{k})\leq
K_{\Omega}(w^{j}_{k},z^{j}_{k})<\delta$, $j=1,2$, it follows from the
completeness of $K_{\Omega}$ that $t_{k}\in[z^{1}_{k},z^{2}_{k}]_{\Omega}$.
But then $t_{k}$ belongs to a $N$-tubular neighborhood of $Q$, which is as
well a relatively compact subset of $\Omega$, and we get a contradiction.

Therefore, $\\{[w^{1}_{k},w^{2}_{k}]_{\Omega}\\}$ eventually escapes every
compact subset of $\Omega$.

Let $a^{j}_{k}>0$ be such that $\gamma^{j}_{k}(a_{k}^{j})=w_{k}^{j}$, $j=0,1$
and consider now the quasi-geodesic rectangle

$[w^{1}_{k},w^{2}_{k}]_{\Omega}\cup\gamma_{k}^{1}([0,a_{k}^{1}])\cup\gamma_{k}^{2}([0,a_{k}^{1}])\cup[\gamma_{k}^{1}(0),\gamma_{k}^{2}(0)]_{\Omega}.$

Note that $\\{[\gamma_{k}^{1}(0),\gamma_{k}^{2}(0)]_{\Omega}\\}$ is relatively
compact in $\Omega$. Hence, by Lemma 2.1, there exists $N>0$ such that for
every point $s_{k}\in[w^{1}_{k},w^{2}_{k}]_{\Omega}$ there exists
$\zeta(s_{k})\in\gamma_{k}^{1}([0,a_{k}^{1}])\cup\gamma_{k}^{2}([0,a_{k}^{1}])$
such that

(3.2) $K_{\Omega}(s_{k},\zeta(s_{k}))\leq N.$

Since $\\{s_{k}\\}$ is compactly divergent, it follows by the completeness of
$K_{\Omega}$ that $\\{\zeta(s_{k})\\}$ is compactly divergent as well. By
(3.1), the cluster set of $\\{\zeta(s_{k})\\}$ is contained in
$\\{p_{1}\\}\cup\\{p_{2}\\}$ and, by (3.2) and Lemma 2.3, so is the cluster
set of $\\{s_{k}\\}$. But this is a contradiction because the cluster set of
$[w^{1}_{k},w^{2}_{k}]_{\Omega}$ has to be connected and contains both $p_{1}$
and $p_{2}$. Hence, $\Omega$ is visible. ∎

##  4\. Geodesic loops in visible domains

######  Definition 4.1.

Let $\Omega\subset\subset\mathbb{C}^{d}$ be a domain such that
$(\Omega,K_{\Omega})$ is complete hyperbolic. We say that two quasi-geodesic
rays $\gamma,\eta$ stay at infinite distance each other provided for every
$C>0$ there exists $t_{C}>0$ such that

$\min_{t\geq t_{C}}K_{\Omega}(\gamma(t),\eta([0,+\infty))>C,\quad\min_{t\geq
t_{C}}K_{\Omega}(\eta(t),\gamma([0,+\infty))>C.$

######  Lemma 4.2.

Let $\Omega\subset\subset\mathbb{C}^{d}$ be a domain such that
$(\Omega,K_{\Omega})$ is complete hyperbolic. Assume
$\gamma:\mathbb{R}\to\Omega$ is a geodesic loop for $\overline{D}$. Let
$\gamma^{+}:[0,+\infty)\to\Omega$ be the geodesic ray defined by
$\gamma^{+}(t):=\gamma(t)$ and let $\gamma^{-}:[0,+\infty)\to\Omega$ be the
geodesic ray defined by $\gamma^{-}(t):=\gamma^{-}(t)$. Then $\gamma^{+}$ and
$\gamma^{-}$ stay at infinite distance each other.

###### Proof.

Assuming by contradiction that $\gamma^{+},\gamma^{-}$ do not stay at infinite
distance each other, we can find $C>0$ such that for every $t\in[0,+\infty)$
there exists $s_{t}\in[0,+\infty)$ such that

$K_{\Omega}(\gamma^{+}(t),\gamma^{-}(s_{t}))\leq C.$

But, since $\gamma$ is a geodesic,

$t+s_{t}=K_{\Omega}(\gamma(t),\gamma(-s_{t}))=K_{\Omega}(\gamma^{+}(t),\gamma^{-}(s_{t}))\leq
C,$

and, for $t\to+\infty$ we have a contradiction. ∎

######  Proposition 4.3.

Let $\Omega\subset\subset\mathbb{C}^{d}$ be a domain such that
$(\Omega,K_{\Omega})$ is complete hyperbolic. Assume that $\Omega$ is visible.
If there exists $p\in\partial\Omega$ and an open neighborhood $U$ of $p$ such
that $U\cap\Omega$ has at least two connected components such that $p$ belongs
to their closure, then there exists a geodesic loop for $\overline{\Omega}$
with vertex $p$.

