# Minimal volume entropy and fiber growth

Ivan Babenko  and  Stéphane Sabourau  Université Montpellier II, CNRS UMR
5149, Institut Montpelliérain Alexander Grothendieck, Place Eugène Bataillon,
Bât. 9, CC051, 34095 Montpellier CEDEX 5, France<EMAIL_ADDRESS>Univ
Paris Est Creteil, CNRS, LAMA, F-94010 Creteil, France Univ Gustave Eiffel,
LAMA, F-77447 Marne-la-Vallée, France<EMAIL_ADDRESS>
###### Abstract.

This article deals with topological assumptions under which the minimal volume
entropy of a closed manifold $M$, and more generally of a finite simplicial
complex $X$, vanishes or is positive. These topological conditions are
expressed in terms of the growth of the fundamental group of the fibers of
maps from a given finite simplicial complex $X$ to lower dimensional
simplicial complexes $P$. We also give examples of finite simplicial complexes
with zero simplicial volume and arbitrarily large minimal volume entropy.

###### Key words and phrases:

Minimal volume entropy, collapsing, exponential and subexponential growth,
fiber growth, Urysohn width.

###### 2020 Mathematics Subject Classification:

Primary 53C23; Secondary 57N65

Partially supported by the ANR project Min-Max (ANR-19-CE40-0014).

##  1\. Introduction

The notion of volume entropy has attracted a lot of attention since the early
works of Efremovich [21], Švarc [55] and Milnor [44]. This Riemannian
invariant describes the asymptotic geometry of the universal cover of a
Riemannian manifold and is related to the growth of its fundamental group; see
[55] and [44]. It is also connected to the dynamics of the geodesic flow. More
specifically, the volume entropy agrees with the topological entropy of the
geodesic flow of a closed nonpositively curved manifold and provides a lower
bound for it in general; see [19] and [41]. In this article, we study the
minimal volume entropy of a closed manifold (and more generally of a finite
simplicial complex), a topological invariant introduced by Gromov [29] related
to the simplicial volume. More precisely, we give topological conditions which
ensure, in one case, that the minimal volume entropy of a finite simplicial
complex is positive and, in the other case, that it vanishes. Before stating
our results, we need to introduce some definitions. Unless stated otherwise,
all spaces are path-connected.

######  Definition 1.1.

The _volume entropy_ of a connected finite simplicial complex $X$ with a
piecewise Riemannian metric $g$ is the exponential growth rate of the volume
of balls in the universal cover of $X$. More precisely, it is defined as

${\rm ent}(X,g)=\lim_{R\to\infty}\frac{1}{R}\log({\rm vol}\,\widetilde{B}(R))$
(1.1)

where $\widetilde{B}(R)$ is a ball of radius $R$ centered at any point in the
universal cover of $X$. The limit exists and does not depend on the center of
the ball. Observe that the volume entropy of a finite simplicial complex with
a piecewise Riemannian metric is positive if and only if its fundamental group
has exponential growth; see Definition 1.2.

The _minimal volume entropy_ of a connected finite simplicial $m$-complex $X$,
also known as _asymptotic volume_ , see [3], is defined as

$\omega(X)=\inf_{g}\,{\rm ent}(X,g)\,{\rm vol}(X,g)^{\frac{1}{m}}$

where $g$ runs over the space of all piecewise Riemannian metrics on $X$. This
topological invariant is known to be a homotopic invariant for closed
manifolds $M$, see [3], and more generally, an invariant depending only on the
image of the fundamental class of $M$ under the classifying map, see [13]. The
exact value of the minimal volume entropy (when nontrivial) of a closed
manifold is only known in a few cases; see [37], [9], [52], [53], [17], [43].
For instance, the minimal volume entropy of a closed $m$-manifold $M$ which
carries a hyperbolic metric is attained by the hyperbolic metric and is equal
to $(m-1)\,{\rm vol}(M,{\rm hyp})^{\frac{1}{m}}$; see [37] for $m=2$ and [9]
for $m\geq 3$.

The _simplicial volume_ of a connected closed orientable $m$-manifold $M$ is
defined as

$\|M\|_{\Delta}=\inf_{c}\sum_{s}|r_{s}|$

where the infimum is taken over all real singular $m$-cycles
$c=\sum_{s}r_{s}\,\sigma_{s}$ representing the fundamental class of $M$ in
$H_{m}(M;{\mathbb{Z}})$. Here, the maps $\sigma_{s}:\Delta^{m}\to M$ are
singular $m$-simplices. The definition extends to finite simplicial
$m$-complexes $X$ whose fundamental class is well-defined, that is, with
$H_{m}(X;{\mathbb{Z}})={\mathbb{Z}}$.

The following inequality of Gromov [29, p. 37] connects the minimal volume
entropy of a connected closed manifold to its simplicial volume (see also [8]
for a presentation of this result). Namely, every connected closed orientable
$m$-manifold $M$ satisfies

$\omega(M)^{m}\geq c_{m}\,\|M\|_{\Delta}$ (1.2)

for some positive constant $c_{m}$ depending only on $m$. Thus, every closed
manifold with positive simplicial volume has positive minimal volume entropy.
In particular, the minimal volume entropy of a closed manifold which carries a
negatively curved metric is positive; see [29]. Other topological conditions
ensuring the positivity of the minimal volume entropy have recently been
obtained in [51] and extended in [7, Section 4]; see [10] for a presentation
of numerous examples and cases where these conditions apply. These conditions
are related to the topology of the loop space of the manifold. In a different
direction, the minimal volume entropy provides a lower bound both on the
minimal volume, see [29], and on the systolic volume of a closed manifold, see
[50] and [13].

A natural question to ask in view of (1.2) is whether every closed orientable
manifold with zero simplicial volume has zero minimal volume entropy. This is
known to be true in dimension two [37] and in dimension three [49] (see also
[2] combined with Perelman’s resolution of Thurston’s geometrization
conjecture), where the cube of the minimal volume entropy is proportional to
the simplicial volume. In dimension four, the same is known to be true but
only for closed orientable geometrizable manifolds; see [54]. The techniques
developed in this article allow us to provide a negative answer for finite
simplicial complexes; see Theorem 1.6. The question for closed orientable
manifolds remains open despite recent progress made with the introduction of
the volume entropy semi-norm; see [6]. This geometric semi-norm in homology
measures the minimal volume entropy of a real homology class throughout a
stabilization process. Namely, given a path-connected topological space $X$,
it is defined for every ${\bf a}\in H_{m}(X;{\mathbb{Z}})$ as

$\|{\bf a}\|_{E}=\lim_{k\to\infty}\frac{\omega(k\,{\bf a})^{m}}{k}$ (1.3)

where $\omega({\bf a})$ is the infimum of the minimal relative volume entropy
of the maps $f:M\to X$ from an orientable connected closed $m$-pseudomanifold
$M$ to $X$ such that $f_{*}([M])={\bf a}$; see [6] for a more precise
definition. The volume entropy semi-norm shares similar functorial features
with the simplicial volume semi-norm. Moreover, the two semi-norms are
equivalent in every dimension. That is,

$c_{m}\,\|{\bf a}\|_{\Delta}\leq\|{\bf a}\|_{E}\leq C_{m}\,\|{\bf
a}\|_{\Delta}$ (1.4)

for some positive constants $c_{m}$ and $C_{m}$ depending only on $m$. Thus, a
closed manifold with zero simplicial volume has zero volume entropy semi-norm,
but its minimal volume entropy may be nonzero _a priori_. See [6] for further
details.

