dust, extinction — planets and satellites: formation — protoplanetary disks —
submillimeter: planetary systems — magnetohydrodynamics (MHD) — telescopes

# Thermal Tomography of the Inner Regions of Protoplanetary Disks with the
ngVLA and ALMA

Satoshi Okuzumi  11affiliation: Tokyo Institute of Technology, 2–12–1
Ookayama,
Meguro, Tokyo 152–8551 Munetake Momose  22affiliation: Ibaraki University,
2–1–1 Bunkyo,
Mito, Ibaraki 310-8512 and Akimasa Kataoka33affiliation: National Astronomical
Observatory of Japan, 2–21–1 Osawa,
Mitaka, Tokyo 181–8588<EMAIL_ADDRESS>
###### Abstract

Understanding the temperature structure of protoplanetary disks is crucial for
answering the fundamental question of when and where in the disks rocky
planets like our own form. However, the thermal structure of the inner few au
of the disks is poorly understood not only because of lack of observational
constraints but also because of the uncertainty of accretion heating
processes. Here, we propose thermal tomography of the inner regions of
protoplanetary disks with the ngVLA and ALMA. The proposed approach is based
on the assumption that the inner disk regions are optically thick at
submillimeter wavelengths but are marginally optically thin at longer
millimeter wavelengths. By combining high-resolution millimeter continuum
images from the ngVLA with submillimeter images at comparable resolutions from
ALMA, we will be able to reconstruct the radial and vertical structure of the
inner few au disk regions. We demonstrate that the thermal tomography we
propose can be used to constrain the efficiency of midplane accretion heating,
a process that controls the timing of snow-line migration to the rocky planet-
forming region, in the few au regions of protoplanetary disks at a distance of
140 pc.

##  1 Introduction

The temperature structure of protoplanetary disks dictates where planets of
different compositions form in the disks. Rocky planets including the Earth
are generally believed to have formed inside of the snow line, where the disk
temperature reaches the sublimation temperature of water ice. Enhancement of
solids outside the snow line due to ice condensation may help gas giants form
within the lifetime of the gas disks (Kokubo & Ida, 2002). The composition of
solids may even determine how planets form because the stickiness of solid
particles generally depends on their surface composition (Chokshi et al.,
1993).

Despite its importance in planet formation, the temperature structure of the
inner regions of protoplanetary disks is poorly understood. Conventionally,
the disk temperature structure has been studied based on the viscous accretion
disk model that assumes vertically uniform viscosity. According to this
classical disk model, accretion heating at the midplane pushes the midplane
snow line in $\sim 1~{}\rm Myr$-old disks out to 2–3 au (Davis, 2005; Garaud &
Lin, 2007; Oka et al., 2011; Bitsch et al., 2015). However, the validity of
the classical viscous model has been questioned by recent
magnetohydrodynamical (MHD) simulations of protoplanetary disks showing that
accretion heating can only take place near the disk surface, where the
ionization fraction is relatively high (Hirose & Turner, 2011; Mori et al.,
2019). A recent disk temperature model based on MHD simulations predicts that
the snow line in $\sim 1~{}\rm Myr$-old disks must lie interior to 1 au (Mori
et al., 2021). This raises the fundamental question of why the Earth is so dry
with a bulk water content likely well below 1 wt% (Abe et al., 2000; Marty,
2012), because the solids outside the snow line in the solar nebula would have
contained water ice in excess of 1 wt% as witnessed by carbonaceous chondrites
and comets (e.g., van Dishoeck et al., 2014). In fact, simulations show that
rocky planetary embryos at 1 au are likely to grow into waterworlds if the
snow line migrates inward of the 1 au orbit within 1 Myr after disk formation
(Sato et al., 2016) unless some mechanism stops the inward migration of the
icy dust particles (e.g., Morbidelli et al., 2016).

To summarize, constraining the thermal structure of the inner regions of
protoplanetary disks from astronomical observations is crucial for fully
understanding how rocky planets form. In this article, we propose a
methodology for testing disk temperature models by multi-wavelength, high-
resolution imaging with the ngVLA and ALMA.

##  2 Theoretical Background

Figure 1: Temperature as a function of the cylindrical radius $r$ and distance
from the midplane $z$ for the viscous and MHD disks (panels (a) and (b),
respectively). The contours mark $T=100$, 200, and 300 K. The maps are
truncated at three scale heights below and above the midplane.

In principle, the efficiency of midplane heating can be studied by measuring
disk temperatures at different vertical depths. In classical accretion disk
models with vertically uniform kinematic viscosity, accretion heating mainly
takes place at the midplane, leading to the midplane temperature higher than
the surface temperature in inner disk regions (e.g., Davis, 2005). In
contrast, in MHD accretion disk models, Joule heating takes place well above
the midplane, leading to vertically uniform temperature structure with the
midplane no hotter than the surface (Hirose & Turner, 2011; Mori et al.,
2019). This difference critically affects when the snow line crosses the 1 au
orbit in the disks (Mori et al., 2021).

Multi-wavelength observations of the disks’ inner regions using the ngVLA and
ALMA will enable us to perform thermal tomography—imaging of the temperature
structure in both the radial and vertical directions—of the rocky planet-
forming regions. The ngVLA will provide access to the midplane temperature
structure of the inner few au of the disks at millimeter to centimeter
wavelengths, at which the inner regions are likely to be optically marginally
thick to optically thin. ALMA is already capable of mapping the dust continuum
emission from well above the midplane at comparable resolutions at optically
thick, submillimeter wavelengths.

##  3 Proof-of-concept Simulations

###  3.1 Models

We examine the feasibility of the disk thermal tomography described above
using two disk temperature models. One is the classical viscous disk model
(Lynden-Bell & Pringle, 1974) with the viscous alpha parameter $\alpha$ taken
to be $10^{-3}$ throughout the disk. The other is the MHD accretion disk model
recently proposed by Mori et al. (2021), which assumes accretion heating to
occur on thin layers near the disk surfaces. For simplicity, we take the depth
of the heating layers to be $0.4~{}\rm g~{}cm^{-2}$ in column mass density
measured from infinity (Mori et al., 2019, 2021). The radial and vertical
temperature profiles for the two accretion disk models are generated using the
plane-parallel radiative transfer model by Hubeny (1990). In addition to
accretion heating, we also include heating by stellar radiation (Chiang &
Goldreich, 1997). For both models, we assume a uniform accretion rate of
$\dot{M}=2\times 10^{-7}~{}M_{\odot}~{}\rm yr^{-1}$, a stellar mass of
$1M_{\odot}$, a stellar luminosity of $1L_{\odot}$, and a uniform infrared
opacity of $5.0~{}\rm cm^{2}~{}g^{-1}$. Figure 1 shows the two-dimensional
temperature maps for the two disk models.

The two-dimensional temperature profiles are used to generate maps of dust
continuum emission from a face-on disk at submillimeter to centimeter
wavelengths. The disk opacity in this wavelength range is assumed to follow a
power law $\kappa(\lambda)=6\times 10^{-3}(\lambda/3~{}\rm mm)^{-1.7}~{}\rm
cm^{2}~{}g^{-1}$, where $\lambda$ is the wavelength. Again, the opacity is
taken to be uniform throughout the disks, which means that we neglect possible
spatial variations of the dust-to-gas mass ratio and grain size distribution
due to dust evolution.

To mimic observations, Gaussian convolution is applied to smooth the
theoretical continuum intensity profiles at angular resolutions of ALMA and
the ngVLA. Specifically, we adopt angular resolutions of 12, 5, and 16 mas for
the ALMA 0.87 mm, ngVLA 3 mm, and ngVLA 1 cm bands, respectively, estimated by
Ricci et al. (2018) assuming 8 and 20 hr observations with ALMA and the ngVLA,
respectively. The modeled disks are assumed to be located a distance of
$d=140$ pc away from the Earth, resulting in spatial resolutions of 1.7, 0.7,
and 2.2 au for the 0.87 mm, 3 mm, and 1 cm band images, respectively. Based on
the sensitivity estimates by Ricci et al. (2018), the rms noise in the
brightness temperature maps for the three bands are estimated to be $\approx
0.6$, 2, and 0.8 K. The sensitivities are high enough for a precise
measurement of thermal emission from around the snow line, where $T\approx
160~{}\rm K$.

###  3.2 Results

Figure 2: Upper panels: maps of the brightness temperatures $T_{\rm B}$ at
$\lambda=0.87$ mm, 3 mm, and 1 cm (left, center, and right panels,
respectively) from a face-on viscous accretion disk. The contours mark $T_{\rm
B}=100$, 200, and 300 K. Lower panels: synthetic brightness temperature maps
after Gaussian smoothing, assuming $d=140$ pc and spatial resolutions of 12,
5, and 16 mas for the ALMA 0.87 mm, ngVLA 3 mm, and ngVLA 1 cm bands,
respectively (Ricci et al., 2018).

Figure 2 presents the maps of the brightness temperatures $T_{\rm B}$ at three
wavelengths from the viscous disk model. The upper and lower panels are the
maps before and after Gaussian smoothing, respectively, Comparison between the
upper and lower panels shows that the angular resolutions of the ALMA
submillimeter band and ngVLA millimeter bands are high enough to resolve the
radial distribution of $T_{\rm B}$ in the inner few au. It is interesting to
note that the brightness temperature of the inner disk region is maximized at
$\lambda=$ 3 mm and decreases both longward and shortward of 3 mm. This
reflects the vertical thermal structure of the uniformly viscous disk, in
which the temperature decreases as we go away from the midplane. At $\lambda=$
0.87 mm, the inner region is optically thick and the thermal emission comes
from $\approx 2$ scale heights above the midplane. At $\lambda=3~{}\rm mm$,
the inner region is marginally optically thick and the thermal emission is
dominated by the midplane. Longward of 3 mm, the inner region is optically
thinner at longer wavelengths, and the brightness temperature of the region
decreases with increasing $\lambda$. This demonstrates the importance of
multi-wavelength observations using ALMA and the ngVLA in studying internal
heating in the inner regions of protoplanetary disks.

Figure 3: Same as Figure 2 but for the MHD disk model.

Figure 3 is the same as Figure 2, but for the MHD accretion disk model. The
brightness temperatures of the MHD disk are overall lower than those of the
viscous disk. Still, the synthetic map for the ngVLA $3~{}\rm mm$ band
accurately captures the brightness temperature profile of the MHD disk up to
$T_{\rm B}\approx 200~{}\rm K$ and thus allows us to study the thermal
structure around the water snow line ($T\approx 160~{}\rm K$). Because of its
vertically uniform temperature structure, the MHD disk model produces nearly
identical brightness temperature maps (except for the unresolved very inner
region) at optically thick wavelengths of $\lambda\lesssim 3~{}\rm mm$.
Therefore, multi-band observations at ALMA submillimeter bands and the ngVLA 3
mm band would allow us to test whether midplane heating is inefficient around
the midplane in a disk.

##  4 Conclusion

We have demonstrated that thermal tomography with the ngVLA and ALMA will
enable us to discriminate between different models for disk thermal structure.
Such observations will provide us with a basis for fully understanding when
and where terrestrial planets form. However, the models we have used here
neglect a number of important dust evolution processes, including coagulation,
vertical sedimentation, and radial drift. In particular, condensation,
sublimation, and sintering of icy particles can significantly affect the size
and spatial distribution of solids around the snow line (e.g., Birnstiel et
al., 2010; Ros & Johansen, 2013; Okuzumi et al., 2016; Schoonenberg & Ormel,
2017). All these dust evolution processes may alter the temperature structure
and emission profiles in the inner regions of protoplanetary disks (e.g.,
Banzatti et al., 2015; Pinilla et al., 2017) and therefore should be included
in future modeling.