###### Proof.

Let $p,U$ as in the statement and let $V_{1},V_{2}$ be two connected
components of $\Omega\cap U$ such that $p\in\overline{V_{j}}$, $j=1,2$. For
$j=1,2$, let $\mathcal{V}_{j}\subset\subset V_{j}$ be an open set such that
$p\in\overline{\mathcal{V}_{j}}$ and $p\not\in\partial\mathcal{V}_{j}$.
Moreover, for $j=1,2$, let $\\{z^{j}_{n}\\}\subset\mathcal{V}_{j}$ be a
sequence converging to $p$. For every $n$, let
$\gamma_{n}:[a_{n},b_{n}]:\to\Omega$ be a geodesic such that
$\gamma_{n}(a_{n})=z^{1}_{n}$ and $\gamma_{n}(b_{n})=z^{2}_{n}$,
$a_{n}<b_{n}$.

We claim that there exists a compact subset $K\subset\Omega$ such that
$\gamma_{n}([a_{n},b_{n}])\cap K\neq\emptyset$ for all $n$. Indeed, if this
were not the case, we can assume that $\\{\gamma_{n}([a_{n},b_{n}])\\}$
escapes every compact subset of $\Omega$ for $n$ large. Let
$r_{n}\in(a_{n},b_{n})$ be such that
$\gamma_{n}(r_{n})\in\partial\mathcal{V}_{2}$, for all $t$. Then, up to
subsequences, $\\{\gamma_{n}(r_{n})\\}$ converges, for $n\to\infty$, to a
point $q\in\partial\Omega\setminus\\{p\\}$. Since $\Omega$ is visible, it
follows that there exists a compact subset $K^{\prime}\subset\Omega$ such that
$\gamma_{n}([r_{n},b_{n}])\cap K^{\prime}\neq\emptyset$, reaching a
contradiction.

Up to an affine change of parameterization, we can thus assume that
$\\{\gamma_{n}(0)\\}$ is relatively compact in $\Omega$ and $a_{n}<0<b_{n}$
for all $n$.

We claim that $\\{\gamma_{n}\\}$ (under the assumption that
$\\{\gamma_{n}(0)\\}$ is relatively compact in $\Omega$) converges, up to
subsequences, to a geodesic loop in $\overline{\Omega}$ with vertex $p$.

Indeed, since $(\Omega,K_{\Omega})$ is complete and $\gamma_{n}(0)\in K$,

$\lim_{n\to\infty}|a_{n}|=\lim_{n\to\infty}K_{\Omega}(\gamma_{n}(0),z_{n}^{1})=+\infty,$

and thus $a_{n}\to-\infty$ for $n\to\infty$. Similarly, $b_{n}\to+\infty$ for
$n\to\infty$. Thus, for every $S,T\in\mathbb{R}$ there exists $n_{0}$ such
that $S,T\in(a_{n},b_{n})$ for $n>n_{0}$. Moreover, setting for $R>0$,

$d_{K}(R):=\max\\{K_{\Omega}(z,w):K_{\Omega}(z,K)\leq R,K_{\Omega}(w,K)\leq
R\\}<+\infty,$

we have, for all $n>n_{0}$,

$K_{\Omega}(\gamma_{n}(S),\gamma_{n}(T))=K_{\Omega}(\gamma_{n}(0),\gamma_{n}(S))+K_{\Omega}(\gamma_{n}(0),\gamma_{n}(T))\leq
d_{K}(|S|)+d_{K}(|T|).$

Hence $\\{\gamma_{n}\\}$ are locally equibounded and locally equi-continuous
and by Arzelà-Ascoli’s theorem, and taking into account that
$K_{D}(\gamma_{n}(T),\gamma_{n}(S))=|T-S|$ for all $n$, we can extract a
subsequence converging on compacta of $\mathbb{R}$ to a geodesic line $\gamma$
such that $\gamma(0)\in K$.