More generally, one may ask for a topological characterization of closed
manifolds or simplicial complexes with positive minimal volume entropy. Such a
topological characterization holds for the systolic volume, a topological
invariant sharing similar properties with the minimal volume entropy; see [3],
[4], [5], [13]. Namely, a closed $m$-manifold or simplicial $m$-complex has
positive systolic volume if and only if it is essential (i.e., its classifying
map cannot be homotoped into the $(m-1)$-skeleton of the target space); see
[30] and [3]. Though this condition is necessary to ensure that a closed
manifold or simplicial complex has positive minimal volume entropy, see [3],
it is not sufficient. Therefore, one should look for stronger or extra
assumptions.

In this article, we present topological conditions in this direction. The
first one implies that the minimal volume entropy of a given simplicial
complex vanishes and the second one ensures it is positive. Both these
conditions are expressed in terms of the exponential/subexponential growth of
the fundamental group of the fibers of maps between a given simplicial complex
and simplicial complexes of lower dimension. We will need the following
notions.

######  Definition 1.2.

Let $G$ be a finitely generated group and $S$ be a finite generating set of
$G$. Denote by $B_{S}(t)\subseteq G$ the ball centered at the identity element
of $G$ and of radius $t$ for the word distance induced by $S$. The group $G$
has _exponential growth_ if the exponential growth rate of the number of
elements in $B_{S}(t)$ defined as

${\rm ent}(G,S)=\lim_{t\to\infty}\frac{1}{t}\log|B_{S}(t)|$

is nonzero for some (and so any) finite generating set $S$. It has _uniform
exponential growth_ at least $h>0$ if the exponential growth rate of the
number of elements in $B_{S}(t)$ is at least $h$ for every finite generating
set $S$. That is, its _algebraic entropy_ satisfies

${\rm ent}(G)=\inf_{S}{\rm ent}(G,S)\geq h.$

The group $G$ has _subexponential growth_ if it does not have exponential
growth. In this case, the _subexponential growth rate_ of $G$ is defined as

$\nu(G)=\limsup_{t\to\infty}\frac{\log\log|B_{S}(t)|}{\log t}.$

Note that the subexponential growth rate does not depend on the chosen finite
generating set $S$.

The group $G$ has _polynomial growth_ if for some (and so any) finite
generating set, there exists a polynomial $P$ such that

$|B_{S}(t)|\leq P(t)$

for every $t\geq 0$. By [28], a finitely generated group has polynomial growth
if and only if it is virtually nilpotent.

The group $G$ has _intermediate growth_ if its growth is subexponential but
not polynomial. The first group of intermediate growth was constructed by
Grigorchuk [25] and [26], answering a question raised by Milnor. Still, it is
an open problem whether _finitely presented_ groups of intermediate growth
exist. Examples of finitely generated groups of exponential growth which do
not have uniform exponential growth were first constructed by Wilson [56],
answering a question asked in [27] and [31]. Still, it is an open question
whether all _finitely presented_ groups of exponential growth have uniform
exponential growth.

For our topological conditions, we consider connected finite simplicial
$m$-complexes $X$ along with simplicial maps $\pi:X\to P$ onto simplicial
complexes $P$ of dimension at most $k<m$, where $m\geq 2$. We denote by
$i_{*}:\pi_{1}(F_{p})\to\pi_{1}(X)$ the homomorphism induced by the inclusion
map $i:F_{p}\hookrightarrow X$ of a connected component $F_{p}$ of a fiber
$\pi^{-1}(p)$ of $\pi$.

The first condition considered for $X$ is the fiber $\pi_{1}$-growth
collapsing assumption (or collapsing assumption for short).

Fiber $\pi_{1}$-growth collapsing assumption. Suppose there exists a
simplicial map $\pi:X\to P$ onto a simplicial complex $P$ of dimension at most
$k<m$ such that for every connected component $F_{p}$ of every fiber
$\pi^{-1}(p)$ with $p\in P$, the finitely generated subgroup
$i_{*}[\pi_{1}(F_{p})]$ of $\pi_{1}(X)$ has subexponential growth. Recall that
$m$ is the dimension of $X$.

The fiber $\pi_{1}$-growth collapsing assumption _with polynomial growth rate_
is defined similarly with the condition that all the finitely generated
subgroup $i_{*}[\pi_{1}(F_{p})]$ of $\pi_{1}(X)$ have polynomial growth.

Likewise, the fiber $\pi_{1}$-growth collapsing assumption _with
subexponential growth rate at most $\nu$_ is defined similarly with the
condition that the subexponential growth rate of all the finitely generated
subgroup $i_{*}[\pi_{1}(F_{p})]$ of $\pi_{1}(X)$ is at most $\nu$.

In these definitions, it is enough to check the condition for every vertex
$p\in P$.

The following result shows that if the subexponential growth rate in the
collapsing assumption is small enough then the minimal volume entropy of $X$
vanishes.

######  Theorem 1.3.

Let $X$ be a connected finite simplicial $m$-complex satisfying the fiber
$\pi_{1}$-growth collapsing assumption with subexponential growth rate at most
$\nu$ onto a simplicial $k$-complex $P$. Suppose that $\nu<\frac{m-k}{m}$.
Then $X$ has zero minimal volume entropy, that is,

$\omega(X)=0.$

In Section 2.7, we give an example of a closed manifold satisfying the
assumption of Theorem 1.3 with a fiber whose image of the fundamental group is
a group of intermediate growth. Note that it is an open question whether
finitely presented groups of intermediate growth exist.

One may also wonder whether a similar result to Theorem 1.3 holds for
connected finite simplicial $m$-complexes $X$ satisfying the fiber
$\pi_{1}$-growth collapsing assumption without any condition on the
subexponential growth rate of the subgroups
$i_{*}[\pi_{1}(F_{p})]\leqslant\pi_{1}(X)$. In Section 2.7, we construct an
example showing that the natural method presented in the proof of Theorem 1.3
consisting of shrinking the fibers of the simplicial map $\pi:X\to P$ involved
in the definition of the fiber $\pi_{1}$-growth collapsing assumption does not
readily apply.

Since the subexponential growth rate of a group with polynomial growth is
zero, we immediately derive the following corollary.

######  Corollary 1.4.

Every connected finite simplicial complex satisfying the fiber
$\pi_{1}$-growth collapsing assumption with polynomial growth rate has zero
minimal volume entropy.

The second condition considered for $X$ is the fiber $\pi_{1}$-growth non-
collapsing assumption (or non-collapsing assumption for short).

Fiber $\pi_{1}$-growth non-collapsing assumption. Suppose that for every
simplicial map $\pi:X\to P$ onto a simplicial complex $P$ of dimension $k<m$,
there exists a connected component $F_{p_{0}}$ of some fiber $\pi^{-1}(p_{0})$
with $p_{0}\in P$ such that the finitely generated subgroup
$i_{*}[\pi_{1}(F_{p_{0}})]$ of $\pi_{1}(X)$ has uniform exponential growth at
least $h$ for some $h=h(X)>0$ depending only on $X$. Recall that $m$ is the
dimension of $X$.

This topological condition ensures that the minimal volume entropy of $X$ does
not vanish.

######  Theorem 1.5.

Every connected finite simplicial $m$-complex $X$ satisfying the fiber
$\pi_{1}$-growth non-collapsing assumption has positive minimal volume
entropy, that is,

$\omega(X)>0.$

As an example, we show in Proposition 3.7 that closed aspherical manifolds
whose fundamental group is a non-elementary word hyperbolic group satisfy the
non-collapsing assumption.

Theorem 1.5 asserts that, under some topological conditions, if the volume of
a simplicial complex is small then its volume entropy is large. The same
conclusion holds true without assuming that the volume of the whole simplicial
complex is small. More specifically, under the same topological conditions, if
the volume of every ball of small radius is small, then the volume entropy is
large; see Theorem 3.6 for a precise statement.