##  5 Acknowledgments

We thank Takahiro Ueda for discussions. This work was supported by JSPS
KAKENHI Grant Numbers JP18H05438, JP19K03926, JP19K03941, and JP20H00182.

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# A Classical Algorithm Which Also Beats
$\frac{1}{2}+\frac{2}{\pi}\frac{1}{\sqrt{D}}$ For High Girth MAX-CUT

Matthew B. Hastings  Station Q, Microsoft Quantum, Santa Barbara, CA
93106-6105, USA  Microsoft Quantum and Microsoft Research, Redmond, WA 98052,
USA

###### Abstract

We give a simple classical algorithm which provably achieves the performance
in the title. The algorithm is a simple modification of the Gaussian wave
process[9].

MAX-CUT is a well-studied optimization problem. On high girth graphs with
degree $D$, there are a variety of classical algorithms which achieve
approximation ratio $\frac{1}{2}+\frac{2}{\pi}\frac{1}{\sqrt{D}}$ up to
subleading corrections. Examples include the Gaussian wave process [9] and
also [12].

There is also a classical algorithm due to Montanari and collaborators[10, 1,
5] which, assuming one conjecture, beats this performance and indeed achieves
arbitrarily close to the optimal performance. This conjecture is generally
believed to hold by the spin-glass community. However, this ratio
$\frac{1}{2}+\frac{2}{\pi}\frac{1}{\sqrt{D}}$ is the best approximation ratio
that we know for a classical algorithm that can be proven without additional
assumptions; see [12].

Recently, it was pointed out[3] that $p=11$ QAOA attains approximation ratio
$\frac{1}{2}+C\frac{1}{\sqrt{D}}$ on high girth graphs for some $C>2/\pi$. The
point of the present short note is to show that a very simple modification of
the Gaussian wave process can also achieve some other $C>2/\pi$.

We make no attempt to optimize the performance. However, since numerical
effort has been expended to optimize the performance of QAOA, it seems that at
least as much effort should be expended to optimize the performance of local
classical algorithms, by performing numerical integrations to determine their
performance and optimize parameters. Indeed, the algorithm of [8] is a single
round local algorithm that was obtained using computer optimization.

##  1 The Problem and Previous Algorithms

We consider a graph with fixed degree $D$. We assume that the girth of the
graph is large. We consider the problem MAX-CUT: maximizing the fraction of
edges which are cut by a bipartition of the vertices.

We define a bipartition of the vertices by assigning a sign, $\pm 1$, to each
vertex. We call that sign the “part" of that vertex and denote it $p_{u}$ for
vertex $u$. An edge is cut if it connects two vertices with opposite sign.

If we assign the signs independently from the uniform distribution, then each
edge is cut with probability $1/2$.

The best one can achieve in general for large $D$ is to find a cut where a
fraction $1/2+C/\sqrt{D}+o(1/\sqrt{D})$ of edges are cut. Since in this short
note we only care about the performance at large $D$ and large girth (doing
anything else would require something more complicated than this simple
calculation), we care only about achieving a large value of $C$.

The algorithm of [5] is shown (under a widely-believed conjecture) to achieve
arbitrarily close to the Parisi[11, 4] value of $C$, $C=0.763\ldots\ $ Note
$2/\pi=0.6366\ldots$

##  2 The Algorithm

We now give the algorithm. In Section 2.1 we give a simple algorithm that
achieves arbitrarily close to $C=2/\pi$. This algorithm is essentially the
same as that in [9]. In Section 2.2, we show how to improve beyond $C=2/\pi$.

###  2.1 Arbitrarily Close to $2/\pi$

There is a real variable for each vertex. Initialize the real variables by
choosing them to be independent Gaussian, all with zero mean and variance $1$.
Label vertices $u,v,\ldots$ and label the corresponding real variables
$x_{u},x_{v},\ldots$.

Then, in parallel update each $x_{u}$ to some new $x^{\prime}_{u}$. A simple
update rule which achieves $C>0$ is

$x^{\prime}_{u}=x_{u}+a\sum_{v,\,\mathrm{dist}(u,v)=1}x_{v},$ (1)

where $a$ is a real scalar and the notation $\mathrm{dist}(u,v)=1$ indicates
the sum over all $v$ which neighbor $u$. Here $\mathrm{dist}(u,v)$ is the
graph metric.

Finally, round each variable $x^{\prime}_{u}$ by assigning each vertex $u$ to
the part $p_{u}\sigma(x^{\prime}_{u})$, where $\sigma(\cdot)$ denotes the sign
function.

Choosing $a$ to be negative and proportional to $1/\sqrt{D}$ gives a positive
value of $C$.

In a slightly more complicated algorithm, everything is the same except we
replace Eq. 1 with

$x^{\prime}_{u}=\sum_{k=0}^{K}a_{k}\sum_{v,\,\mathrm{dist}(u,v)=k}x_{v},$ (2)

for some given integer $k$ and some sequence of reals $a_{k}$. Note that for
$a_{0}=1,a_{1}=a$, Eq. 2 reduces to Eq. 1.

We claim that by choosing the $a_{k}$ appropriately, we may take

$\frac{\mathbb{E}[x^{\prime}_{u}x^{\prime}_{v}]}{\mathbb{E}[(x^{\prime}_{u})^{2}]}=-\frac{2}{\sqrt{D}}\cdot(1-o(1)),$
(3)

for neighboring $u,v$, where the $o(1)$ is a quantity tending to zero as
$D,K\rightarrow\infty$. One may see this by considering the spectrum of the
adjacency matrix of an infinite tree. However, a simple explicit choice is
$a_{k}=(-1/\sqrt{D})^{k}$, up to a multiplicative factor for normalization. In
this case,
$\mathbb{E}[(x^{\prime}_{u})^{2}]=\sum_{k=0}^{K}D^{-k}N_{k}=K\cdot(1-o(1))$,
where $N_{k}=D^{k}\cdot(1-o(1))$ is the number of vertices at distance $k$. At
the same time, for neighboring $u,v$,

$\displaystyle\mathbb{E}[x^{\prime}_{u}x^{\prime}_{v}]$ $\displaystyle=$
$\displaystyle\sum_{w,\,\mathrm{dist}(u,w)=\mathrm{dist}(v,w)+1\leq
K}(-1/\sqrt{D})^{\mathrm{dist}(u,w)+\mathrm{dist}(v,w)}+\sum_{w,\,\mathrm{dist}(v,w)=\mathrm{dist}(u,w)+1\leq
K}(-1/\sqrt{D})^{\mathrm{dist}(u,w)+\mathrm{dist}(v,w)}$ $\displaystyle=$
$\displaystyle-(2/\sqrt{D})(K-1)\cdot(1-o(1))=(-2/\sqrt{D})K\cdot(1-o(1)).$

From Eq. 3 it follows that[6]

$\mathbb{E}[\sigma(x^{\prime}_{u})\sigma(x^{\prime}_{v})]=-\frac{2}{\pi}\frac{2}{\sqrt{D}}\cdot(1-o(1)).$
(5)

Finally, the expected cut fraction equals
$1/2-(1/2)\mathbb{E}[\sigma(x^{\prime}_{u})\sigma(x^{\prime}_{v})]=1/2+(2/\pi)\frac{1}{\sqrt{D}}\cdot(1-o(1)).$

###  2.2 And One More Step

We now modify the algorithm further. The initialization is as before, and we
use Eq. 2 to define $x^{\prime}_{u}$. However we then choose an
$\epsilon$-fraction of vertices uniformly at random, for some $\epsilon>0$,
and “mark" those vertices. For each unmarked vertex $u$, we let
$p_{u}=\sigma(x^{\prime}_{u})$. For each marked vertex $u$, we compute the
majority of $\sigma(x^{\prime}_{v})$ over $v$ which neighbor that vertex (if
there is a tie, choose the majority arbitrarily) and then return $-1$ times
that majority as the part.

In words: round all $x^{\prime}_{u}$ to signs. Then, on an $\epsilon$-fraction
of vertices, perform a greedy update, flipping the part if it improves the
cut.

Let us analyze the performance of this algorithm.

Neither vertex marked— For any neighboring pair $u,v$, there is a
$(1-\epsilon)^{2}$ probability that both vertices are unmarked, in which case
$\mathbb{E}[p_{u}p_{v}]=-\frac{2}{\pi}\frac{2}{\sqrt{D}}\cdot(1-o(1))$.

One vertex marked— There is a probability $2\epsilon(1-\epsilon)$ that one
vertex is marked and the other is unmarked. Suppose $u$ is marked and $v$ is
unmarked. Then $\mathbb{E}[p_{u}p_{v}]$ is _at least as negative_ as if both
were unmarked, i.e., $\leq-\frac{2}{\pi}\frac{2}{\sqrt{D}}\cdot(1-o(1))$,
because the flip of the marked vertex $u$ is done greedily and so it cannot
increase
$\sum_{w,\,\mathrm{dist}(u,w)=1}\mathbb{E}[p_{u}\sigma(x^{\prime}_{w})]$, and
since $v$ is unmarked, $p_{v}=\sigma(x^{\prime}_{v})$.

Further, we claim that
$\mathbb{E}[p_{u}p_{v}]=-c^{\prime}\frac{2}{\sqrt{D}}\cdot(1-o(1))$, for some
$c$ strictly greater than $2/\pi$. To see this, consider the distribution of
$x^{\prime}_{w}$ for $w$ which neighbor $u$. These $x^{\prime}_{w}$ are
correlated Gaussian variables. Normalize so that
$\mathbb{E}[(x^{\prime}_{u})^{2}]=1$. The correlation matrix
$\mathbb{E}[x^{\prime}_{w_{1}}x^{\prime}_{w_{2}}]$ is a $(D-1)$-by-$(D-1)$
matrix, with diagonal entries equal to $1$ and all off-diagonal entries equal
to each other. Off-diagonal entries are equal to $\Theta(1/D)$; in general,
for arbitrary vertices $y,z$ the correlation is
$(-1)^{\mathrm{dist}(y,z)}\Theta(1/\sqrt{D})^{\mathrm{dist}(y,z)}$. We can
generate such a correlation matrix by the following process: generate
independent identically distributed random Gaussian with zero mean and
variance $1-o(1)$ for each $w$ neighboring $u$, and generate a random Gaussian
$g$ with zero mean and variance $\Theta(1/D)$. Then, add $g$ to the variable
on each vertex, giving $x^{\prime}_{w}$ for the given vertex $w$.

So, conditioned on $g$, the signs $\sigma(x^{\prime}_{w})$ for $w$ which
neighbor $u$ are independent random variables. They are identically
distributed, with a bias which is $\Theta(g)$, i.e., they have a probability
$1/2+\Theta(g)$ of being the same as $\sigma(x^{\prime}_{u})$. With
probability at least $0.99$, $g$ is bounded by some constant multiple of
$1/\sqrt{D}$, in which case the sum of biases over neighbors is comparable to
the root-mean-square variance in the sum of the signs, i.e., both are
$\Theta(\sqrt{D})$. Hence, there is a probability $\Omega(1)$ that we flip
$p_{u}$ in the last round of the algorithm, i.e., that
$p_{u}=-\sigma(x^{\prime}_{u})$, and that further, when we do flip, the
expectation value of the absolute value of sum of signs is $\Theta(\sqrt{D})$.
Hence, the improvement in cut, summed over neighbors, by flipping $p_{u}$ is
$\Theta(\sqrt{D})$.