By Lemma 2.2, $\lim_{t\to\pm\infty}\gamma(t)=p$. Hence, $\gamma$ is a geodesic
loop in $\overline{\Omega}$ with vertex $p$, and we are done. ∎

For Gromov hyperbolic domains the existence of a geodesic loop is equivalent
to the existence of two geodesic rays landing at the same point which stay at
infinite distance each other:

######  Proposition 4.4.

Let $\Omega\subset\subset\mathbb{C}^{d}$ be a domain such that
$(\Omega,K_{\Omega})$ is complete hyperbolic and Gromov hyperbolic. Assume
that $\Omega$ is visible. Then there exists a geodesic loop for
$\overline{\Omega}$ with vertex $p\in\partial\Omega$ if and only if there
exists two geodesic rays in $\Omega$ landing at $p$ which stay at infinite
distance each other.

###### Proof.

If $\gamma$ is a geodesic loop for $\overline{\Omega}$ with vertex $p$, then
$\gamma^{+}$ and $\gamma^{-}$ are geodesic rays at infinite distance each
other, landing at $p$ (see Lemma 4.2).

Conversely, if $\alpha,\beta:[0,+\infty)\to\Omega$ are two geodesic rays which
stay at infinite distance each other, for each $T>0$, we consider the geodesic
rectangle given by

$[\alpha(0),\beta(0)]_{\Omega}\cup\alpha([0,T])\cup[\alpha(T),\beta(T)]_{\Omega}\cup\beta([0,T]).$

Since $(\Omega,K_{\Omega})$ is Gromov hyperbolic, by Lemma 2.1 there exists
$\delta>0$ such that $[\alpha(T),\beta(T)]_{\Omega}$ is contained in the
$\delta$-tubular neighborhood (with respect to $K_{\Omega}$) of
$[\alpha(0),\beta(0)]_{\Omega}\cup\alpha([0,T])\cup\beta([0,T])$.

Since $\alpha$ and $\beta$ stay at infinite distance each other, it follows
that there exists $T_{0}>0$ such that $K_{\Omega}(\alpha(t),\beta(s))>4\delta$
for all $t,s\geq T_{0}$.

We claim that for all $T>T_{0}$ there exists
$z_{T}\in[\alpha(T),\beta(T)]_{\Omega}$ such that $z_{T}$ belongs to the
$\delta$-tubular neighborhood $\mathcal{N}$ of
$[\alpha(0),\beta(0)]_{\Omega}\cup\alpha([0,T_{0})]\cup\beta([0,T_{0}])$.

Indeed, if this were not the case, then every
$\xi\in[\alpha(T),\beta(T)]_{\Omega}$ would belong to a $\delta$-tubular
neighborhood $U$ of $\alpha([T_{0},T])\cup\beta([T_{0},T])$. However, since
$K_{\Omega}(\alpha(t),\beta(s))>4\delta$ for all $t,s\geq T_{0}$, it follows
that $U$ is the disjoint union of two open sets. But then
$[\alpha(T),\beta(T)]_{\Omega}$ is the disjoint union of the two open sets
given by $[\alpha(T),\beta(T)]_{\Omega}\cap U$, hence, not connected, a
contradiction and the claim follows.

Now, since $K_{\Omega}$ is a complete distance, $K:=\overline{\mathcal{N}}$ is
a compact subset of $\Omega$ and

$[\alpha(T),\beta(T)]_{\Omega}\cap K\neq\emptyset\quad\forall T\geq 0.$

Choosing a parametrization $\gamma_{T}$ of $[\alpha(T),\beta(T)]_{\Omega}$
such that $\gamma_{T}(0)\in K$ and arguing as in the proof of Proposition 4.3,
we see that we can extract a sequence $\\{\gamma_{T_{n}}\\}$ converging to a
geodesic loop for $\overline{\Omega}$ with vertex $p$. ∎

Now we are in good shape to prove Theorem 1.4:

###### Proof of Theorem 1.4.

First, assume that $\Omega$ has no geodesic loops. Then by Proposition 3.2
(and its proof), the family $\mathcal{F}$ of all geodesics steaming from a
given point $z_{0}\in\Omega$ is a geodesic frame. Such a geodesic frame is
non-looping by Proposition 4.4.

Suppose now that $\Omega$ has a non-looping frame of quasi-geodesics
$\mathcal{F}$. Assume by contradiction that $\gamma:\mathbb{R}\to\Omega$ is a
geodesic loop. Since $\Omega$ is visible, it follows that there exists
$p\in\partial\Omega$ such that $\lim_{t\to\pm\infty}\gamma(t)=p$. Let
$\delta>0,\epsilon>0$ be given as in (5) of Definition 1.1 and let

$V:=\\{z\in\Omega:\hbox{dist}(z,\partial\Omega)<\epsilon\\}.$

Let $\gamma^{+}(t):=\gamma(t)$ for $t\geq 0$ and $\gamma^{-}(t):=\gamma(-t)$
for $t\geq 0$.