Note that the fibers of the simplicial map $\pi:X\to P$ in the definition of
the collapsing and non-collapsing conditions can always be assumed to be
connected; see Proposition 2.1. We will also give alternative formulations of
both the collapsing and non-collapsing assumptions in terms of open covers of
the simplicial complex $X$; see Proposition 2.13 and Proposition 3.10.

The new techniques developed in this article allow us to investigate the
relationship between the minimal volume entropy and the simplicial volume of
simplicial complexes whose fundamental class is well-defined. In view of the
lower and upper bounds (1.4), one can ask whether there is a complementary
inequality to the bound (1.2). Namely, does there exist a positive constant
$C_{m}$ such that

$\omega(M)^{m}\leq C_{m}\,\|M\|_{\Delta}$

for every connected closed orientable $m$-manifold $M$? The question also
makes sense for every connected finite simplicial $m$-complex $X$ whose
fundamental class is well-defined. Our next result provides a negative answer
in this case.

######  Theorem 1.6.

There exists a sequence of connected finite simplicial complexes $X_{n}$ with
a well-defined fundamental class such that the simplicial volume of $X_{n}$
vanishes and the minimal volume entropy of $X_{n}$ tends to infinity.

We emphasize that both Theorem 1.3 and Theorem 1.5 hold for the class of
finite simplicial complexes (including compact ${\rm CAT}(0)$ simplicial or
cubical complexes) and not solely for closed manifolds. This contrasts with
all previous works, which focus on closed manifolds. In particular, the
topological conditions ensuring the positivity of the minimal volume entropy,
see Theorem 1.5, apply to simplicial complexes for which the simplicial volume
is zero and the inequality (1.2) does not readily extend. This is exemplified
by Theorem 1.6.

The results established in this article have recently found applications in
[11].

_Acknowledgment._ The second author would like to thank the Fields Institute
and the Department of Mathematics at the University of Toronto for their
hospitality while this work was completed. We express our gratitude to
Rostislav Grigorchuk for multiple stimulating discussions. Finally, we thank
Corey Bregman and Matt Clay who pointed out a mistake in a previous version of
this article and drew our attention on their recent work [11].

##  2\. Simplicial complexes with zero minimal volume entropy

In this section, we show that the minimal volume entropy of a finite
simplicial complex satisfying the fiber $\pi_{1}$-growth collapsing assumption
with small subexponential growth rate vanishes. We also give a
characterization of the collapsing assumption in terms of open covers.
Finally, we show that the simplicial volume of a closed manifold satisfying
the collapsing assumption vanishes. Several examples and counter-examples are
presented throughout this section.

###  2.1. Connected and non-connected fibers

The following result shows that we can assume that the fibers of the
simplicial map $\pi:X\to P$ in the definition of the collapsing and non-
collapsing conditions are connected.

######  Proposition 2.1.

Let $\pi:X\to P$ be a simplicial map between two finite simplicial complexes.
Denote by $k$ the dimension of $P$. Then there exists a surjective simplicial
map $\bar{\pi}:X\to\bar{P}$ to a finite simplicial complex $\bar{P}$ of
dimension at most $k$ such that the fibers of $\bar{\pi}:X\to\bar{P}$ agree
with the connected components of the fibers of $\pi:X\to P$.

###### Proof.

Without loss of generality, we can assume that the simplicial map $\pi:X\to P$
is onto. Define $\bar{P}=X/\\!\sim$ as the quotient space of $X$, where $x\sim
y$ if $x$ and $y$ lie in the same connected component of a fiber of $\pi:X\to
P$. Since the map $\pi:X\to P$ is simplicial, the quotient space $\bar{P}$ is
a simplicial complex of the same dimension as $P$. By construction, the map
$\pi:X\to P$ factors out through a simplicial map $\bar{\pi}:X\to\bar{P}$
whose fibers agree with the connected components of the fibers of $\pi$. ∎

###  2.2. Construction of a family of piecewise flat metrics

Let $\pi:X\to P$ be simplicial map from a connected finite simplicial
$m$-complex $X$ to a simplicial $k$-complex $P$ with $k<m$. We will assume
that the map $\pi:X\to P$ is onto and that its fibers $F_{p}$ are connected;
see Proposition 2.1.

The goal of this section is to construct a family of piecewise flat metrics
$g_{t}$ on $X$ which collapses onto $P$. The construction relies on some
simplicial embeddings of $X$ and $P$ into an Euclidean space $E$ of large
dimension.

Let $\Delta^{s}=\Delta^{s}(p_{0},\dots,p_{s})$ be the abstract $s$-simplex
with the same vertices $p_{0},\dots,p_{s}$ as $P$. Fix an $(s+1)$-dimensional
Euclidean space $H$ with an orthonormal basis $e_{0},\dots,e_{s}$. Identify
the abstract $s$-simplex $\Delta^{s}$ with the regular $s$-simplex of $H$ with
vertices $\frac{1}{\sqrt{2}}e_{0},\dots,\frac{1}{\sqrt{2}}e_{s}$. Define the
subcomplex

$R_{i}=\pi^{-1}(p_{i})\subseteq X.$

As previously, let $\Delta(R_{i})$ be the abstract simplex with the same
vertices as $R_{i}$. Denote by $m_{i}$ the dimension of $\Delta(R_{i})$. Fix
an $(m_{i}+1)$-dimensional Euclidean space $H_{i}$ with an orthonormal basis
$e_{0}^{i},\dots,e_{m_{i}}^{i}$. Identify the abstract $m_{i}$-simplex
$\Delta(R_{i})$ with the regular $m_{i}$-simplex of $H_{i}$ with vertices
$\frac{1}{\sqrt{2}}e^{i}_{0},\dots,\frac{1}{\sqrt{2}}e^{i}_{m_{i}}$.

Consider the orthogonal sum

$E=H\oplus H_{0}\oplus\dots\oplus H_{s}.$ (2.1)

Denote by $g_{E}$ the scalar product on $E$. There is a natural simplicial
embedding $\chi:X\hookrightarrow E$ taking every vertex $v\in X$, identified
with some element $\frac{1}{\sqrt{2}}e_{j}^{i}$ with $0\leq i\leq s$ and
$0\leq j\leq m_{i}$, to

$\chi(v)=\tfrac{1}{\sqrt{2}}e_{i}+\tfrac{1}{\sqrt{2}}e_{j}^{i}.$

(Here, a simplicial embedding means an embedding whose restriction to each
simplex is an affine map.) Thus, the whole space $R_{i}$ is sent under
$\chi:X\hookrightarrow E$ to a point lying in the subspace
$H^{\prime}_{i}=\frac{1}{\sqrt{2}}e_{i}+H_{i}$ orthogonal to $H$, parallel to
$H_{i}$ and passing through $\frac{1}{\sqrt{2}}e_{i}$. By our choices of
identification, the composition of $\chi:X\hookrightarrow E$ with the
orthogonal projection $p_{H}:E\to H$ onto $H$ coincides with the simplicial
map $\pi:X\to P$, that is,

$\pi=p_{H}\circ\chi.$

The piecewise flat metric on $X$ induced by the simplicial embedding
$\chi:X\hookrightarrow E$ can be deformed as follows. Let $h_{t}:E\to E$ be
the endomorphism of $E$ preserving each factor of the decomposition (2.1)
whose restriction to $H$ is the identity map and restriction to each $H_{i}$
is the homothety with coefficient $t$. For every $t\in(0,1]$, the map
$\chi_{t}:X\hookrightarrow E$ defined as

$\chi_{t}=h_{t}\circ\chi$

is a simplicial embedding. Note that $h_{t}$ preserves the subspaces
$H^{\prime}_{i}$. By construction, we still have

$\pi=p_{H}\circ\chi_{t}.$

Endow $X$ with the piecewise flat metric $g_{t}$ induced by the simplicial
embedding $\chi_{t}:X\hookrightarrow E$ defined as

$g_{t}=\chi_{t}^{*}(g_{E}).$

Endow also $P$ with the natural piecewise flat metric where all its simplices
are isometric to the standard Euclidean simplex induced by the simplicial
embedding $P\subseteq E$. By construction, the map $\pi:X\to P$ is
$1$-Lipschitz. Observe also that the $g_{t}$-length of every edge lying in
some fiber $\pi^{-1}(p_{i})\subseteq X$ over a vertex $p_{i}\in P$ is equal to
$t$. Since $P$ is a $k$-dimensional simplicial complex, we conclude that

${\rm vol}(X,g_{t})=O(t^{m-k})$ (2.2)

as $t$ goes to zero. Note also that $\mathop{\rm
diam}\nolimits_{g_{t}}(\Delta_{X}^{p})\leq\sqrt{2}$.