So, the claim follows that
$\mathbb{E}[p_{u}p_{v}]=-c^{\prime}\frac{2}{\sqrt{D}}\cdot(1-o(1))$, for some
$c^{\prime}$ strictly greater than $2/\pi$.

Both Vertices Marked— With probability $\epsilon^{2}$, both $u,v$ are marked.
We claim that $\mathbb{E}[p_{u}p_{v}]\leq O(1/\sqrt{D})$. Likely the
expectation value is negative and the expectation value could be estimated by
some numerical integration, but we are doing the minimum amount of work here.

Indeed, suppose $\mathbb{E}[p_{u}p_{v}]=\omega(1/\sqrt{D})$. Then, we could
define a different algorithm which took $\epsilon=1$ (i.e., it marks all
vertices) and which replaced our given choice of $a_{k}$ in Eq. 2 by
$a_{k}(-1)^{k}$, and then we claim
$\mathbb{E}[p_{u}p_{v}]=-\omega(1/\sqrt{D})$, asymptotically better than the
optimum!

To see this, imagine flipping the sign of $x_{w}$ for $\mathrm{dist}(u,w)$
odd. This has no effect on the distribution of $x_{w}$. Then, the change in
$a_{k}$ also flips the sign of $x^{\prime}_{w}$ for $\mathrm{dist}(u,w)$ odd,
and the final greedy update flips the sign of $p_{w}$ for $\mathrm{dist}(u,w)$
_even_.

Performance of Algorithm— We have

$\mathbb{E}[p_{u}p_{v}]\leq-(1-\epsilon)^{2}\frac{2}{\pi}\frac{2}{\sqrt{D}}\cdot(1-o(1))-2\epsilon(1-\epsilon)c^{\prime}\frac{2}{\sqrt{D}}\cdot(1-o(1))+\epsilon^{2}O(1/\sqrt{D}).$
(6)

Up to subleading terms, this is a quadratic function of $\epsilon$, with all
coefficients proportional to $1/\sqrt{D}$ and the linear term is negative.
Hence, for some constant $\epsilon>0$, the algorithm achieves
$\mathbb{E}[p_{u}p_{v}]\leq-C\frac{2}{\sqrt{D}}\cdot(1-o(1))$ for some $C$
strictly greater than $2/\pi$.

###  2.3 Related Algorithms

This algorithm is similar to [7, 2]. There, a variable was initialized between
$[-1,+1]$ on each vertex. One then repeats the following two-step process:
update each variable with a linear function of its value and the sum of the
values on its neighbors, and then replace the value of the variable $x_{u}$ on
each vertex $u$ with $\tanh(\beta x_{u})$ for some $\beta$. After some number
of rounds, finally round each variable by replacing it with its sign. The
linear functions and $\beta$ could be taken to be functions of the round.

Suppose we replace the initialization with Gaussian variables instead of
uniformly distributed ones. Keeping $\beta$ small for the early rounds, this
dynamics remains in the linear regime, and by iterating it we reproduce Eq. 2
up to small nonlinearities. Taking a large value of $\beta$ for the next round
implements the rounding to a value close to $\pm 1$. The “one more step" can
be done if we allow the injection of randomness: add random noise to each
variable so that there is a small chance that each variable changes from close
to $\pm 1$ to close to $0$. Then, implement one more linear transform and
round again.

Perhaps even without modification, this algorithm achieves $C>2/\pi$. Other
natural variants of the algorithm considered here include, for example,
rounding the $x^{\prime}_{u}$, and then following with a threshold algorithm
which flips the sign if sufficiently many (compared to $\sqrt{D}$) of the
neighbors have the same part, or adding some low order nonlinearities before
rounding. Numerical calculation can help optimize performance.

Acknowledgments— I thank K. Marwaha for pointing out [5] and other comments.

## References

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Given $r_{0},D>0$, $C>1$ and $K\geq 0$, define ${\cal M}(r_{0},C,K,D)$ as the
set of arcwise connected length spaces $(X,d)$, with diameter $\leq D$, whose
fundamental group $\Gamma_{X}$ is torsion free and belongs to ${\rm Hyp}_{\rm
act}$, and which admit a metric universal cover
$\big{(}\widetilde{X},\tilde{d}\big{)}$ satisfying the following properties:
$\big{(}\widetilde{X},\tilde{d}\big{)}$ is Busemann with the property of
extension of local geodesics and admits a $\Gamma_{X}$-invariant measure which
verifies a Bishop–Gromov inequality at scale $r_{0}$, with factor $C$ and
exponent $K$. The homotopy class of a space $(X,d)$ being the subset of all
the spaces $(Y,d)\in{\cal M}(r_{0},C,K,D)$ which are homotopically equivalent
to $(X,d)$ and $N_{3}\big{(}C,Kr_{0},\frac{D}{r_{0}}\big{)}$ being the
universal constant introduced in Corollary 3.46, we obtain the

######  Theorem 3.51.

For every $r_{0}$, $D>0$, $C>1$ and $K\geq 0$, there are less than
$N_{3}\big{(}C,Kr_{0},\frac{D}{r_{0}}\big{)}$ homotopy classes of spaces in
${\cal M}(r_{0},C,K,D)$.

Let us now define ${\cal H}(\delta,K,D)$ as the set of arcwise connected
length spaces $(X,d)$, with diameter $\leq D$, whose fundamental group
$\Gamma_{X}$ is torsion free, and which admit a metric universal cover
$\big{(}\widetilde{X},\tilde{d}\big{)}$ satisfying the following properties:
$\big{(}\widetilde{X},\tilde{d}\big{)}$ is Busemann, $\delta$-hyperbolic, with
the property of extension of local geodesics and entropy $\leq K$. Recalling
that $N_{4}\left(\delta,K,D\right)$ is the universal constant introduced in
Theorem 3.47, we obtain the

######  Theorem 3.52.

For every $\delta,K,D>0$, there are less than $N_{4}\left(\delta,K,D\right)$
homotopy classes of spaces in ${\cal H}(\delta,K,D)$.

The proof of Theorem 3.51 (resp. of Theorem 3.52) is a consequence of the fact
that, modulo isomorphisms, the number of fundamental groups of spaces
belonging to ${\cal M}(r_{0},C,K,D)$ (resp. to ${\cal H}(\delta,K,D)$) is less
than $N_{3}\big{(}C,Kr_{0},\frac{D}{r_{0}}\big{)}$ (resp. than
$N_{4}(\delta,K,D)$) by Corollary 3.46 (resp. by Theorem 3.47), and of the
contractibility of the universal cover of every $(X,d)$ in ${\cal
M}(r_{0},C,K,D)$ (resp. in ${\cal H}(\delta,K,D)$) along the geodesics issued
from one point.

The problem of trying to prove that, on some families of metric spaces which
are moreover compact for the Gromov–Hausdorff distance, a homotopical
finiteness result implies a finiteness theorem for topologies has a quite long
history, it would thus be interesting to prove that the family ${\cal
M}(r_{0},C,K,D)$ (resp. ${\cal H}(\delta,K,D)$) is compact with respect to the
Gromov–Hausdorff distance. A section of [5] is devoted to this problem,
however we already proved the

######  Theorem 3.53.

The family ${\cal H}_{0}(\delta,K,D)$ of the elements of ${\cal
H}(\delta,K,D)$ which are locally ${\rm CAT}(0)$ is compact with respect to
the Gromov–Hausdorff distance. Moreover there exists a universal constant
$\operatorname{\varepsilon}_{0}=\operatorname{\varepsilon}_{0}(\delta,K,D)>0$
such that every ball of radius $\operatorname{\varepsilon}_{0}$, in any
$(X,d)\in{\cal H}(\delta,K,D)$ which is not a circle, is contractible.

In addition to the compactness property, this last theorem proves the
existence of a lower bound of the contractibility radius; this is an important
tool in proving that homotopical finiteness implies topological finiteness.
This lower bound of the contractibility radius is a consequence of the lower
bound of the systole given by Theorems 3.42 and 3.15.

######  Question 3.54.

Does $\,{\cal H}_{0}(\delta,H,D)\,$ and ${\cal M}(r_{0},C,K,D)$ only contain a
finite number of topologies?

In [5], we give a partial answer to this question (this is only a work in
progress): the subset of the elements of ${\cal H}_{0}(\delta,K,D)$ (resp. of
${\cal M}(r_{0},C,K,D)$) which are topological manifolds only contains a
finite number of topologies

A (very rough) sketch of the proof is as follows:

  * •

In dimension $\neq 3$, it is a consequence of the following result of S. Ferry
[23, Theorem 1]: in the set of compact metric spaces (endowed with the
Gromov–Hausdorff distance), any precompact subset, whose elements all have
contractibility radius uniformly bounded from below, contains a finite number
of topologies. Ferry’s result concludes a series of results of L. Siebenmann
[63], M. Gromov [28], F. Farrell–L. Jones [22], K. Grove–P. Petersen–J. Wu
[36] (corrected in [37]), R. Greene–P. Petersen [26], … this list being non
exhaustive. We thus have to prove first that ${\cal H}_{0}(\delta,K,D)$ (resp.
${\cal M}(r_{0},C,K,D)$) is precompact, secondly that the contractibility
radii of all the elements of ${\cal H}_{0}(\delta,K,D)$ (resp. of ${\cal
M}(r_{0},C,K,D)$) are bounded from below. The precompactness is a corollary of
M. Gromov’s precompactness theorem (see [31, Proposition 5.2]) and of Theorem
3.39$(iii)$, which bounds the number of $\operatorname{\varepsilon}$-balls in
a packing of every element of ${\cal H}_{0}(\delta,K,D)$ (resp. of ${\cal
M}(r_{0},C,K,D)$). The lower bound of the contractibility radius is given, in
the Busemann case, by Theorem 3.53.

  * •

In dimension 3, the same finiteness result is a corollary of G. Perelman’s
proof of Thurston’s geometrisation conjecture and Poincaré’s conjecture, with
the addition of a result of M. Kreck and W. Lück [45, Theorem 0.7].

###  3.6 Appendices

####  3.6.1 Geodesics

The following definitions are classical (see for example [8, Définitions
I.1.3, p. 4]):

######  Definitions 3.55.

In any metric space $(X,d)$

  * •

a (normal) “geodesic” is a map $c$ from some interval $I\subset\mathbb{R}$ to
$X$ verifying the property: for every $t,\,t^{\prime}\in I$
$d(c(t),c(t^{\prime}))=|t-t^{\prime}|$,

  * •

when $I$ is a closed interval (resp. $]{-}\infty,+\infty[$) the geodesic is
called a geodesic segment (resp. a geodesic line), the image of a geodesic
segment $c$ with origin $x$ and endpoint $y$ is often denoted by $[x,y]$
(though this does not suppose that this geodesic segment is unique),

  * •

a (normal) “local geodesic” is a map $c$ from some interval
$I\subset\mathbb{R}$ to $X$ verifying the property: for every $t\in I$, there
exists $\operatorname{\varepsilon}>0$ such that
$d(c(t^{\prime}),c(t^{\prime\prime}))=|t^{\prime}-t^{\prime\prime}|$ for every
$t^{\prime},\,t^{\prime\prime}\in\,]t-\operatorname{\varepsilon},t+\operatorname{\varepsilon}[\,\cap
I$,

  * •

a metric space $(X,d)$ is geodesic if any two points can be joined by at least
one geodesic.