Hence, there exists $t_{0}>0$ such that for $t\geq t_{0}$ sufficiently large,
$\gamma^{\pm}(t)\in V$.

By hypothesis, for each $k\geq t_{0}$, $k\in\mathbb{N}$, there exist
$\gamma^{\pm}_{k}\in\mathcal{F}$ and some $s^{\pm}_{k}$ in the domain of
definition of $\gamma^{\pm}_{k}$ such that

(4.1) $K_{\Omega}(\gamma^{\pm}(k),\gamma^{\pm}_{k}(s^{\pm}_{k}))\leq\epsilon.$

As $k\to+\infty$, since $K_{\Omega}$ is a complete distance and by Lemma 2.3,
it follows that $s^{\pm}_{k}\to+\infty$ and $\gamma^{\pm}_{k}(s^{\pm}_{k})\to
p$. By definition of quasi-geodesic frame, up to subsequences,
$\\{\gamma^{\pm}_{k}\\}$ converges uniformly to some quasi-geodesic ray
$\gamma^{\pm}_{\infty}\in\mathcal{F}$ such that

$\lim_{t\to+\infty}\gamma^{\pm}_{\infty}(t)=p.$

Since $\mathcal{F}$ is non-looping, $\gamma_{\infty}^{+}$ and
$\gamma_{\infty}^{-}$ stay at finite distance each other.

We claim that this implies that there exists $C>0$ such that for all $T\geq
0$,

(4.2) $K_{\Omega}(\gamma^{+}(T),\gamma^{-}(T))\leq C,$

reaching a contradiction to Lemma 4.2.

In order to prove (4.2), for every $k\geq t_{0}$, consider the quasi-geodesic
rectangle given by

$[\gamma^{+}(k),\gamma^{+}_{k}(s_{k})]_{\Omega}\cup\gamma^{+}([0,k])\cup\gamma^{+}_{k}([0,s_{k}^{+}])\cup[\gamma(0),\gamma^{+}_{k}(0)]_{\Omega}.$

Fix $T\geq t_{0}$. Hence, by Lemma 2.1, there exists some $R>0$ such that for
every $k>T$ there exists
$z_{k}\in[\gamma^{+}(k),\gamma^{+}_{k}(s_{k})]_{\Omega}\cup\gamma^{+}_{k}([0,s_{k}^{+}])\cup[\gamma(0),\gamma^{+}_{k}(0)]_{\Omega}$
such that

(4.3) $K_{\Omega}(z_{k},\gamma^{+}(T))\leq R.$

Since by hypothesis $\\{\gamma^{+}_{k}(0)\\}$ is relatively compact in
$\Omega$—and so is $\\{[\gamma(0),\gamma^{+}_{k}(0)]_{\Omega}\\}$—and
$\gamma^{+}(t)\to p\in\partial\Omega$ as $t\to+\infty$, it follows that, if
$t_{0}$ (and hence $T$) is large enough,
$z_{k}\not\in[\gamma(0),\gamma^{+}_{k}(0)]_{\Omega}$.

Moreover, $\\{[\gamma^{+}(k),\gamma^{+}_{k}(s_{k})]_{\Omega}\\}$ is compactly
divergent. Indeed, if this is not the case, there is
$y_{k}\in[\gamma^{+}(k),\gamma^{+}_{k}(s_{k})]_{\Omega}$ such that
$\\{y_{k}\\}$ is relatively compact, thus

$K_{\Omega}(\gamma^{+}(k),\gamma^{+}_{k}(s_{k}))=K_{\Omega}(\gamma^{+}(k),y_{k})+K_{\Omega}(y_{k},\gamma^{+}_{k}(s_{k})),$

and the latter quantity tends to $+\infty$ as $k\to+\infty$ because
$K_{\Omega}$ is a complete distance, against (4.1).

Hence, if $k$ is sufficiently large, $z_{k}\in\gamma^{+}_{k}([0,s_{k}^{+}])$.
By (4.3), $\\{z_{k}\\}$ is relatively compact in $\Omega$.

Since $\gamma_{\infty}^{-}$ and $\gamma_{\infty}^{+}$ stay at finite distance
each other and $\\{\gamma_{k}^{\pm}\\}$ converges uniformly to
$\gamma_{\infty}^{\pm}$, it follows that there exist $D>0$ and
$w_{k}\in\gamma_{k}^{-}$ for $k$ sufficiently large, so that

$K_{\Omega}(z_{k},w_{k})\leq D.$

In particular, this implies that also $\\{w_{k}\\}$ is relatively compact in
$\Omega$.