###  2.3. Construction of Lipschitz retractions around each fiber

Using the same notations as in the previous section, we construct a Lipschitz
retraction from a neighborhood of each fiber of $\pi:X\to P$ onto the fiber
itself. This is an important technical result which will be used in Section
2.4 to deform paths of $X$ into the $1$-skeleton of $X$ without increasing
their $g_{t}$-length too much (uniformly in $t$).

Let $\Delta^{q}=\Delta_{P}^{q}$ be a $q$-simplex of $P$. Recall that
$\Delta^{q}$ lies in $H$. Fix a vertex $v$ of $\Delta^{q}$, say $v=p_{0}$.
Denote by $\Delta_{v}^{q-1}$ the $(q-1)$-face of $\Delta^{q}$ opposite to $v$.
Consider a $p$-simplex $\Delta_{X}^{p}$ of $X$ mapped onto $\Delta_{P}^{q}$
under $\pi:X\to P$. Denote by $\delta_{0}^{r}=\pi^{-1}(v)\cap\Delta_{X}^{p}$
the $r$-simplex of $\Delta_{X}^{p}$ sent to $v$ by $\pi:X\to P$. Construct a
retraction

$\bar{\varrho}_{t}:\Delta_{X}^{p}\setminus\pi^{-1}(\Delta_{v}^{q-1})\to\delta_{0}^{r}$

onto $\delta_{0}^{r}$ as follows. First, embed $\Delta_{X}^{p}$ into the
Euclidean space $E$ through $\chi_{t}:X\hookrightarrow E$. Under this
identification, the image $h_{t}(\delta_{0}^{r})$ of $\delta_{0}^{r}$ lies in
the subspace $H^{\prime}_{0}$ orthogonal to $H$, parallel to $H_{0}$ and
passing through $v$. Then, take the orthogonal projection to $H\oplus H_{0}$.
Note that the image of $\Delta_{X}^{p}$ under the composition of these maps
agrees with the convex hull ${\rm
Conv}(h_{t}(\delta_{0}^{r})\cup\Delta_{v}^{q-1})$. Thus, every point
$x\in\Delta_{X}^{p}\setminus\pi^{-1}(\Delta_{v}^{q-1})$ is sent to a point
$\bar{x}\in{\rm Conv}(h_{t}(\delta_{0}^{r})\cup\Delta_{v}^{q-1})$. Then, for
every $\bar{x}\in{\rm
Conv}(h_{t}(\delta_{0}^{r})\cup\Delta_{v}^{q-1})\setminus\Delta_{v}^{q-1}$ not
lying in $h_{t}(\delta_{0}^{r})$, take the orthogonal projection
$\bar{x}^{\prime}\in\Delta^{q}$ of $\bar{x}$ to $\Delta^{q}$, send
$\bar{x}^{\prime}$ to the point $\bar{x}^{\prime\prime}\in\Delta_{v}^{q-1}$
where the ray arising from $v$ and passing through $\bar{x}^{\prime}$ meets
$\Delta_{v}^{q-1}$, and map $\bar{x}$ to the point $y^{\prime}\in
h_{t}(\delta_{0}^{r})$ where the ray arising from $\bar{x}^{\prime\prime}$ and
passing through $\bar{x}$ intersects $h_{t}(\delta_{0}^{r})$. The map taking
$\bar{x}$ to $y^{\prime}$ extends by continuity into the identity map on
$h_{t}(\delta_{0}^{r})$. Finally, take the image $y\in\delta_{0}^{r}$ of
$y^{\prime}$ under the inverse map
$\chi_{t}^{-1}:h_{t}(\delta_{0}^{r})\to\delta_{0}^{r}$. The resulting map
$\bar{\varrho}_{t}:\Delta_{X}^{p}\setminus\pi^{-1}(\Delta_{v}^{q-1})\to\delta_{0}^{r}$
sending $x$ to $y$ is a retraction onto $\delta_{0}^{r}$.

The map
$\bar{\varrho}_{t}:\Delta_{X}^{p}\setminus\pi^{-1}(\Delta_{v}^{q-1})\to\delta_{0}^{r}$
is not Lipschitz as the Lipschitz constant at a point goes to infinity when
the point moves to $\Delta_{X}^{p}\cap\pi^{-1}(\Delta_{v}^{q-1})$. For the map
to be Lipschitz, we need to restrict it to a domain away from
$\pi^{-1}(\Delta_{v}^{q-1})\cap\Delta_{X}^{p}$. In order to use the map as a
building block to construct further maps on simplicial complexes, we also need
to take domains that are coherent in terms of face inclusion. Extend
$\Delta^{q}$ into a regular Euclidean $m$-simplex $\Delta^{m}\subseteq H$,
where $\Delta^{q}$ is a face of $\Delta^{m}$. The perpendicular bisector
hyperplane of the segment joining the barycenters of $\Delta^{m}$ and
$\Delta_{v}^{m}$ intersects $\Delta^{q}$ along a subspace $\mathcal{H}$. Let
$\tau_{q,m}=d(v,\mathcal{H})$ be the distance from $v$ to $\mathcal{H}$ in
$\Delta^{q}$. Observe that the sequence $\tau_{q,m}$ is decreasing in $q$. In
particular,

$\tau_{q,m}\geq\tau_{m}:=\tau_{m,m}.$ (2.3)

Note also that $\tau_{m}\geq\frac{1}{2}$. See Figure 1 below.

$\tau_{q,m}$
---
$\Delta^{q}$
---
$\mathcal{H}$
---
$\Delta^{m}$
---
$\mathcal{H}^{\prime}$
---
$v$
---
Figure 1. Construction of $\mathcal{H}$

Consider the domain $\Delta^{q}(v)$ of $\Delta^{q}$ containing $v$ delimited
by $\mathcal{H}$. The restriction

$\varrho_{t}:\pi^{-1}(\Delta^{q}(v))\cap\Delta_{X}^{p}\to\delta_{0}^{r}$

of $\bar{\varrho}_{t}$ is $\kappa_{m}$-Lipschitz for some constant
$\kappa_{m}\geq 1$ depending only on $m$. Note that this construction is
coherent. That is, if $\Delta_{P}$ and $\Delta^{\prime}_{P}$ are two simplices
of $P$ containing $v$, and $\Delta_{X}$ and $\Delta^{\prime}_{X}$ are two
simplices of $X$ mapped onto $\Delta_{P}$ and $\Delta^{\prime}_{P}$ under
$\pi:X\to P$, then the retractions $\varrho_{t}$ defined on
$\pi^{-1}(\Delta_{P}(v))\cap\Delta_{X}$ and
$\pi^{-1}(\Delta^{\prime}_{P}(v))\cap\Delta^{\prime}_{X}$ coincide with the
intersection of their domains of definition. This will allows us to put
together the retractions $\varrho_{t}$.