######  Definition 3.56.

Given a geodesic segment $[x_{0},x_{1}]$, the “natural parametrization” of
this segment is the map $t\mapsto x_{t}$ from $[0,1]$ to $[x_{0},x_{1}]$,
defined by $d(x_{0},x_{t})=t\,d(x_{0},x_{1})$.

######  Definitions 3.57 (cf. [8, Definition II.5.7, p. 208]).

A geodesic metric space is said to verify the “property of extension of local
geodesics” if, for every local geodesic $c\colon[a,b]\to X$ ($a<b$), there
exists $\operatorname{\varepsilon}>0$ and a local (eventually not unique)
geodesic $c^{\prime}\colon[a,b+\operatorname{\varepsilon}]\to X$ which extends
$c$ $\big{(}$i.e., $c^{\prime}_{|_{[a,b]}}=c\big{)}$.

This space is said to be “geodesically complete” if every local geodesic
$c\colon[a,b]\to X$ ($a<b$), can be extended as a local geodesic
$\bar{c}\colon]{-}\infty,+\infty[{}\to X$.

Notice that every complete or open Riemannian manifold verifies the property
of extension of local geodesics.

######  Lemma 3.58 (cf. [8, Lemma II.5.8(1), p. 208]).

A _complete_ geodesic metric space verifies the property of extension of local
geodesics if and only if it is geodesically complete.

####  3.6.2 About Gromov-hyperbolic spaces

Given three nonnegative numbers $\alpha$, $\beta$, $\gamma$, we define the
tripod $T:=T(\alpha,\beta,\gamma)$ as the metric simplicial tree with $3$
vertices $x^{\prime}$, $y^{\prime}$, $z^{\prime}$ of valence $1$ (the
“endpoints”), one vertex $c$ of valence $3$ $($the “branching point”$)$, and
$3$ edges $[cx^{\prime}]$, $[cy^{\prime}]$, $[cz^{\prime}]$ of respective
lengths $\alpha$, $\beta$, $\gamma$ (the “branches”). We denote by
$d_{T}(u,v)$ the distance on this tree between two points $u,v\in T$, i.e.,
the minimal length of a path contained in $T$ and joining $u$ to $v$.

For the sake of simplicity, we only consider geodesic metric spaces (see
definition in Section 3.6.1). In such a space a geodesic triangle
$\Delta=[x,y,z]$ is the union of three geodesics $[x,y]$, $[y,z]$ and $[z,x]$.
Given three points $x,y,z$ in a geodesic metric space, there exists at least
one geodesic triangle $\Delta=[x,y,z]$ whose sides have respective lengths
$d(x,y),\,d(y,z)$ and $d(x,z)$.

######  Lemma 3.59.

To any geodesic triangle $\Delta$ corresponds a metric tripod
$(T_{\Delta},d_{T})$ and a surjective map $f_{\Delta}\colon\Delta\rightarrow
T_{\Delta}$ $($called the “approximation of $\Delta$ by a tripod”$)$ such
that, in restriction to each side of $\Delta$, $f_{\Delta}$ is an isometry,

Indeed, $T_{\Delta}$ is constructed as the tripod $T(\alpha,\beta,\gamma)$,
where (by the triangle inequality) $(\alpha,\beta,\gamma)$ is the unique
element of $[0,+\infty[^{3}$ such that $d(x,y)=\alpha+\beta$,
$d(x,z)=\alpha+\gamma$ and $d(y,z)=\beta+\gamma$. This choice of
$(\alpha,\beta,\gamma)$ implies the existence of the map
$f_{\Delta}\colon\Delta\rightarrow T_{\Delta}$ as asserted in Lemma 3.59.

######  Definitions 3.60.

A geodesic triangle $\Delta$ of $(X,d)$ is said to be $\delta$-thin if, for
every $u\in T$ and every $x,y\in f_{\Delta}^{-1}(\\{u\\})$, one has
$d(x,y)\leq\delta$.

In this article, a metric space is said to be $\delta$-hyperbolic if it is
geodesic, proper, and if all its geodesic triangles are $\delta$-thin.

For more informations about Gromov-hyperbolic metric spaces see the original
publication [29] and, for more explanations, [18, 24] and [8].

####  3.6.3 Busemann spaces

######  Definition 3.61.

A Busemann space is a geodesic and proper metric space (not reduced to one
point) whose distance between geodesics is convex, i.e. for every pair of
geodesic segments $c_{1}$, $c_{2}$ such that $c_{1}(0)=c_{2}(0)$ (with their
natural parametrization, see Definition 3.56), the function $t\mapsto
d(c_{1}(t),c_{2}(t))$ is convex.

Examples of Busemann spaces are Hadamard spaces and, more generally,
$\operatorname{Cat}(0)$-spaces.

The following results are classical and based on the facts that every Cauchy
sequence is bounded and that every Busemann space is proper:

######  Remark 3.62.

Every Busemann space is a complete metric space.

######  Remark 3.63.

On every Busemann space, any two points are joined by a single geodesic.

######  Lemma 3.64.

On any Busemann space, every local geodesic is a $($minimizing$)$ geodesic.

######  Lemma 3.65.

Every Busemann space $(X,d)$ satisfying the property of extension of local
geodesics is non compact and every geodesic extends as a $($minimizing$)$
geodesic line $]{-}\infty,+\infty[{}\to X$.

####  3.6.4 Examples

######  Example 3.66.

A family
$(X_{\operatorname{\varepsilon}},d_{\operatorname{\varepsilon}})_{\operatorname{\varepsilon}>0}$
of $\delta_{0}$-hyperbolic spaces with entropy $\leq H_{0}$, which admit a
proper isometric action of a group $\Gamma$ such that
$\operatorname{diam}(\Gamma\backslash X_{\operatorname{\varepsilon}})\leq D$
($\delta_{0},H_{0},D$ being independent of $\operatorname{\varepsilon}$) such
that $(X_{\operatorname{\varepsilon}},d_{\operatorname{\varepsilon}})$ is
Busemann, though there exists $x\in X$ such that
$\big{(}X_{\operatorname{\varepsilon}},d_{\operatorname{\varepsilon}},\mu^{\Gamma}_{x}\big{)}$
do not verify any weak Bishop–Gromov inequality at any scale $r_{0}<2D$ when
$\operatorname{\varepsilon}$ is small enough.

###### Proof.

Choose any $r_{0},D$ such that $0<r_{0}<2D$; for every
$\operatorname{\varepsilon}\leq 2D-r_{0}$,
$(X_{\operatorname{\varepsilon}},d_{\operatorname{\varepsilon}})$ is obtained
by gluing to $\mathbb{R}$ an interval $I_{k}=\big{[}0,\frac{r_{0}}{2}\big{]}$
at each point
$\operatorname{\varepsilon}k\in\operatorname{\varepsilon}\mathbb{Z}$,
identifying the origin of $I_{k}$ with the point
$\operatorname{\varepsilon}k$. Obviously, the group
$\Gamma:=\operatorname{\varepsilon}\cdot\mathbb{Z}$ acts on
$(X_{\operatorname{\varepsilon}},d_{\operatorname{\varepsilon}})$ by
translations sending $I_{k}$ onto $I_{p+k}$ for every
$\operatorname{\varepsilon}\cdot
p\in\operatorname{\varepsilon}\cdot\mathbb{Z}$.
$(X_{\operatorname{\varepsilon}},d_{\operatorname{\varepsilon}})$ is then
$0$-hyperbolic (thus Busemann), has trivial Entropy, verifies
$\operatorname{diam}(\Gamma\backslash
X)=\frac{r_{0}+\operatorname{\varepsilon}}{2}\leq D$. Let $x$ be the endpoint
of the interval $I_{0}$, let $r:=r_{0}+\operatorname{\varepsilon}$, it is
obvious that $\mu_{x}^{\Gamma}\big{(}B_{X}(x,r)\big{)}=1$ and that
$\mu_{x}^{\Gamma}\big{(}B_{X}(x,2r)\big{)}\geq
2\big{[}\frac{r_{0}}{\operatorname{\varepsilon}}\big{]}+3$. Hence, for every
choice of $C,K$, the inequality
$\frac{\mu_{x}^{\Gamma}(B_{X}(x,2r))}{\mu_{x}^{\Gamma}(B_{X}(x,r))}\leq
C\,{\rm e}^{Kr}$ is not verified when $\operatorname{\varepsilon}$ is small
enough. ∎

In the following examples, saying that almost all the elements of a sequence
$\big{(}\overline{M}_{i},\bar{g}_{i}\big{)}_{i\in\mathbb{N}^{*}}$ of
Riemannian manifolds do not verify any weak Bishop–Gromov inequality at a
given scale $r_{0}$ means that, for every $C,D>0$, there exists $i_{0}$ such
that, for every $i\geq i_{0}$, the Riemannian measure of
$\big{(}\overline{M}_{i},\bar{g}_{i}\big{)}$ do not verify the weak
Bishop–Gromov inequality at scale $r_{0}$, with factor $C$ and exponent $K$.
Similarly, saying that almost all the Riemannian manifolds of this sequence do
not verify any weak Bishop–Gromov inequality means that, for every $r,C,D>0$,
there exists $i_{0}$ such that, for every $i\geq i_{0}$, the Riemannian
measure of $\big{(}\overline{M}_{i},\bar{g}_{i}\big{)}$ do not verify the weak
Bishop–Gromov inequality at scale $r$, with factor $C$ and exponent $K$.

######  Example 3.67.

There exist positive constants $\delta$ and $K_{1}$ and sequences of
$\delta$-hyperbolic Riemannian manifolds
$\big{(}\overline{M}_{i},\bar{g}_{i}\big{)}_{i\in\mathbb{N}^{*}}$, whose
entropy is $\leq K_{1}$ and such that each
$\big{(}\overline{M}_{i},\bar{g}_{i}\big{)}$ admits a compact quotient by a
discrete group of isometries, though almost all the
$\big{(}\overline{M}_{i},\bar{g}_{i}\big{)}$’s do not verify any weak
Bishop–Gromov inequality.

######  Example 3.68.

There exist positive constants $\delta$, $K_{1}$ and $D$ and a sequence of
$\delta$-hyperbolic Riemannian manifolds
$\big{(}\overline{M}_{i},\bar{g}_{i}\big{)}_{i\in\mathbb{N}^{*}}$, such that
each $\big{(}\overline{M}_{i},\bar{g}_{i}\big{)}$ has entropy $\leq K_{1}$,
admits a quotient of diameter $\leq D$ by a discrete group of isometries, and
satisfies the following properties: on each
$\big{(}\overline{M}_{i},\bar{g}_{i}\big{)}$, the Riemannian measure verifies
a weak Bishop–Gromov inequality at scale $\frac{5}{2}(7D+4\delta)$, with
exponent $\frac{36}{5}K_{1}$ and factor $C_{0}(K_{1},D)$, though almost all
the $\big{(}\overline{M}_{i},\bar{g}_{i}\big{)}$’s do not verify any weak
Bishop–Gromov inequality at scale $\frac{3}{4}D$.