Using again Lemma 2.1 and arguing as before with the quasi-geodesic rectangle

$[\gamma^{-}(k),\gamma^{-}_{k}(s_{k})]_{\Omega}\cup\gamma^{-}([0,k])\cup\gamma^{-}_{k}([0,s_{k}^{-}])\cup[\gamma(0),\gamma^{-}_{k}(0)]_{\Omega},$

we can find $S\geq 0$ and $E>0$ such that for all $k$ sufficiently large

$K_{\Omega}(\gamma^{-}(S),w_{k})\leq E.$

Then, by the triangle inequality, $K_{\Omega}(\gamma^{+}(T),\gamma^{-}(S))\leq
C:=R+D+E$. Therefore,

$\begin{split}K_{\Omega}(\gamma^{+}(T),\gamma^{-}(T))&\leq
K_{\Omega}(\gamma^{+}(T),\gamma^{-}(S))+K_{\Omega}(\gamma^{-}(S),\gamma^{-}(T))\\\
&=K_{\Omega}(\gamma^{+}(T),\gamma^{-}(S))+|T-S|\\\
&=K_{\Omega}(\gamma^{+}(T),\gamma^{-}(S))+|K_{\Omega}(\gamma^{+}(T),\gamma(0))-K_{\Omega}(\gamma^{-}(S),\gamma(0))|\\\
&\leq 2K_{\Omega}(\gamma^{+}(T),\gamma^{-}(S))\leq C.\end{split}$

Thus, (4.2) follows, and we are done. ∎

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CSI and performs data detection using centralized combining. Since the
compression is implemented at the APs, ECF reduces the fronthaul load in a
distributed fashion.

Estimate-multiply-compress-forward (EMCF) [140] Each AP first estimates the
channels, then multiplies the received data signal by the local combining
vector (computed based on the local channel estimate) and compresses and
forwards the results to the CPU. Thus, CPU only performs data detection.
Therefore, when using EMCF, the design of combining vectors and the
compression is implemented at the APs in a distributed way.

Estimate–multiply-compress-forward-weight (EMCFW) [140, 138, 139] Similar to
EMCF, the signal is further multiplied by receiver filter coefficients at the
CPU to improve the performance. The design of combining vectors and the
compression is also implemented in a distributed way.

When it comes to the downlink, two approaches have been considered:

Compress-after-precoding (CAP) [141, 142, 143] The centralized signal with
precoding is first computed and then compressed at the CPU before being sent
to the APs, which makes CAP suitable for the centralized precoding.

Precoding-after-compress (PAC) [144] A simple compression is done at the CPU
where the symbol for each of the UEs is separately quantized. Then, each AP
receives the symbols and designs the precoding vectors for each UE, which
makes PAC suitable for the distributed precoding.

Figure 10: A C&F framework for CF mMIMO systems.

####  IV-A2 Compute-and-Forward

Another possible solution could be the compute-and-forward (C&F) schemes [145,
146], which can reduce the fronthaul load efficiently by decreasing the
cardinality of symbols transmitted to the CPU. As shown in Fig. 10, in the C&F
scheme, each AP forwards an integer-linear combination of the transmitted
signals of all UEs with the same cardinality as each UE’s signal. The served
UEs can be determined by selecting the coefficient vector and are limited by
the computation rate. Although the C&F scheme can obtain the theoretical
minimum fronthaul requirement of CF mMIMO with achieving lossless
transmission, it is only suitable for the symmetric scenario where all UEs
transmit with equal power. However, in CF mMIMO, the UEs can be allocated with
unequal transmit power to compensate for pathloss differences. Hence, the
expanded compute-and-forward (E-C&F) scheme [147], which is versatile to
distribute UE’s power unequally and can tolerate different noise at the
targeted AP for ensuring the minimum loss in the computation rate, offers
significant performance improvement over the C&F scheme. Fig. 11 shows the CDF
of the achievable sum-rate obtained via C&F and E-C&F schemes versus the
number of APs with $K=8$, $L=20$, and ${p=200}$ mW. The E-C&F scheme
outperforms the conventional C&F scheme in terms of achievable sum-rate
noticeably since the E-C&F framework enables optimal transmit power of UEs
which facilitates the exploitation of performance gain.

Figure 11: CDF of the achievable sum-rate with C&F [146] and E-C&F [147]
schemes. Figure 12: Average SE per UE versus the number of quantization bits
with EMCF [140], EMF, EMCFW [138], and EMFW schemes.

Although quantizing is an efficient and straightforward solution to reduce the
fronthaul load, the uniform quantization method can induce the quantization
error and achieves a considerable performance loss. Besides, it is shown that
performances of the different types of transmission are not the same because
they require different fronthaul rate allocations for CSI and/or data signals
transmitted to CPU, and their AP signal processing capabilities are different.