Given a point $z$ of $\Delta_{X}^{p}$ lying in $\mathcal{H}$, let $z_{-}$ be a
vertex of $\Delta_{X}^{p}$ lying in $\delta_{0}^{r}$ and $z_{+}$ be a vertex
of $\Delta_{X}^{p}$ not lying in $\delta_{0}^{r}$ at minimal $g_{t}$-distance
from $z$. Recall that $\Delta_{X}^{p}$ collapses onto $\Delta_{P}^{q}$ in $E$
as $t$ goes to zero. By our choice of $\mathcal{H}$, there exist
$\varepsilon_{m},\sigma_{m}\in(0,1)$ depending only on $m$ such that for $t$
small enough

$d(z,z_{+})\leq d(z,z_{-})-\varepsilon_{m}$ (2.4)

and

$d(z,z_{+})+\sigma_{m}\leq\tau_{m}.$ (2.5)

Now, define

$P_{v}=\cup\Delta_{P}^{q}(v)\subseteq P$ (2.6)

as the union over all the domains $\Delta_{P}^{q}(v)\subseteq\Delta_{P}^{q}$,
where $\Delta_{P}^{q}$ is a simplex of $P$ of any dimension $q$ containing
$v$. Denote also

$X_{v}=\pi^{-1}(P_{v})\subseteq X.$ (2.7)

Putting together the retractions
$\varrho_{t}:\pi^{-1}(\Delta^{q}(v))\cap\Delta_{X}^{p}\to\delta_{0}^{r}$ where
$\Delta_{X}^{p}$ is a simplex of $X_{v}$ projecting to a simplex
$\Delta_{P}^{q}$ of $P$ containing $v$ and
$\delta_{0}^{r}=\pi^{-1}(v)\cap\Delta_{X}^{p}$, we obtain a
$\kappa_{m}$-Lipschitz retraction of $X_{v}$ onto $\pi^{-1}(v)$, still denoted
by

$\varrho_{t}:X_{v}\to\pi^{-1}(v).$ (2.8)

###  2.4. Deforming arcs into edge-arcs

Considering the family of piecewise flat metrics $g_{t}$ on $X$ previously
defined, we show the following result about the deformation of arcs of $X$
into its $1$-skeleton. This result will allow us to apply combinatorial
techniques to count homotopy classes in Section 2.5.

######  Proposition 2.2.

Every arc $\gamma$ of $X$ joining two vertices can be deformed into an arc
$\gamma_{e}$ lying in the $1$-skeleton of $X$ (i.e., $\gamma_{e}$ is an edge-
arc), while keeping its endpoints fixed, with

${\rm length}_{g_{t}}(\gamma_{e})\leq C_{m}\,{\rm length}_{g_{t}}(\gamma)$
(2.9)

for every $t\in(0,1]$, where $C_{m}$ is a positive constant depending only on
$m$.

###### Proof.

Let us start with a simple observation. Every arc of a regular Euclidean
simplex $\Delta^{m}$ with endpoints on $\partial\Delta^{m}$ can be deformed
into an arc of $\partial\Delta^{m}$ with the same endpoints at the cost of
multiplying its length by a factor bounded by a constant $\lambda_{m}$
depending only on $m$. Applying this observation successively on the simplices
of the skeleta of $X$, we deduce by induction that the inequality (2.9) holds
with $C_{m}=\lambda^{\prime}_{m}$ for $t=1$, where
$\lambda^{\prime}_{m}=\prod_{i=2}^{m}\lambda_{i}$, and, more generally, when
every simplex of $X$ is isometric to a regular Euclidean simplex of the same
size.

Endow $X$ with the piecewise flat metric $g_{t}$ and $P$ with its natural
piecewise flat metric where all simplices are isometric to the standard
Euclidean simplex of the same dimension. Denote by $v$ the image of the
starting point of $\gamma$ by $\pi:X\to P$. Note that $v$ is a vertex of $P$.
Consider the domains $P_{v}$ and $X_{v}$ introduced in (2.6) and (2.7). For
every $q$-simplex $\Delta^{q}\subseteq P_{v}$ containing $v$, the distance
between $v$ and its opposite side in $\Delta^{q}(v)$ is at least $\tau_{m}$.
Since the map $\pi:X_{v}\to P_{v}$ is $1$-Lipschitz, we deduce that if
$\gamma$ leaves $X_{v}$ then its $g_{t}$-length is greater than $\tau_{m}$.

Let us argue by induction on the integer $n\geq 0$ such that

$n\tau_{m}\leq{\rm length}_{g_{t}}(\gamma)<(n+1)\tau_{m}$

where $\tau_{m}$ is defined in (2.3). The value of $C_{m}$ in (2.9) can be
taken to be equal to
$C_{m}=12\tfrac{\lambda^{\prime}_{m}\kappa_{m}}{\sigma_{m}}$, where
$\kappa_{m}$ is the Lipschitz constant of the map $\varrho_{t}$ defined in
(2.8), $\sigma_{m}$ is defined in (2.5), and $\lambda^{\prime}_{m}$ is defined
above.

Suppose that $\gamma$ lies in $X_{v}$. (This is the case for instance if ${\rm
length}_{g_{t}}(\gamma)<\tau_{m}$, i.e., $n=0$.) The image $\gamma^{\prime}$
of $\gamma$ under the $\kappa_{m}$-Lipschitz retraction
$\varrho_{t}:X_{v}\to\pi^{-1}(v)$ satisfies

${\rm length}_{g_{t}}(\gamma^{\prime})\leq\kappa_{m}\,{\rm
length}_{g_{t}}(\gamma).$

By construction, the fiber $\pi^{-1}(v)$ is a simplicial complex of dimension
at most $m$ composed of regular Euclidean simplices of size $t$. As observed
at the beginning of the proof, the arc $\gamma^{\prime}$ lying in
$\pi^{-1}(v)$ can be deformed into an arc $\gamma_{e}$ lying in the
$1$-skeleton of $\pi^{-1}(v)$, and so of $X$, with the same endpoints
multiplying its length by a factor bounded by at most $\lambda^{\prime}_{m}$.
This concludes the proof of the proposition in this case with
$C_{m}=\kappa_{m}\lambda^{\prime}_{m}$.

Suppose that $\gamma$ leaves $X_{v}$. Denote by $z$ the first point where
$\gamma$ leaves $X_{v}$. The point $z$ splits $\gamma$ into two subarcs,
$\gamma^{\prime}$ and $\gamma^{\prime\prime}$, with $\gamma^{\prime}\subseteq
X_{v}$. Let $\Delta_{X}^{p}$ be the simplex of $X$ containing $z$. (This
simplex is not necessarily unique.) Pick a vertex $z_{-}$ of $\Delta_{X}^{p}$
lying in $\pi^{-1}(v)$ and a vertex $z_{+}$ of $\Delta_{X}^{p}$ not lying in
$\pi^{-1}(v)$ at minimal distance from $z$. By (2.4), we have

$d(z,z_{+})\leq d(z,z_{-})-\varepsilon_{m}\leq{\rm
length}_{g_{t}}(\gamma^{\prime})-\varepsilon_{m}.$ (2.10)

Since $z$ and $z_{\pm}$ lie in the same simplex $\Delta_{X}^{p}$, the arc
$\gamma$ is homotopic to
$\alpha^{\prime}\cup[z_{-},z_{+}]\cup\alpha^{\prime\prime}$, where the two
arcs

$\alpha^{\prime}=\gamma^{\prime}\cup[z,z_{-}]\quad\mbox{ and
}\quad\alpha^{\prime\prime}=[z_{+},z]\cup\gamma^{\prime\prime}$

start and end at vertices of $P$. As previously observed, we have ${\rm
length}_{g_{t}}(\gamma^{\prime})\geq\tau_{m}$. Recall also that $\mathop{\rm
diam}\nolimits_{g_{t}}(\Delta_{X}^{p})\leq\sqrt{2}$. Thus,