The factor $C_{0}(K_{1},D)$ is computed at the end of the following proof.

###### Construction and proof of Examples 3.67 and 3.68.

We consider a sequence of closed Riemannian manifolds of dimension $n$,
$(N_{i},h_{i})_{i\in\mathbb{N}^{*}}$, with Ricci curvature
$\geq-\frac{K_{0}^{2}}{n-1}$ and sectional curvature
$\leq-\frac{K_{0}^{2}}{100(n-1)^{2}}$, such that
$\operatorname{diam}(N_{i},h_{i})=D_{i}$, and, on each $N_{i}$, we choose a
pair of points $x_{i}$, $y_{i}$ at distance $D_{i}$. On the other hand, for
each $i\in\mathbb{N}^{*}$, let $(Y_{i},k_{i})$ be any closed $n$-dimensional
Riemannian manifold, with small diameter $\leq\frac{1}{i}$ and big volume
$\operatorname{Vol}(Y_{i},k_{i})\geq C_{0}\big{(}{\rm e}^{3i^{2}K_{0}}\big{)}$
(such metrics exist on every closed manifold). We construct a new sequence of
closed $n$-dimensional Riemannian manifolds
$(M_{i},g_{i})_{i\in\mathbb{N}^{*}}$, each $(M_{i},g_{i})$ being obtained as
the connected sum of $(N_{i},h_{i})$ with $(Y_{i},k_{i})$; the construction is
made by gluing $Y_{i}$ to $N_{i}$ inside the ball
$B_{N_{i}}\big{(}y_{i},\frac{1}{i}\big{)}$. More precisely, defining
$r_{i}:=\frac{1}{2}{\rm
e}^{-i}\cdot\min\big{(}\operatorname{inj}(Y_{i},k_{i}),\operatorname{inj}(N_{i},h_{i})\big{)}$,
we choose a point $y^{\prime}_{i}\in Y_{i}$, we excise one ball
$B_{Y_{i}}(y^{\prime}_{i},r_{i})$ (resp. $B_{N_{i}}(y_{i},r_{i})$) from
$Y_{i}$ (resp. from $N_{i}$) and glue $N_{i}\setminus B_{N_{i}}(y_{i},r_{i})$
and $Y_{i}\setminus B_{Y_{i}}(y^{\prime}_{i},r_{i})$ by identification of
their boundaries.101010To be isometric, this identification may require to
slighly modify the two metrics $h_{i}$ and $k_{i}$ on
$B_{N_{i}}(y_{i},2r_{i})$ and $B_{Y_{i}}(y^{\prime}_{i},2r_{i})$ respectively,
in order that they become flat on these two balls. As $N_{i}\setminus
B_{N_{i}}\big{(}y_{i},\frac{1}{i}\big{)}$ is included in both spaces $N_{i}$
and $M_{i}$, one can construct a continuous map $f_{i}\colon M_{i}\rightarrow
N_{i}$, such that $f_{i}^{-1}(y_{i})=Y_{i}\setminus
B_{Y_{i}}(y^{\prime}_{i},r_{i})$), which coincides with the identity map on
$N_{i}\setminus B_{N_{i}}\big{(}y_{i},\frac{1}{i}\big{)}$, the new metric
$g_{i}$ coinciding with $h_{i}$ on $N_{i}\setminus
B_{N_{i}}\big{(}y_{i},\frac{1}{i}\big{)}$. We call $(f_{i})_{*}$ the induced
morphism from $\pi_{1}(M_{i},x_{i})$ to $\pi_{1}(N_{i},x_{i})$.

Let us now define
$\pi_{i}\colon(\widetilde{N}_{i},\tilde{h}_{i})\rightarrow(N_{i},h_{i})$ as
the Riemannian universal covering of $(N_{i},h_{i})$ and choose
$\tilde{x}_{i}\in\pi_{i}^{-1}(x_{i})$ and
$\tilde{y}_{i}\in\pi_{i}^{-1}(y_{i})$ such that
$d_{\tilde{h}_{i}}(\tilde{x}_{i},\tilde{y}_{i})=D_{i}$. Let now
$p_{i}\colon\big{(}\overline{M}_{i},\bar{g}_{i}\big{)}\rightarrow(M_{i},g_{i})$
be the Riemannian covering of $(M_{i},g_{i})$ such that
$(p_{i})_{*}\big{(}\pi_{1}\big{(}\overline{M}_{i},\bar{x}_{i}\big{)}\big{)}=\operatorname{Ker}(f_{i})_{*}$
for some point $\bar{x}_{i}\in p_{i}^{-1}(x_{i})$ (see [25, Corollary 6.9]).
As $\operatorname{Ker}(f_{i})_{*}$ is normal in $\pi_{1}(M_{i},x_{i})$,
$p_{i}$ is a Galois covering, and the group ${\rm Aut}(p_{i})$ of
automorphisms of this covering is the group
$\pi_{1}(M_{i},x_{i})/\operatorname{Ker}(f_{i})_{*}$, hence $(f_{i})_{*}$
induces an isomorphism $f_{i}^{*}$ from
$\pi_{1}(M_{i},x_{i})/\operatorname{Ker}(f_{i})_{*}$ to the fundamental group
$\Gamma_{i}$ of $N_{i}$, viewed as the group of automorphisms of the universal
covering $\pi_{i}$. Classicaly (see [25, Theorem (6.1)]), there then exists a
map $\bar{f}_{i}\colon\overline{M}_{i}\rightarrow\widetilde{N}_{i}$ such that
$\pi_{i}\circ\bar{f}_{i}=f_{i}\circ p_{i}$ and
$\bar{f}_{i}(\operatorname{\gamma}\cdot\bar{x})=f_{i}^{*}(\operatorname{\gamma})\cdot\bar{f}_{i}(\bar{x})$
for every $\bar{x}\in\overline{M}_{i}$ and every $\operatorname{\gamma}\in{\rm
Aut}(p_{i})$.

As these two actions of $\Gamma_{i}$ commute with $\bar{f}_{i}$ and as $f_{i}$
is the identity map from $N_{i}\setminus
B_{N_{i}}\big{(}y_{i},\frac{1}{i}\big{)}$ onto itself, direct computations
give first that $p_{i}^{-1}\big{(}f_{i}^{-1}(y_{i})\big{)}$ is the union of
the left-translates $\operatorname{\gamma}\bar{f}_{i}^{-1}(\tilde{y}_{i})$ of
$\bar{f}_{i}^{-1}(\tilde{y}_{i})$, all isometric to
$f_{i}^{-1}(y_{i})=Y_{i}\setminus B_{Y_{i}}(y^{\prime}_{i},r_{i})$ via the map
$p_{i}$, and secondly that $\bar{f}_{i}$ is isometric from
$p_{i}^{-1}\big{(}N_{i}\setminus
B_{N_{i}}\big{(}y_{i},\frac{1}{i}\big{)}\big{)}\subset\overline{M}_{i}$ to
$\widetilde{N}_{i}\setminus\bigcup_{\operatorname{\gamma}\in\Gamma_{i}}B_{\widetilde{N}_{i}}\big{(}\operatorname{\gamma}\tilde{y}_{i},\frac{1}{i}\big{)}$.

This shows that, from a geometric point of view,
$\big{(}\overline{M}_{i},\bar{g}_{i}\big{)}$ is the connected sum of
$\big{(}\widetilde{N}_{i},\tilde{h}_{i}\big{)}$ with an infinite family
$\big{(}Y_{i}^{\gamma},k_{i}^{\gamma}\big{)}_{\operatorname{\gamma}\in\Gamma_{i}}$
of copies of $(Y_{i},k_{i})$, each
$Y_{i}^{\gamma}=\bar{f}_{i}^{-1}(\operatorname{\gamma}\tilde{y}_{i})$ being
glued to $\widetilde{N}_{i}$ inside the ball
$B_{\widetilde{N}_{i}}\big{(}\operatorname{\gamma}\tilde{y}_{i},\frac{1}{i}\big{)}$.
Recall that $(\widetilde{N}_{i},\tilde{h}_{i})$ is $\delta_{0}$-hyperbolic,
with $\delta_{0}=\frac{10(n-1)}{K_{0}}\ln 3$, by [18, Proposition 1.4.3, p.
12], for its sectional curvature is $\leq-\frac{K_{0}^{2}}{100(n-1)^{2}}$, and
that its entropy is bounded above by $K_{0}$, by the Bishop–Gromov Theorem
3.13. From this and the fact that $\big{(}\overline{M}_{i},\bar{g}_{i}\big{)}$
is quasi-isometric to $\big{(}\widetilde{N}_{i},\tilde{h}_{i}\big{)}$, we
deduce that there exist constants $\delta$ and $K_{1}$ (independent of $i$)
such that $\big{(}\overline{M}_{i},\bar{g}_{i}\big{)}$ is $\delta$-hyperbolic
and has entropy bounded above by $K_{1}$.

As $\big{(}\widetilde{N}_{i},\tilde{h}_{i}\big{)}$ has Ricci curvature
$\geq-\frac{K_{0}^{2}}{n-1}$, it verifies
$\operatorname{Vol}\big{(}B_{\widetilde{N}_{i}}(\tilde{x},r)\big{)}\leq
C_{0}\cdot{\rm e}^{K_{0}r}$ for every $r>0$. From this and from the fact that
$d_{\tilde{h}_{i}}(\tilde{x}_{i},\operatorname{\gamma}\tilde{y}_{i})\geq
D_{i}=d_{\tilde{h}_{i}}(\tilde{x}_{i},\tilde{y}_{i})$ for every
$\operatorname{\gamma}\in\Gamma_{i}$, setting $r_{i}:=\frac{3}{4}D_{i}$, we
deduce

$\displaystyle\dfrac{\operatorname{Vol}\big{(}B_{\overline{M}_{i}}(\tilde{x}_{i},2r_{i})\big{)}}{\operatorname{Vol}\big{(}B_{\overline{M}_{i}}(\tilde{x}_{i},r_{i})\big{)}}=\dfrac{\operatorname{Vol}\big{(}B_{\overline{M}_{i}}\big{(}\tilde{x}_{i},\frac{3}{2}D_{i}\big{)}\big{)}}{\operatorname{Vol}\big{(}B_{\overline{M}_{i}}\big{(}\tilde{x}_{i},\frac{3}{4}D_{i}\big{)}\big{)}}>\dfrac{C_{0}\,{\rm
e}^{3i^{2}K_{0}}}{\operatorname{Vol}\big{(}B_{\widetilde{N}_{i}}\big{(}\tilde{x}_{i},\frac{3}{4}D_{i}\big{)}\big{)}}\geq\dfrac{C_{0}\,{\rm
e}^{3i^{2}K_{0}}}{C_{0}\,{\rm e}^{\frac{3}{4}K_{0}\,D_{i}}}.$ (3.8)

  * •

If we choose $D_{i}=i$, then (3.8) gives
$\frac{\operatorname{Vol}(B_{\overline{M}_{i}}(\tilde{x}_{i},2r_{i}))}{\operatorname{Vol}(B_{\overline{M}_{i}}(\tilde{x}_{i},r_{i}))}>{\rm
e}^{i(3i-1)K_{0}}$; hence, for every choice of the scale $r_{0}$, of the
factor $C>0$ and of the exponent $K$, and for every
$i>\operatorname{Max}\big{(}\frac{4}{3}r_{0},\frac{\ln
C}{K_{0}},\frac{K}{K_{0}}\big{)}$, one obtains that
$\frac{\operatorname{Vol}(B_{\overline{M}_{i}}(\tilde{x}_{i},2r_{i}))}{\operatorname{Vol}(B_{\overline{M}_{i}}(\tilde{x}_{i},r_{i}))}>C{\rm
e}^{Kr_{i}}$, despite the fact that $r_{i}\geq r_{0}$. It thus follows that
almost all the $\big{(}\overline{M}_{i},\bar{g}_{i}\big{)}$’s do not verify
any weak Bishop–Gromov inequality. This ends the proof of Example 3.67.