To compare the six transmission strategies by highlighting the required
processing at APs and the CPU, Table VIII is provided. Besides, in Fig. 12, we
plot the average SE with MR combining at the APs, as a function of the number
of the quantization bits for EMCF, EMF, EMCFW, and EMFW, where EMF and EMFW
refer to the case of using perfect ADCs in EMCF and EMCFW, respectively, and
hence no compression is done at the APs. It can be seen from Fig. 12 that
EMCFW provides a larger SE than EMCF. This can be explained that the optimized
receiver filter coefficients step in EMCFW maximizes the SNR. Besides, via
numerical results in [140, 138], it is shown that the EMCF can outperform the
other schemes when they apply UatF bounding (not the CSI-based ones), and ECF
strategy also outperforms the CFE. Thus, to decrease the deployment cost of
the network, dummy APs which only compress and forward the received signals
are preferred at the cost of performance loss. So, there is a tradeoff between
performance gain and implementation costs which must be taken into account.

TABLE VIII: Comparison of the Quantization Schemes. Scheme |  Processing at the APs |  Processing at the CPU |  Comb./prec. Design |  Compression
---|---|---|---|---
CFE [135, 136, 130, 137, 134, 138, 139] |  CSI compression Data compression |  Channel estimation Combining design |  Centralized |  Distributed
ECF [135, 136] |  Channel estimation CSI compression Data compression |  Combining design |  Centralized |  Distributed
EMCF [140] |  Channel estimation Combining design |  – |  Distributed |  Distributed
EMCFW [140, 138, 139] |  Channel estimation Combining design |  Receiver filter design |  Distributed |  Distributed
CAP [141, 142, 143] |  Precoding design Compression |  – |  Centralized |  Centralized
PAC [144] |  Compression |  Precoding design |  Distributed |  Distributed
|  |  |  |

###  IV-B Hardware Impairments

Figure 13: A generalized CF mMIMO system with/without HIs.

To enable a ubiquitous deployment of CF mMIMO, a tradeoff between cost and
quality of the transceiver hardware in CF mMIMO should be considered since
many antenna elements in APs are deployed, which might increase the deployment
cost and energy consumption. A possible countermeasure is to make use of
compact low-cost components, which introduce power amplifier non-linearities,
phase noise in local oscillators, amplitude/phase imbalance in I/Q mixers, and
finite-resolution quantization in ADCs. All these non-idealities are referred
to as _hardware impairments_ (HIs). Most existing works on CF mMIMO neglect
the impacts of the HIs by modeling the wireless communication channels as
linear filters, as shown in Fig. 13(a). Although this ideal model can be used
to devise analog or digital compensation algorithms that could substantially
mitigate the impacts of the HIs, the residual HIs will still exist due to
modeling inaccuracies and the destructive nature of some HIs. Instead, the
non-ideal transceiver hardware can be modeled as non-linear memoryless filters
to provide better insights for the practical implementation, as shown in Fig.
13(b).

Research on the impact that residual HIs have on the data rate performance of
the CF mMIMO systems has been made in [126, 148, 149, 150, 151, 152]. To be
specific, authors in [126] quantified the performance of both uplink and
downlink CF mMIMO with the classical additive hardware distortion model. By
exploiting the MR processing, SE and EE were derived in closed-form. With
these tractable expressions, hardware-quality scaling laws were presented,
which proved that the detrimental effect of the HIs at the APs vanishes as the
number of APs increases. Moreover, a max-min power control algorithm was
proposed to maximize the minimum UE data rate. Mitigating the impacts of the
HIs by using different levels of AP cooperation was investigated in [148],
where the APs could perform data decoding fully distributively or by
exploiting LSFD. The results revealed that the LSFD could provide the largest
SE under the HIs. [149, 150] focused on the impact of employing low-resolution
ADCs on the CF mMIMO. A simple asymptotic approximation for the achievable
rate was derived by considering the effects of AP and UE number, antenna
number per AP, and ADC resolution. The results showed that the achievable rate
of an arbitrary UE converges to a finite limit, independent of the ADC
resolution of the APs, as the number of APs goes to infinity. Additionally, an
ADC resolution bits allocation scheme was proposed to maximize the sum rate
given a fixed total ADC resolution bits. Authors in [151] considered the
physical layer security in a CF mMIMO system with HIs, where a lower bound for
the ergodic secrecy rate in the presence of pilot spoofing attack and
imperfect CSI is derived. Moreover, an optimal power allocation scheme was
obtained to maximize the achievable secrecy rate using the continuous
approximation and path-following algorithms. Analytical results revealed that
the hardware-quality scaling law is almost inapplicable for secure
transmission in CF mMIMO system except for some particular scaling factors. An
article that predates the CF mMIMO area but applies to the same scenario is
[152], which assessed the impact of HIs on scalable CF mMIMO systems by also
considering the effect of phase noise.