${\rm length}_{g_{t}}(\alpha^{\prime})\leq{\rm
length}_{g_{t}}(\gamma^{\prime})+\sqrt{2}\leq\left(1+\tfrac{\sqrt{2}}{\tau_{m}}\right)\,{\rm
length}_{g_{t}}(\gamma^{\prime})$

for $t$ small enough. The arc $\alpha^{\prime}$ lies in $X_{v}$ and is sent to
an arc of $\pi^{-1}(v)$ with the same endpoints under the
$\kappa_{m}$-Lipschitz retraction $\varrho_{t}:X_{v}\to\pi^{-1}(v)$. In turn,
this arc can be deformed into an arc $\alpha_{e}^{\prime}$ lying in the
$1$-skeleton of $X$ with the same endpoints with

$\displaystyle{\rm length}_{g_{t}}(\alpha_{e}^{\prime})$
$\displaystyle\leq\lambda^{\prime}_{m}\kappa_{m}\,{\rm
length}_{g_{t}}(\alpha^{\prime})$
$\displaystyle\leq\lambda^{\prime}_{m}\kappa_{m}\left(1+\tfrac{\sqrt{2}}{\tau_{m}}\right)\,{\rm
length}_{g_{t}}(\gamma^{\prime}).$ (2.11)

Now, by (2.10), we have

$\displaystyle{\rm length}_{g_{t}}(\alpha^{\prime\prime})$
$\displaystyle\leq{\rm length}_{g_{t}}(\gamma^{\prime\prime})+d(z,z_{+})$
$\displaystyle\leq{\rm length}_{g_{t}}(\gamma)-\varepsilon_{m}.$

By induction, the arc $\alpha^{\prime\prime}$ can be deformed into an edge-arc
$\alpha^{\prime\prime}_{e}$ with the same endpoints with

$\displaystyle{\rm length}_{g_{t}}(\alpha^{\prime\prime}_{e})$
$\displaystyle\leq C_{m}\,{\rm length}_{g_{t}}(\alpha^{\prime\prime})$
$\displaystyle\leq C_{m}\,{\rm
length}_{g_{t}}(\gamma^{\prime\prime})+C_{m}\,d(z,z_{+}).$ (2.12)

As a result of (2.11) and (2.12), the arc $\gamma$ can be deformed into the
edge-arc
$\gamma_{e}=\alpha_{e}^{\prime}\cup[z_{-},z_{+}]\cup\alpha^{\prime\prime}_{e}$,
where

${\rm
length}_{g_{t}}(\gamma_{e})\leq\lambda^{\prime}_{m}\kappa_{m}\left(1+\tfrac{\sqrt{2}}{\tau_{m}}\right)\,{\rm
length}_{g_{t}}(\gamma^{\prime})+\sqrt{2}+C_{m}\,{\rm
length}_{g_{t}}(\gamma^{\prime\prime})+C_{m}\,d(z,z_{+}).$

In order to have ${\rm length}_{g_{t}}(\gamma_{e})\leq C_{m}\,{\rm
length}_{g_{t}}(\gamma)$, it is enough to have

$A_{m}\,{\rm length}_{g_{t}}(\gamma^{\prime})+\sqrt{2}+C_{m}\,d(z,z_{+})\leq
C_{m}\,{\rm length}_{g_{t}}(\gamma^{\prime})$

where
$A_{m}=\lambda^{\prime}_{m}\kappa_{m}\left(1+\tfrac{\sqrt{2}}{\tau_{m}}\right)\leq
4\lambda^{\prime}_{m}\kappa_{m}$ (recall that $\tau_{m}\geq\frac{1}{2}$). That
is,

$\frac{C_{m}\,d(z,z_{+})+\sqrt{2}}{C_{m}-A_{m}}\leq{\rm
length}_{g_{t}}(\gamma^{\prime}).$

Recall that $d(z,z_{+})+\sigma_{m}\leq\tau_{m}$; see (2.5). Thus, for $C_{m}$
large enough (e.g., $C_{m}\geq
12\frac{\lambda^{\prime}_{m}\kappa_{m}}{\sigma_{m}}\geq\frac{(1+\sqrt{2}+\sigma_{m})A_{m}}{\sigma_{m}}$),
we have

$\frac{C_{m}\,d(z,z_{+})+\sqrt{2}}{C_{m}-A_{m}}\leq
d(z,z_{+})+\sigma_{m}\leq\tau_{m}\leq{\rm length}_{g_{t}}(\gamma^{\prime})$

as desired. ∎

###  2.5. Edge-loop entropy

In this section, we introduce the edge-loop entropy – a discrete substitute
for the volume entropy – and show that the growth of the edge-loop entropy of
$(X,g_{t})$ is controlled as $t$ goes to zero.

######  Definition 2.3.

Let $X$ be a connected finite simplicial complex with a piecewise Riemannian
metric $g$. The volume entropy of $(X,g)$, see (1.1), can also be defined as
the exponential growth rate of the number of homotopy classes induced by loops
of length at most $T$. Namely,

${\rm ent}(X,g)=\lim_{T\to\infty}\frac{1}{T}\log\mathcal{N}(X,g;T)$ (2.13)

where $\mathcal{N}(X,g;T)={\rm
card}\\{[\gamma]\in\pi_{1}(X,\star)\mid\gamma\mbox{ loop of }g\mbox{-length at
most }T\\}$. See [50, Lemma 2.3] for instance, for a proof of this classical
result.

It will be convenient to consider a similar notions for edge-loops. Define the
edge-loop entropy of $(X,g)$ as

${\rm ent}_{e}(X,g)=\lim_{T\to\infty}\frac{1}{T}\log\mathcal{N}_{e}(X,g;T)$

where $\mathcal{N}_{e}(X,g;T)={\rm
card}\\{[\gamma]\in\pi_{1}(X,\star)\mid\gamma\mbox{ edge-loop of
}g\mbox{-length at most }T\\}$. Clearly, one has ${\rm ent}_{e}(X,g)\leq{\rm
ent}(X,g)$.

Let $A$ be a subcomplex of $X$ with basepoint $a$. We also define

$\mathcal{N}(A\subseteq X;T)={\rm
card}\\!\left\\{[\gamma]\in\pi_{1}(X,a)\mid\gamma\subseteq A\mbox{ and }{\rm
length}(\gamma)\leq T\right\\}$

of homotopy classes (in $X$) of loops of $A$ based at $a$ of $g$-length at
most $T$.

The edge-loop entropy of $(X,g_{t})$ can be bounded as follows.

######  Proposition 2.4.

Suppose that the subexponential growth rate of all the subgroups
$i_{*}[\pi_{1}(F_{p})]$ of $\pi_{1}(X)$ is at most $\nu$, where
$F_{p}=\pi^{-1}(p)$ is a (connected) fiber of $\pi:X\to P$ and
$i:F_{p}\hookrightarrow X$ is the inclusion map. Then

${\rm ent}_{e}(X,g_{t})=O\left(\tfrac{1}{t^{\nu}}\right).$ (2.14)

ss $t$ goes to zero.

###### Proof.

Let us introduce a couple of definitions. An edge of $X$ is said to be _long_
if it is sent to an edge of $P$ by the simplicial map $\pi:X\to P$. It is said
to be _short_ otherwise (in which case, it is sent to a vertex of $P$). By
construction, every long edge of $X$ is of length $\sqrt{1+t^{2}}$ and every
short edge of $X$ is of length $t$. Denote also by $n_{e}$ the number of edges
of $X$.