  * •

If we choose $D_{i}=D$, we have $r_{i}=r_{1}=\frac{3}{4}D$ and, for every
choice of $C$ and $K$ and for every $i>\frac{1}{\sqrt{3\,K_{0}}}\big{(}\ln
C+(K+K_{0})D\big{)}^{1/2}$, inequality (3.8) gives
$\frac{\operatorname{Vol}(B_{\overline{M}_{i}}(\tilde{x}_{i},2r_{1}))}{\operatorname{Vol}(B_{\overline{M}_{i}}(\tilde{x}_{i},r_{1}))}\geq{\rm
e}^{3K_{0}(i^{2}-D/4)}>C{\rm e}^{Kr_{1}}$. This proves that almost all the
$\big{(}\overline{M}_{i},\bar{g}_{i}\big{)}$’s do not verify any weak
Bishop–Gromov inequality at scale $r_{1}=\frac{3}{4}D$. On the contrary,
applying Theorem 3.15$(i)$, we know that every $\Gamma_{i}$-invariant measure
$\mu_{i}$ satisfies the weak Bishop–Gromov inequality at scale
$\frac{5}{2}(7D+4\delta)$, with factor $C_{0}(K_{1},D)=3\cdot 2^{25/4}{\rm
e}^{(1+6\ln 2)K_{1}D}$ and exponent $\frac{36}{5}K_{1}$. This ends the proof
of Example 3.68. ∎

######  Example 3.69.

An infinite family $\big{(}\overline{M}_{i},\bar{g}_{i}\big{)}_{i\in I}$ of
complete, $n$-dimensional, $\delta$-hyperbolic Riemannian manifolds, with
distinct topologies and distinct local topologies, each of them admitting a
co-compact proper action, by isometries, of the same group $\Gamma$ such that
the diameter of $\Gamma\backslash\overline{M}_{i}$ and the entropy of
$\big{(}\overline{M}_{i},\bar{g}_{i}\big{)}$ are respectively bounded by
constants $D$ and $K$ independent on $i$. However, on each
$\big{(}\overline{M}_{i},\bar{g}_{i}\big{)}$, every $\Gamma$-invariant measure
$\mu_{i}$ verifies a weak Bishop–Gromov inequality at scale
$\frac{5}{2}(7D+4\delta)$, with exponent $\frac{36}{5}K$ and factor
$C_{0}(K,D)$ (defined in Example 3.68).

In this example, the set of indices $I$ may be chosen as the set
$\\{Y_{i}\\}_{i\in I}$ of all closed, $n$-dimensional manifolds modulo
homeomorphisms. A consequence is that the local topology of the quotient
spaces $M_{i}:=\Gamma_{i}\backslash\overline{M}_{i}$, which is the connected
sum of $\mathbb{B}^{n}$ with $Y_{i}$, is arbitrary.

###### Construction and proof.

We start from any fixed closed $n$-dimensional Riemannian manifold
$(M_{0},g_{0})$ with Ricci curvature $\geq-\frac{K_{0}^{2}}{n-1}$ and
sectional curvature $\leq-\frac{K_{0}^{2}}{100(n-1)^{2}}$. Let $D$ be an upper
bound of $\operatorname{diam}(M_{0},g_{0})+1$. We still denote by
$\\{Y_{i}\\}_{i\in I}$ the set of all closed, $n$-dimensional manifolds modulo
homeomorphisms. For any choice of $\operatorname{\varepsilon}\in{}]0,1]$, we
can endow each manifold $Y_{i}$ with a Riemannian metric $h_{i}$ such that
$\operatorname{diam}(Y_{i},h_{i})\leq\operatorname{\varepsilon}$.

To each choice of such a manifold $Y_{i}$, we associate a new closed
$n$-dimensional Riemannian manifold $(M_{i},g_{i})$, by connected sum of
$(M_{0},g_{0})$ and $(Y_{i},h_{i})$, constructed as follows: defining
$\operatorname{\varepsilon}_{i}:=10^{-6}\cdot\min\big{(}\operatorname{inj}(Y_{i},h_{i}),\operatorname{inj}(M_{0},g_{0})\big{)}\leq
10^{-6}\operatorname{\varepsilon}$, we choose points $y_{i}\in Y_{i}$ and
$x_{0}\in M_{0}$, we excise balls
$B_{Y_{i}}(y_{i},\operatorname{\varepsilon}_{i})$ and
$B_{M_{0}}(x_{0},\operatorname{\varepsilon}_{i})$ from $Y_{i}$ and $M_{0}$
(respectively) and glue $M_{0}\setminus
B_{M_{0}}(x_{0},\operatorname{\varepsilon}_{i})$ and $Y_{i}\setminus
B_{Y_{i}}(y_{i},\operatorname{\varepsilon}_{i})$ by identification of their
boundaries.111111To be isometric, this identification may require to slighly
modify the two metrics $g_{0}$ and $h_{i}$ on
$B_{M_{0}}(x_{0},2\operatorname{\varepsilon}_{i})$ and
$B_{Y_{i}}(y_{i},2\operatorname{\varepsilon}_{i})$ (respectively), in order
that they become flat on these two balls. It is classical that, as $Y_{i}$ and
$Y_{j}$ are not homeomorphic when $i\neq j$, then $M_{i}=M_{0}\\#Y_{i}$ and
$M_{j}=M_{0}\\#Y_{j}$ are not homeomorphic. Moreover, when
$4\operatorname{\varepsilon}$ is smaller than the injectivity radius of
$(M_{0},g_{0})$, for every $x$ on the boundary of $M_{0}\setminus
B_{M_{0}}(x_{0},\operatorname{\varepsilon}_{i})$ the balls
$B_{M_{i}}(x,2\operatorname{\varepsilon})$ and
$B_{M_{j}}(x,2\operatorname{\varepsilon})$ have different topologies.

One can construct a continuous map $f_{i}\colon M_{i}\rightarrow M_{0}$ such
that $f_{i}^{-1}(x_{0})=Y_{i}\setminus
B_{Y_{i}}(y_{i},\operatorname{\varepsilon}_{i})$ and which is the identity map
on the subset $M_{0}\setminus
B_{M_{0}}(x_{0},2\operatorname{\varepsilon}_{i})$ (included in both spaces
$M_{0}$ and $M_{i}=M_{0}\\#Y$), the metric $g_{i}$ coinciding with $g_{0}$ on
$M_{0}\setminus B_{M_{0}}(x_{0},2\operatorname{\varepsilon}_{i})$. We call
$(f_{i})_{*}$ the induced morphism from $\pi_{1}(M_{i},x_{1})$ to
$\pi_{1}(M_{0},x_{1})$, where $x_{1}$ is any point of $M_{0}\setminus
B_{M_{0}}(x_{0},2\operatorname{\varepsilon}_{i})$.

Let now
$\pi\colon\big{(}\widetilde{M}_{0},\tilde{g}_{0}\big{)}\rightarrow(M_{0},g_{0})$
be the Riemannian universal covering of $(M_{0},g_{0})$ and choose
$\tilde{x}_{0}\in\pi^{-1}(x_{0})$. Let
$p_{i}\colon\big{(}\overline{M}_{i},\bar{g}_{i}\big{)}\rightarrow(M_{i},g_{i})$
be the Riemannian covering of $(M_{i},g_{i})$ such that
$(p_{i})_{*}\big{(}\pi_{1}\big{(}\overline{M}_{i},\bar{x}_{1}\big{)}\big{)}=\operatorname{Ker}(f_{i})_{*}$
for some point $\bar{x}_{1}\in p_{i}^{-1}(x_{1})$ (see [25, Corollary (6.9)]).
As $\operatorname{Ker}(f_{i})_{*}$ is normal in $\pi_{1}(M_{i},x_{1})$,
$p_{i}$ is a Galois covering, and the group ${\rm Aut}(p_{i})$ of
automorphisms of this covering is
$\pi_{1}(M_{i},x_{1})/\operatorname{Ker}(f_{i})_{*}$, hence $(f_{i})_{*}$
induces an isomorphism $f_{i}^{*}$ from
$\pi_{1}(M_{i},x_{1})/\operatorname{Ker}(f_{i})_{*}$ onto the group $\Gamma$
of automorphisms of the universal covering $\pi$ of $M_{0}$.

Classically (see [25, Theorem (6.1)]), there then exists a map
$\bar{f}_{i}\colon\overline{M}_{i}\rightarrow\widetilde{M}_{0}$ such that
$\pi\circ\bar{f}_{i}=f_{i}\circ p_{i}$ and
$\bar{f}_{i}(\operatorname{\gamma}\cdot\bar{x})=f_{i}^{*}(\operatorname{\gamma})\cdot\bar{f}_{i}(\bar{x})$
for every $\bar{x}\in\overline{M}_{i}$ and every $\operatorname{\gamma}\in{\rm
Aut}(p_{i})$.

Arguing as in the proof of Example 3.67, we prove that
$p_{i}^{-1}\big{(}f_{i}^{-1}(x_{0})\big{)}$ is the union of the left-
translates $\operatorname{\gamma}\bar{f}_{i}^{-1}(\tilde{x}_{0})$ of
$\bar{f}_{i}^{-1}(\tilde{x}_{0})$, all isometric to
$f_{i}^{-1}(x_{0})=Y_{i}\setminus
B_{Y_{i}}(y_{i},\operatorname{\varepsilon}_{i})$ via the map $p_{i}$. Using
the same arguments as in the proof of Example 3.67, we also obtain that the
restriction of $\bar{f}_{i}$ to $p_{i}^{-1}\big{(}M_{0}\setminus
B_{M_{0}}(x_{0},2\operatorname{\varepsilon}_{i})\big{)}\subset\overline{M}_{i}$
is isometric onto
$\widetilde{M}_{0}\setminus\bigcup_{\operatorname{\gamma}\in\Gamma}B_{\widetilde{M}_{0}}(\operatorname{\gamma}\tilde{x}_{0},2\operatorname{\varepsilon}_{i})$.
Revisiting these arguments, one can say that, from a geometric point of view,
$\big{(}\overline{M}_{i},\bar{g}_{i}\big{)}$ is a connected sum of
$\big{(}\widetilde{M}_{0},\tilde{g}_{0}\big{)}$ with an infinite number of
copies
$\big{(}Y_{i}^{\gamma},h_{i}^{\gamma}\big{)}_{\operatorname{\gamma}\in\Gamma}$
of $(Y_{i},h_{i})$, each copy
$Y_{i}^{\gamma}=\bar{f}_{i}^{-1}(\operatorname{\gamma}\tilde{x}_{0})$ of
$Y_{i}$ being glued to $\widetilde{M}_{0}$ inside the ball
$B_{\widetilde{M}_{0}}(\operatorname{\gamma}\tilde{x}_{0},2\operatorname{\varepsilon}_{i})$.