##  V Future Research Directions

In this section, we briefly highlight some major open problems and research
challenges to be addressed in future work.

###  V-A Multiple CPUs and Practical Fronthaul Topology

Much of the algorithmic design for CF mMIMO has been developed to be
transparent to the topology of the underlying network architecture [4], to
make it applicable when having one or multiple CPUs, and having parallel or
sequential fronthaul connections. In the canonical case, the network comprises
many distributed APs with independent cables to the single CPU, also known as
a star topology. Still, it is unlikely that geographically large networks will
be deployed in that manner. There might be multiple CPUs that are connected to
disjoint subsets of the APs [153], thus if two APs that belong to different
CPUs are cooperating, the fronthaul signaling will have to involve multiple
APs. Moreover, the radio-stripes topology has been proposed in [46], where a
set of APs are deployed along a fronthaul cable and, thus, have a sequential
connection to the CPU. This design is motivated by the practical need for
limiting the total cable lengths and opens up research questions related to
how the communication algorithms can be adapted to exploit the finer details
of the fronthaul topology. For example, some centralized processing schemes
can be implemented sequentially [154]. There are many open research challenges
related to distributing the signal processing over multiple CPUs and adapting
the algorithms to exploit the structure.

Each fronthaul connection will have a limited capacity. The leading theory for
CF mMIMO has been developed under the assumption of infinite fronthaul
connections, but with a general awareness that one needs to limit the number
of signals that are transmitted between APs and CPUs to achieve scalability
[4]. When dealing with a practical capacity-limited fronthaul, one must
consider the tradeoff between precision and the number of conveying signals.
As discussed in Section IV-A, it then matters where the processing is done:
Signals that are measured at the APs can either be processed there at full
precision or elsewhere with reduced precision. When it comes to signal
compression for fronthaul signaling, one can either consider model-aided or
data-driven methods, where the latter can make use of autoencoder
methodologies.

###  V-B Synchronization

Coherent signal processing is possible only if the APs maintain a sufficiently
accurate relative timing and phase synchronization. The network might have an
absolute time (phase) reference, but the APs are unsynchronized. Therefore, AP
synchronization and TDD reciprocity calibration are two critical problems to
enable CF mMIMO. Suppose each AP has a local oscillator, and the wired
fronthaul network cannot provide a sufficiently accurate common time and
frequency reference. In that case, such synchronization must occur via OTA
signaling [46]. AirSync, which provides timing and phase synchronization
accuracy, has been implemented in distributed mMIMO [155]. Specifically, it
detects the slot boundary such that all APs are time-synchronous within a
cyclic prefix (CP) of the OFDM modulation and predicts the instantaneous
carrier phase correction along with the transmit slot such that all
transmitters maintain their coherence. To limit the reciprocity and
synchronization errors, a synchronization process needs to be applied at
regular intervals. High-precision inter-node clock synchronization is a
prerequisite for joint processing of distributed mMIMO [156]. All radio access
unit (RAU) clocks in the system are assigned by the master node through IEEE
1588 PTPv2. In practice, a global position system (GPS) can also be used for
more precise synchronization. More generally, the communication theory that
underpins CF mMIMO assumes a perfect timing synchronization, which is
physically impossible over a large network, even if the clocks are
synchronized. Hence, there is room for theoretical advancements as well.

###  V-C Mobile Edge Computing

The previous subsections focused on where they carry out the lower-layer
processing in a communication network. A related concept is mobile edge
computing (MEC), where the computation/storage resources of the higher layers
in the network are pushed to the edge to alleviate the burden of core networks
[157, 158]. MEC will naturally reduce the latency since processing is moved
closer to the UEs and use general-purpose cloud computing hardware that can be
co-located with CPUs. The user-edge-cloud architecture conceived for MEC
perfectly matches the UE-AP-CPU architecture of CF mMIMO, making MEC and CF
mMIMO a perfect fit [159, 160]. Edge nodes (APs and CPUs) equipped with
computation/storage capability could deal with UEs’ computation and content
requests, and consequently, reduce the transmission delay and requirement of
the fronthaul/backhaul connection capacities. The research into this direction
is in its infancy.