Observe that $g_{t}=t^{2}g_{1}$ on every (connected) fiber $F_{p}=\pi^{-1}(p)$
of $\pi:X\to P$. Hence,

$\mathop{\rm diam}\nolimits(F_{p},g_{t})=t\cdot\mathop{\rm
diam}\nolimits(F_{p},g_{1})\xrightarrow[t\to 0]{}0.$

where the diameter is measured with respect to the length structure induced by
$g_{t}$ on $F_{p}$. Thus, by taking $t$ small enough, we can assume that
$\mathop{\rm diam}\nolimits(F_{p},g_{t})<\frac{1}{2}$ for every vertex $p\in
P$.

Let us estimate the number of homotopy classes of edge-loops in $X$ of
$g_{t}$-length at most $T$. Every edge-loop $\gamma$ in $X$ of $g_{t}$-length
at most $T$ decomposes as

$\gamma=\alpha_{1}\cup\beta_{1}\cup\alpha_{2}\cup\cdots\cup\beta_{N}$ (2.15)

where $\alpha_{i}$ is a long edge of $X$ and $\beta_{i}$ is a possibly
constant edge-path lying in a (connected) fiber $F_{i}=\pi^{-1}(p_{i})$ of
$\pi:X\to P$ over a vertex $p_{i}\in P$, which joins the terminal endpoint of
$\alpha_{i}$ to the initial endpoint of $\alpha_{i+1}$.

Fix a basepoint $a_{i}\in F_{i}$. Denote by $\ell_{i}$ the $g_{t}$-length of
$\beta_{i}$. Let $\bar{\beta}_{i}$ be the loop of $F_{i}$ based at $a_{i}$
obtained by connecting the endpoints $x_{i}$ and $y_{i}$ of $\beta_{i}$ to the
basepoint $a_{i}$ along two paths of $F_{i}$ of $g_{t}$-length at most
$\mathop{\rm diam}\nolimits(F_{i},g_{t})<\frac{1}{2}$. The number
$\mathcal{N}^{e}_{{x_{i}},{y_{i}}}(F_{i}\subseteq(X,g_{t});\ell_{i})$ of
homotopy classes (relative to the endpoints) in $X$ of edge-paths in $F_{i}$
with endpoints $x_{i}$ and $y_{i}$, and $g_{t}$-length at most $\ell_{i}$ is
bounded by the number of homotopy classes in $X$ of loops in $F_{i}$ based at
$a_{i}$ of $g_{t}$-length at most $\ell_{i}+2\,\mathop{\rm
diam}\nolimits(F_{i},g_{t})$. Thus,

$\displaystyle\mathcal{N}^{e}_{{x_{i}},{y_{i}}}(F_{i}\subseteq(X,g_{t});\ell_{i})$
$\displaystyle\leq\mathcal{N}(F_{i}\subseteq(X,g_{t});\ell_{i}+2\,\mathop{\rm
diam}\nolimits(F_{i},g_{t}))$
$\displaystyle\leq\mathcal{N}\left(F_{i}\subseteq(X,g_{1});\tfrac{\ell_{i}+1}{t}\right)$
(2.16)

since $g_{t}=t^{2}g_{1}$ on the fiber $F_{i}$, where we refer to Definition
2.3 for the definition of $\mathcal{N}$.

By assumption, the subgroups $i_{*}[\pi_{1}(F_{p})]\leqslant\pi_{1}(X)$ have a
subexponential growth at most $\nu$ and the same holds for
$\mathcal{N}(F_{p}\subseteq(X,g_{1});T)$; see [42]. More specifically, there
exists a function $Q(T)=A\exp(T^{\nu})$ with $A>0$ such that

$\mathcal{N}(F_{p}\subseteq(X,g_{1});T)\leq Q(T)$ (2.17)

for every vertex $p\in P$ and every $T\geq 0$.

It follows from (2.16) and (2.17) that the number of homotopy classes in $X$
induced by the different possibilities for the edge-path $\beta_{i}$ of length
$\ell_{i}$ is at most

$\mathcal{N}^{e}_{{x_{i}},{y_{i}}}(F_{i}\subseteq(X,g_{t});\ell_{i})\leq
Q\left(\tfrac{\ell_{i}+1}{t}\right)$

where $\ell_{i}$ is the $g_{t}$-length of $\beta_{i}$.

Now, there are at most $n_{e}$ possible choices for each long edge
$\alpha_{i}$. (Recall that $n_{e}$ is the number of edges of $X$.) Hence, the
number of homotopy classes of edge-loops in $X$ of $g_{t}$-length at most $T$
which decomposes as in (2.15) with $\beta_{i}$ of $g_{t}$-length
$\ell_{i}\leq\theta_{i}$, where
$\theta_{i}=\left\lceil{\ell_{i}}\right\rceil$, is bounded by

$n_{e}^{N}\,\prod_{i=1}^{N}Q\left(\tfrac{\theta_{i}+1}{t}\right).$

Since every edge $\alpha_{i}$ is of $g_{t}$-length at least $1$, we have
$N\leq T$ and $\sum_{i=1}^{N}\ell_{i}\leq T-N$. Since
$\theta_{i}=\left\lceil{\ell_{i}}\right\rceil$, we also have
$\sum_{i=1}^{N}\theta_{i}\leq T$. Therefore, the number
$\mathcal{N}_{e}(X,g_{t};T)$ of homotopy classes of edge-loops in $X$ of
$g_{t}$-length at most $T$ is bounded by

$\mathcal{N}_{e}(X,g_{t};T)\leq\sum_{N\leq\left\lfloor{T}\right\rfloor}\sum_{(\theta_{i})_{N}\leq\left\lfloor{T}\right\rfloor}n_{e}^{N}\,\prod_{i=1}^{N}Q\left(\tfrac{\theta_{i}+1}{t}\right)$
(2.18)

where the second sum is over all $N$-tuples $(\tau_{1},\dots,\tau_{N})$ of
positive integers such that
$\sum_{i=1}^{N}\theta_{i}\leq\left\lfloor{T}\right\rfloor$.

The double sum (2.18) has at most $T\,2^{T}$ terms (the first sum has
$\left\lfloor{T}\right\rfloor$ terms and the second sum has
$2^{\left\lfloor{T}\right\rfloor-1}$ terms given by the distinct decomposition
of the integer $\left\lfloor{T}\right\rfloor$). Consider the largest term of
(2.18) attained by some $N\leq T$ and $(\theta_{i})_{N}\leq T$. We have

$\displaystyle\mathcal{N}_{e}(X,g_{t};T)$ $\displaystyle\leq
T\,2^{T}n_{e}^{T}\,\prod_{i=1}^{N}Q\left(\tfrac{\theta_{i}+1}{t}\right)$
(2.19) $\displaystyle\leq
T\,2^{T}n_{e}^{T}\,A^{T}\exp\left(\frac{1}{t^{\nu}}\sum_{i=1}^{N}(\theta_{i}+1)^{\nu}\right).$

Applying Hölder’s inequality to the sum $\sum_{i=1}^{N}(\theta_{i}+1)^{\nu}$
with $p=\frac{1}{1-\nu}$ and $q=\frac{1}{\nu}$, we obtain

$\sum_{i=1}^{N}(\theta_{i}+1)^{\nu}\leq\left(\sum_{i=1}^{N}1^{p}\right)^{1/p}\cdot\left(\sum_{i=1}^{N}(\theta_{i}+1)\right)^{1/q}\leq
T^{1-\nu}\cdot 2^{\nu}T^{\nu}\leq 2T$

since $\nu q=1$, $N\leq T$ and
$\sum_{i=1}^{N}(\theta_{i}+1)\leq\sum_{i=1}^{N}\theta_{i}+N\leq 2T$. Hence,