Recall that $\big{(}\widetilde{M}_{0},\tilde{g}_{0}\big{)}$, whose sectional
curvature is $\leq-\frac{K_{0}^{2}}{100(n-1)^{2}}$, is
$\delta_{0}$-hyperbolic, with $\delta_{0}=\frac{10(n-1)}{K_{0}}\ln 3$, by [18,
Proposition 1.4.3, p. 12], and that its entropy is bounded above by $K_{0}$,
by the Bishop–Gromov Theorem 3.13. From this and from the fact that
$\big{(}\overline{M}_{i},\bar{g}_{i}\big{)}$ is quasi-isometric to
$\big{(}\widetilde{M}_{0},\tilde{g}_{0}\big{)}$, we deduce that there exist
constants $\delta$ and $K$ (independent of $i$) such that
$\big{(}\overline{M}_{i},\bar{g}_{i}\big{)}$ is $\delta$-hyperbolic and has
entropy bounded above by $K$. As $\Gamma$ acts on $\overline{M}_{i}$ via the
isomorphic representation of $\Gamma$ as ${\rm Aut}(p_{i})$, one has
$\operatorname{diam}\big{(}\Gamma\backslash\overline{M}_{i}\big{)}=\operatorname{diam}(M_{i},g_{i})\leq\operatorname{diam}(M_{0},g_{0})+1\leq
D$.

Now, applying Theorem 3.15$(i)$, we know that every $\Gamma$-invariant measure
$\mu_{i}$ satisfies the weak Bishop–Gromov inequality at scale
$\frac{5}{2}(7D+4\delta)$, with factor $C_{0}(K,D)=3\cdot 2^{25/4}{\rm
e}^{(1+6\ln 2)KD}$ and exponent $\frac{36}{5}K$. This ends the proof. ∎

######  Example 3.70.

For every $n\geq 4$ and every integer $k$ such that $2\leq k\leq n-2$, there
exists a sequence $(M_{i},g_{i})_{i\in\mathbb{N}}$ of closed, connected,
$n$-dimensional Riemannian manifolds, with diameter $\leq D$, whose universal
covers $\big{(}\widetilde{M}_{i},\tilde{g}_{i}\big{)}$ are $\delta$-hyperbolic
with entropy $\leq K$ (where the constants $D$, $\delta$, $K$ are independent
on $i$), and such that
$\dim\big{(}H_{k}(M_{i},\mathbb{R})\big{)}\rightarrow+\infty$ when
$i\rightarrow+\infty$.

###### Construction and proof.

We start from any fixed closed $n$-dimensional Riemannian manifold
$(M_{0},g_{0})$ with Ricci curvature $\geq-\frac{K_{0}^{2}}{n-1}$ and
sectional curvature $\leq-\frac{K_{0}^{2}}{100(n-1)^{2}}$. Let $D$ be an upper
bound of $\operatorname{diam}(M_{0},g_{0})+2$.

Let $Y$ be any closed, simply connected, $n$-dimensional manifold. For any
choice of $\operatorname{\varepsilon}\in{}]0,1]$, we can endow $Y$ with a
Riemannian metric $h_{\operatorname{\varepsilon}}$ such that
$\operatorname{diam}(Y,h_{\operatorname{\varepsilon}})\leq\operatorname{\varepsilon}$.
To every $i\in\mathbb{N}$, we associate a new closed $n$-dimensional
Riemannian manifold $(M_{i},g_{i})$, by connected sum of $(M_{0},g_{0})$ with
$i$ copies of $(Y,h_{\operatorname{\varepsilon}})$, constructed as follows:
defining $\operatorname{\varepsilon}_{i}:=\frac{1}{100}\,{\rm
e}^{-2i}\min\big{(}\operatorname{inj}(Y,h_{\operatorname{\varepsilon}})$;
$(\operatorname{inj}(M_{0},g_{0}))^{2};1\big{)}\leq{\rm
e}^{-i}\frac{\operatorname{\varepsilon}}{10}$, we choose points
$x_{1},\dots,x_{i}\in M_{0}$ such that, for any pair of distinct indices
$j,l\leq i$, one has $d_{M_{0}}(x_{j},x_{l})\geq
2\sqrt{\operatorname{\varepsilon}_{i}}\,(1+\sqrt{\operatorname{\varepsilon}_{i}})^{2}\gg
2\operatorname{\varepsilon}_{i}$. To $M_{0}\setminus\cup_{1\leq j\leq
i}B_{M_{0}}(x_{j},\operatorname{\varepsilon}_{i})$, we glue $i$ copies
$\breve{Y}_{1},\dots,\breve{Y}_{i}$ of
$(Y,h_{\operatorname{\varepsilon}})\setminus
B_{Y}(y_{0},\operatorname{\varepsilon}_{i})$ identifying the boundary of each
$\breve{Y}_{j}$ with $\partial
B_{M_{0}}(x_{j},\operatorname{\varepsilon}_{i})$ (see the proofs of Examples
3.67 and 3.69).

Let us define $M_{0}^{i{\rm\,holes}}:=M_{0}\setminus\cup_{1\leq j\leq
i}B_{M_{0}}(x_{j},2\operatorname{\varepsilon}_{i})$, which is considered as
included in both spaces $M_{0}$ and $M_{i}$, then $M_{i}\setminus
M_{0}^{i{\rm\,holes}}$ has $i$ connected components, the connected component
which contains $\breve{Y}_{j}$ (denoted by $\widehat{Y}_{j}$) being the result
of the connected sum (constructed above) of
$B_{M_{0}}(x_{j},2\operatorname{\varepsilon}_{i})$ with the $j$-th copy of
$Y$. One can construct a continuous map $f_{i}\colon M_{i}\rightarrow M_{0}$
such that $f_{i}^{-1}(x_{j})=\breve{Y}_{j}$, which maps each connected
component $\widehat{Y}_{j}$ of $M_{i}\setminus M_{0}^{i{\rm\,holes}}$ onto
$B_{M_{0}}(x_{j},2\operatorname{\varepsilon}_{i})$, and which is the identity
map on the subset $M_{0}^{i{\rm\,holes}}$, the metric $g_{i}$ coinciding with
$g_{0}$ on $M_{0}^{i{\rm\,holes}}$.

As $Y$ is simply connected, the converse of the induced morphism
$\pi_{1}(M_{i},x_{0})\rightarrow\pi_{1}(M_{0},x_{0})$, where $x_{0}\in
M_{0}^{i{\rm\,holes}}$, induces an isomorphism, denoted by $\varrho_{i}$,
between the groups $\Gamma$ and $\Gamma_{i}$ of deck-transformations of the
universal coverings
$\pi_{0}\colon\big{(}\widetilde{M}_{0},\tilde{g}_{0}\big{)}\rightarrow(M_{0},g_{0})$
and
$\pi_{i}\colon\big{(}\widetilde{M}_{i},\tilde{g}_{i}\big{)}\rightarrow(M_{i},g_{i})$.
As in the proofs of Examples 3.67 and 3.69, there exists a map
$\tilde{f}_{i}\colon\widetilde{M}_{i}\rightarrow\widetilde{M}_{0}$ such that
$\pi_{0}\circ\tilde{f}_{i}=f_{i}\circ\pi_{i}$ and
$\tilde{f}_{i}\big{(}\varrho_{i}(\operatorname{\gamma})\cdot\tilde{x}\big{)}=\operatorname{\gamma}\cdot\tilde{f}_{i}(\tilde{x})$
for every $\tilde{x}\in\widetilde{M}_{i}$ and every
$\operatorname{\gamma}\in\Gamma$.

Arguing as in the proof of Examples 3.67 and 3.69, we prove that, for each $j$
($1\leq j\leq i$), denoting by $\tilde{x}_{j}$ any point of
$\pi_{0}^{-1}(x_{j})$, the set $\pi_{i}^{-1}\big{(}f_{i}^{-1}(x_{j})\big{)}$
is the union of the left-translates
$\varrho_{i}(\operatorname{\gamma})\tilde{f}_{i}^{-1}(\tilde{x}_{j})$ of
$\tilde{f}_{i}^{-1}(\tilde{x}_{j})$ for all the
$\operatorname{\gamma}\in\Gamma$, each of these translates being isometric to
$f_{i}^{-1}(x_{j})=\breve{Y}_{j}$ via the map $\pi_{i}$. As in the proof of
Examples 3.67 and 3.69, we prove that the restriction of $\tilde{f}_{i}$ to
$\pi_{i}^{-1}\big{(}M_{0}^{i{\rm\,holes}}\big{)}\subset\widetilde{M}_{i}$ is
one to one isometric onto
$\pi_{0}^{-1}\big{(}M_{0}^{i{\rm\,holes}}\big{)}=\widetilde{M}_{0}\setminus\bigcup_{\operatorname{\gamma}\in\Gamma}\big{(}\cup_{1\leq
j\leq
i}B_{\widetilde{M}_{0}}(\operatorname{\gamma}\tilde{x}_{j},2\operatorname{\varepsilon}_{i})\big{)}$
that we shall denote by $\widetilde{M}_{0}^{i{\rm\,holes}}$ for the sake of
simplicity.

Revisiting these arguments, one can say that, from a geometric point of view,
$\big{(}\widetilde{M}_{i},\tilde{g}_{i}\big{)}$ is a connected sum of
$(\widetilde{M}_{0},\tilde{g}_{0})$ with an infinite number of copies
$\big{(}Y_{j}^{\gamma},h_{j}^{\gamma}\big{)}_{\operatorname{\gamma}\in\Gamma}$
of $(Y,h_{\varepsilon})$, each copy
$Y_{j}^{\gamma}=\tilde{f}_{i}^{-1}(\operatorname{\gamma}\tilde{x}_{j})=\varrho_{i}(\operatorname{\gamma})\tilde{f}_{i}^{-1}(\tilde{x}_{j})$
being glued to $\widetilde{M}_{0}$ inside the ball
$B_{\widetilde{M}_{0}}\big{(}\operatorname{\gamma}\tilde{x}_{j},\operatorname{\varepsilon}_{i}\big{)}$.

Considering that $\widetilde{M}_{0}^{i{\rm\,holes}}$ is also included in
$\widetilde{M}_{i}$ via the inverse of the isometric map $\tilde{f}_{i}$:
$\pi_{i}^{-1}\big{(}M_{0}^{i{\rm\,holes}}\big{)}\rightarrow\widetilde{M}_{0}^{i{\rm\,holes}}$,
the two inclusion maps from $\widetilde{M}_{0}^{i{\rm\,holes}}$ into
$\widetilde{M}_{0}$ and $\widetilde{M}_{i}$ preserve the lengths of paths. On
$\widetilde{M}_{0}^{i{\rm\,holes}}$, we compare the length-distance
$d_{\widetilde{M}_{0}^{i{\rm\,holes}}}$ $\big{(}$associated to paths lying in
$\widetilde{M}_{0}^{i{\rm\,holes}}\big{)}$ with the restriction of the two
distances $d_{\widetilde{M}_{0}}$ and $d_{\widetilde{M}_{i}}$ of
$\widetilde{M}_{0}$ and $\widetilde{M}_{i}$ $\big{(}$which are length-
distances associated to paths lying in $\widetilde{M}_{0}$ and
$\widetilde{M}_{i}$ respectively$\big{)}$; it is clear that
$d_{\widetilde{M}_{0}}$, $d_{\widetilde{M}_{i}}\leq
d_{\widetilde{M}_{0}^{i{\rm\,holes}}}$.