###  V-D Enabling Federated Learning

Apart from using downlink to improve the channel estimation or resource
allocation in the CF mMIMO system, as described earlier in this survey, a
wireless network can also be part of the infrastructure used when implementing
machine learning algorithms for other applications. The federated learning
(FL) concept can facilitate collaborative learning processes of complex models
among the distributed devices, keeping their local training data and control
privacy. The signals sent from the UEs over the wireless network are suggested
local model updates (i.e., the model’s weights), aggregated at the core of the
network, where global model updates are determined and broadcasted to the UEs.
The distributed processing manner of FL naturally fits CF mMIMO, which makes
CF mMIMO an enabler of FL [161, 162].

###  V-E Multi-Antenna UEs

The main theory for CF mMIMO has been developed for single-antenna UEs, even
though contemporary UEs have at least two antennas, and future devices will
feature even larger arrays when operating in the mmWave bands. First steps
towards considering CF mMIMO with multi-antenna UEs are found in [163, 72,
81]. In general, the multiple antennas can either be used for spatial
multiplexing of multiple streams per UE (up to one per antenna) or for
improved precoding/combining that mitigates interference [164]. While the
achievable SE can be quantified using existing methods, there are many open
resource allocation questions related to power allocation, pilot assignment,
and precoding/combining design.

###  V-F Channel Estimation and Prediction Beyond the Block Fading Model

In practice, the wireless channels vary continuously over time and frequency,
not in the block-fading manner described in this survey and most of the
theoretical works on this topic. On the one hand, the coherence block size can
always be dimensioned in a conservative manner such that the channels are
indeed approximately constant within each block. On the other hand, underlying
physical rules dictate how the channel can evolve and frequently. By
exploiting such properties, using model-aided or data-driven approaches [165],
the communication performance can be significantly improved: pilots can be
transmitted less frequently, and/or the CSI quality can be increased. Prior
work on this topic has been done in the cellular mMIMO field [166], which
potentially can be adapted to cover CF mMIMO.

###  V-G Integrated Sensing and Computing

In future wireless communications, integrated sensing and communication (ISAC)
will be a paradigm change. Some promising ISAC-like dual-functional radar-
communication (DFRC) system has attracted substantial attention, where joint
radar sensing and multi-user communication can be simultaneously implemented
[167, 168]. And it is also interesting to investigate a DFRC-based system with
CF mMIMO for the feature of uniform coverage. However, signal processing and
system design will be the key challenges. Besides, the great demand for
computation is also an issue to address. Therefore, the DFRC system based on
scalable CF mMIMO will be investigated in the future.

##  VI Conclusion and Lessons learned

Exploiting densification and decentralization to boost the user-experienced
data rates and realize a ubiquitous service is an irresistible general trend
for future wireless communications. CF mMIMO represents an attempt to reach
this promising prospect by coordinating dense serving antennas in a
decentralized CF approach, which greatly squeezes the potential of the
multiple antenna technology so that it dynamically achieves the best
performance with the available resources. In particular, the dense deployment
of the serving antennas in CF mMIMO will result in strong macro-diversity from
a significantly smaller average distance between a UE and its closest
antennas, while the joint signal processing and scheduling among the
distributed antennas achieve array gains and spatial interference suppression
which substantially reduce the QoS variations within the coverage area.

In this paper, we have presented a comprehensive review of the concepts and
techniques proposed for CF mMIMO systems. First, we gave the motivation for CF
mMIMO and provide a brief introduction of CF mMIMO itself and the other
technologies related to it. Then we used a section to briefly provide a
tutorial about the technical foundations of CF mMIMO including the
transmission procedure and mathematical system model. The core of the paper
provides an extensive survey on the state-of-the-art schemes and algorithms
available in the literature for the resource allocation and signal processing
(i.e., channel estimation, combining and precoding, user access and
association, and power control) and practical implementation (i.e., fronthaul
limitation and hardware impairment) in CF mMIMO systems. We then highlighted
the open research areas for CF mMIMO (e.g., multiple CPUs cooperation, MEC,
enabling machining learning, etc.) and proposed potential approaches for
solutions.

Although the research on this topic is still in the exploratory phase, the
primary demonstrations, field tests, and prototypes of CF mMIMO systems have
been ongoing across different projects in academia and industry. How to
realize the scalable intelligent system deployment with new mathematical
tools, new applications, and new standardizations becomes a very attractive
open issue for all researchers in this field. Though many challenges remain to
address, CF mMIMO shows great potential to meet the ubiquitous high QoS
demands of the 6G communications. In the foreseeable future, research on CF
mMIMO will continue to mature. With no doubt, this technology with its
concepts will open up new frontiers in wireless services and applications.

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