Between two points $\tilde{x},\tilde{y}$ of
$\widetilde{M}_{0}^{i{\rm\,holes}}$, let $c$ be a minimizing geodesic of
$\big{(}\widetilde{M}_{0},\tilde{g}_{0}\big{)}$, let $\bar{c}$ be the
disconnected union of geodesic segments which is the intersection of $c$ with
$\widetilde{M}_{0}^{i{\rm\,holes}}$ and $\tilde{c}$ the path of minimal length
among all the continuous paths which coincide with $\bar{c}$ on the interior
of $\widetilde{M}_{0}^{i{\rm\,holes}}$ and which connect the different
connected components of $\bar{c}$ by means of several arcs of circle, each of
these arcs lying in one of the $i$ geodesic spheres which are the connected
components of the boundary of $\widetilde{M}_{0}^{i{\rm\,holes}}$; as the
length of each of these arcs of circles is at most
$2\pi\operatorname{\varepsilon}_{i}(1+\operatorname{\varepsilon}_{i})$ (when
$i$ is large enough), and as the length of each connected component of
$\bar{c}$ (except the first and the last ones) is at least
$2\sqrt{\operatorname{\varepsilon}_{i}}(1+\operatorname{\varepsilon}_{i})$, we
get that ${\rm
length}(\tilde{c})\leq(1+\pi\sqrt{\operatorname{\varepsilon}_{i}})\,{\rm
length}(\bar{c})+2\pi\operatorname{\varepsilon}_{i}(1+\operatorname{\varepsilon}_{i})$
and it follows that
$d_{\widetilde{M}_{0}^{i{\rm\,holes}}}(\tilde{x},\tilde{y})\leq(1+\pi\sqrt{\operatorname{\varepsilon}_{i}})\,d_{\widetilde{M}_{0}}(\tilde{x},\tilde{y})+2\pi\operatorname{\varepsilon}_{i}(1+\operatorname{\varepsilon}_{i})$.
We similarly prove that
$d_{\widetilde{M}_{0}^{i{\rm\,holes}}}(\tilde{x},\tilde{y})\leq(1+\pi\sqrt{\operatorname{\varepsilon}_{i}})\,d_{\widetilde{M}_{i}}(\tilde{x},\tilde{y})+2\pi\operatorname{\varepsilon}_{i}(1+\operatorname{\varepsilon}_{i})$.
A first consequence is that

$\displaystyle\operatorname{Ent}\big{(}\widetilde{M}_{i},\tilde{g}_{i}\big{)}\leq\big{(}1+\pi\sqrt{\operatorname{\varepsilon}_{i}}\big{)}\operatorname{Ent}\big{(}\widetilde{M}_{0}^{i{\rm\,holes}},d_{\widetilde{M}_{0}^{i{\rm\,holes}}}\big{)}\leq\big{(}1+\pi\sqrt{\operatorname{\varepsilon}_{i}}\big{)}\operatorname{Ent}\big{(}\widetilde{M}_{0},\tilde{g}_{0}\big{)}.$

Let now $\tilde{x}$, $\tilde{y}$ be any pair of points of $\widetilde{M}_{i}$,
$c$ be a minimizing geodesic of
$\big{(}\widetilde{M}_{i},\tilde{g}_{i}\big{)}$, and $t_{0}$ (resp. $t_{1}$)
be the infimum (resp. the supremum) of the values of $t$ such that
$c(t)\in\widetilde{M}_{0}^{i{\rm\,holes}}$. We then have

$\displaystyle d_{\widetilde{M}_{i}}(\tilde{x},\tilde{y})\leq
d_{\widetilde{M}_{0}^{i{\rm\,holes}}}(c(t_{0}),c(t_{1}))+2(\operatorname{\varepsilon}+4\operatorname{\varepsilon}_{i})$
$\displaystyle\hphantom{d_{\widetilde{M}_{i}}(\tilde{x},\tilde{y})}{}\leq(1+\pi\sqrt{\operatorname{\varepsilon}_{i}})\,d_{\widetilde{M}_{0}}(c(t_{0}),c(t_{1}))+2\pi\operatorname{\varepsilon}_{i}(1+\operatorname{\varepsilon}_{i})+2\operatorname{\varepsilon}+8\operatorname{\varepsilon}_{i}$
$\displaystyle\hphantom{d_{\widetilde{M}_{i}}(\tilde{x},\tilde{y})}{}\leq(1+\pi\sqrt{\operatorname{\varepsilon}_{i}})\big{(}d_{\widetilde{M}_{0}}\big{(}\tilde{f}_{i}(\tilde{x}),\tilde{f}_{i}(\tilde{y})\big{)}+8\operatorname{\varepsilon}_{i}\big{)}+2\pi\operatorname{\varepsilon}_{i}(1+\operatorname{\varepsilon}_{i})+2\operatorname{\varepsilon}+8\operatorname{\varepsilon}_{i},$

where the last inequality uses the fact that $c(t_{0})$ and $\tilde{x}$ (resp.
$c(t_{1})$ and $\tilde{y}$), if they do not coincide, belong to the same
connected component of
$\widetilde{M}_{i}\setminus\widetilde{M}_{0}^{i{\rm\,holes}}$, thus that
$\tilde{f}_{i}(c(t_{0}))$ and $\tilde{f}_{i}(\tilde{x})$ $\big{(}$resp.
$\tilde{f}_{i}(c(t_{1}))$ and $\tilde{f}_{i}(\tilde{y})\big{)}$ belong to the
same connected component of
$\widetilde{M}_{0}\setminus\widetilde{M}_{0}^{i{\rm\,holes}}$, thus to the
same ball of radius $2\operatorname{\varepsilon}_{i}$. Analogous arguments
yield

$\displaystyle
d_{\widetilde{M}_{0}}\big{(}\tilde{f}_{i}(\tilde{x}),\tilde{f}_{i}(\tilde{y})\big{)}\leq\big{(}1+\pi\sqrt{\operatorname{\varepsilon}_{i}}\big{)}\big{(}d_{\widetilde{M}_{i}}(\tilde{x},\tilde{y})+2\operatorname{\varepsilon}+8\operatorname{\varepsilon}_{i}\big{)}+2\pi\operatorname{\varepsilon}_{i}(1+\operatorname{\varepsilon}_{i})+8\operatorname{\varepsilon}_{i}.$

This proves that $\tilde{f}_{i}$ is a quasi-isometry from
$\big{(}\widetilde{M}_{i},\tilde{g}_{i}\big{)}$ onto
$\big{(}\widetilde{M}_{0},\tilde{g}_{0}\big{)}$. Recall that
$\big{(}\widetilde{M}_{0},\tilde{g}_{0}\big{)}$ is $\delta_{0}$-hyperbolic
with $\delta_{0}=\frac{10(n-1)}{K_{0}}\ln 3$, because its sectional curvature
is $\leq-\frac{K_{0}^{2}}{100(n-1)^{2}}$ (by [18, Proposition 1.4.3, p. 12]),
and that it has entropy bounded above by $K_{0}$ (by the Bishop–Gromov Theorem
3.13). This quasi-isometric comparison implies that there exist constants
$\delta$ and $K$ (independent on $i$) such that
$\big{(}\widetilde{M}_{i},\tilde{g}_{i}\big{)}$ is $\delta$-hyperbolic and has
entropy bounded above by $K$. As $\Gamma$ acts on $\widetilde{M}_{i}$ via the
isomorphic representation $\varrho_{i}$, one has
$\operatorname{diam}\big{(}\Gamma\backslash\widetilde{M}_{i}\big{)}=\operatorname{diam}(M_{i},g_{i})\leq\operatorname{diam}(M_{0},g_{0})+2\leq
D$. This ends the first part of the proof.

In the sequel the homology groups are with coefficients in $\mathbb{R}$. Let
us prove that $\dim\big{(}H_{k}(M_{i})\big{)}$ $\rightarrow+\infty$ with $i$
when $2\leq k\leq n-2$. From the topological viewpoint, as
$M_{i+1}=M_{i}\\#Y$, it is the union of two open subsets $M_{i}^{*}$ and
$Y^{*}$, where $M_{i}^{*}$ (resp. $Y^{*}$) is obtained by removing from
$M_{i}$ (resp. from $Y$) a small ball contained in
$B_{M_{i}}(x_{i+1},\operatorname{\varepsilon}_{i})$ (resp. in
$B_{Y}(y_{0},\operatorname{\varepsilon}_{i})$); then, using the fact that
$M_{i}^{*}\cap Y^{*}$ is homeomorphic to
$]{-}\operatorname{\varepsilon}_{i},\operatorname{\varepsilon}_{i}[{}\times\mathbb{S}^{n-1}$,
the Mayer–Vietoris exact sequence associated to the covering $M_{i}^{*}\cup
Y^{*}$ of $M_{i+1}$ reads

$\displaystyle\cdots\rightarrow H_{k+1}(M_{i}\\#Y)\rightarrow
H_{k}\big{(}\mathbb{S}^{n-1}\big{)}\rightarrow H_{k}(M_{i}^{*})\oplus
H_{k}(Y^{*})\rightarrow H_{k}(M_{i}\\#Y)$
$\displaystyle\hphantom{\cdots}\rightarrow
H_{k-1}(\mathbb{S}^{n-1})\rightarrow\cdots.$

As $n\geq 4$ and $2\leq k\leq n-2$, one has
$H_{k}\big{(}\mathbb{S}^{n-1}\big{)}=\\{0\\}=H_{k-1}\big{(}\mathbb{S}^{n-1}\big{)}$
and $H_{k}(\mathbb{B}^{n})=\\{0\\}$, hence

$\displaystyle H_{k}(M_{i+1})=H_{k}(M_{i}\\#Y)\simeq H_{k}(M_{i}^{*})\oplus
H_{k}(Y^{*})\simeq H_{k}(M_{i})\oplus H_{k}(Y).$

Let us consider three examples of manifolds $Y$:

  * •

when $n$ is even ($n=2d\geq 4$), if we choose $Y=\mathbb{C}P^{d}$, then
$H_{k}(Y)\simeq\mathbb{R}$ for every even integer $k$ such that $2\leq k\leq
n-2$ and we have, in this case,
$\dim\big{(}H_{k}(M_{i+1})\big{)}=\dim\big{(}H_{k}(M_{i})\big{)}+1$; this ends
the proof in this case,

  * •

when $n$ is odd ($n=2d+1\geq 5$), if we choose
$Y=\mathbb{C}P^{d-1}\times\mathbb{S}^{3}$, we have $\dim H_{k}(Y)\geq 1$ for
every $k$ such that $2\leq k\leq n-2$ and thus
$\dim\big{(}H_{k}(M_{i+1})\big{)}\geq\dim\big{(}H_{k}(M_{i})\big{)}+1$, and
this ends the proof in this case,

  * •

when $n$ is even ($n=2d\geq 8$), if we choose
$Y=\mathbb{C}P^{d-3}\times\mathbb{S}^{3}\times\mathbb{S}^{3}$, then $\dim
H_{k}(Y)\geq 1$ for every $k$ such that $2\leq k\leq n-2$ (except when $d=4$
and $k=4$) and then
$\dim\big{(}H_{k}(M_{i+1})\big{)}\geq\dim\big{(}H_{k}(M_{i})\big{)}+1$, and
this ends the proof in this case. ∎

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