$\mathcal{N}_{e}(X,g_{t};T)\leq
T\,2^{T}\,n_{e}^{T}\,A^{T}\,\exp\left(\frac{2T}{t^{\nu}}\right).$

This implies that

${\rm ent}_{e}(X,g_{t})\leq\log(2n_{e}A)+\frac{2}{t^{\nu}}.$

∎

######  Remark 2.5.

If $X$ satisfies the collapsing assumption with polynomial growth rate, we can
derive a stronger bound than (2.14). Namely, the edge-loop entropy of
$(X,g_{t})$ has a logarithmic growth when $t$ goes to zero, that is,

${\rm ent}_{e}(X,g_{t})=O\left(\log\left(\tfrac{1}{t}\right)\right).$

The argument is similar to the proof of Proposition 2.4 until the inequality
(2.19), except that $Q$ should be replaced by a polynomial of the form
$Q(T)=a(T+1)^{d}$ with $a>0$. Now, using the expression of $Q$, the concavity
of the nondecreasing function $\log(1+\cdot)$, and the inequalities $N\leq T$
and $\sum_{i=1}^{N}(\theta_{i}+1)\leq 2T$, we obtain

$\displaystyle\log\left(\prod_{i=1}^{N}Q\left(\tfrac{\theta_{i}+1}{t}\right)\right)$
$\displaystyle\leq
T\,\log(a)+d\,\sum_{i=1}^{N}\log\left(1+\tfrac{\theta_{i}+1}{t}\right)$
$\displaystyle\leq T\,\log(a)+d\,N\,\log\left(1+\tfrac{2T}{Nt}\right)$ (2.20)

Introduce $f_{t}(x)=x\,\log(1+\frac{1}{xt})$ with $x\in[0,1]$. For
$t\leq\frac{1}{e-1}$, we have

$f^{\prime}_{t}(x)=\log(1+\tfrac{1}{xt})-\tfrac{1}{xt+1}\geq\log(1+\tfrac{1}{t})-1\geq
0.$

Thus, for $x=\frac{N}{2T}$ and $t$ small enough, we deduce that

$\tfrac{1}{2}\cdot\tfrac{N}{T}\,\log\left(1+\tfrac{2T}{Nt}\right)=f_{t}(\tfrac{N}{2T})\leq
f_{t}(1)=\log\left(1+\tfrac{1}{t}\right).$ (2.21)

Taking the log in (2.19), dividing by $T$ and letting $T$ go to infinity, we
obtain from (2.20) and (2.21) that

${\rm ent}_{e}(X,g_{t})=O\left(\log\left(\tfrac{1}{t}\right)\right)$

as $t$ goes to zero.

###  2.6. Collapsing assumption and zero minimal volume entropy

We show the following result.

######  Theorem 2.6.

Let $X$ be a connected finite simplicial $m$-complex. Suppose there exists a
simplicial map $\pi:X\to P$ to a simplicial $k$-complex $P$ with $k<m$ such
that for every connected component $F_{p}$ of every fiber $\pi^{-1}(p)$ with
$p\in P$, the finitely generated subgroup $i_{*}[\pi_{1}(F_{p})]$ of
$\pi_{1}(X)$ has subexponential growth rate at most $\nu$. Suppose that
$\nu<\frac{m-k}{m}$. Then $X$ has zero minimal volume entropy.

###### Proof.

By Proposition 2.1, we can assume that the simplicial map $\pi:X\to P$ in
Theorem 2.6 is onto and that its fibers $F_{p}$ are connected. Consider the
family of piecewise flat metrics $g_{t}$ on $X$ defined in Section 2.2. Recall
that ${\rm ent}_{e}(X,g_{t})\leq{\rm ent}(X,g_{t})$; see Definition 2.3. By
Proposition 2.2, a reverse inequality holds true. Namely, there exists
$C_{m}>0$ such that

${\rm ent}(X,g_{t})\leq C_{m}\,{\rm ent}_{e}(X,g_{t})$ (2.22)

for every $t\in(0,1]$. By (2.2) and (2.14), we deduce that

${\rm ent}(X,g_{t})\,{\rm
vol}(X,g_{t})^{\frac{1}{m}}=O\left(t^{\frac{m-k}{m}-\nu}\right).$

Since $\nu<\frac{m-k}{m}$, we conclude that ${\rm ent}(X,g_{t})\,{\rm
vol}(X,g_{t})^{\frac{1}{m}}$ converges to zero as $t$ goes to zero. ∎

We conclude with an application. Let us recall the definition of an
$F$-structure, first introduced by Cheeger-Gromov in a different context; see
[15] and [16]

######  Definition 2.7.

A closed manifold $M$ admits an _$F$ -structure_ if there are a locally finite
open cover $\\{U_{i}\\}$ of $M$, finite normal coverings
$\pi_{i}:\tilde{U}_{i}\to U_{i}$ and effective smooth actions of tori
${\mathbb{T}}^{k_{i}}$ on $\tilde{U}_{i}$ which extend the action of the deck
transformation group such that if $U_{i}$ and $U_{j}$ intersect each other,
then $\pi_{i}^{-1}(U_{i}\cap U_{j})$ and $\pi_{j}^{-1}(U_{i}\cap U_{j})$ have
a common covering space on which the lifting actions of ${\mathbb{T}}^{k_{i}}$
and ${\mathbb{T}}^{k_{j}}$ commute. We also assume that some orbits have
positive dimension. See [15] or [16] for a more precise definition. The _rank_
of an $F$-structure is the minimal dimension of the orbits.

As an application of Corollary 1.4, we derive the following result, which is
also a consequence of Paternain and Petean’s work on the connection between
the topological entropy of the geodesic flow and $F$-structures; see [48,
Theorem A].

######  Corollary 2.8.

Every closed manifold admitting an $F$-structure (of possibly zero rank) has
zero minimal volume entropy.

###### Proof.

By the Slice Theorem and its consequences, see [32, Appendix B], we derive the
following properties. The orbits of the $F$-structure of a closed $m$-manifold
$M$ partition the manifold into closed submanifolds covered by tori; see also
[15] and [48]. The trivial orbits form a submanifold of codimension at least
one (at least two if the manifold is orientable) and the orbit space is an
orbifold of dimension at most $m-1$. Since the fibers of the natural
projection from $M$ to the orbit space have almost abelian fundamental groups,
the manifold $M$ satisfies the collapsing assumption with polynomial growth
rate and has zero minimal volume entropy by Corollary 1.4. ∎

###  2.7. Examples of manifolds satisfying the collapsing assumption

In this section, we construct a closed orientable manifold with fundamental
group of exponential growth satisfying the collapsing assumption with fibers
of subexponential growth which do not have polynomial growth. Furthermore,
this example satisfies the condition on the subexponential growth rate of the
subgroups $i_{*}[\pi_{1}(F_{p})]$ of Theorem 2.6 (which implies that their
minimal volume entropy is zero). We also show through another example which
satisfies the collapsing assumption but does not satisfy the condition of
Theorem 2.6 that the natural method presented in the proof of Theorem 2.6
consisting of shrinking the fibers does not readily apply.

The first Grigorchuk group $G$ was defined in [24]. It is the first example of
a finitely generated group of intermediate growth, that is, its growth is
subexponential but not polynomial; see [25] and [26]. The exact value of the
subexponential growth rate of $G$ has recently been computed in [22]. It is
roughly equal to

$\nu(G)\simeq 0.7674\in[\tfrac{3}{4},\tfrac{4}{5}].$

The group $G$ is a finitely generated recursively presented group – a
description of its presentation can be found in [40] – but it is not finitely
presented. It is an open question whether finitely presented groups of
intermediate growth exist. By Higman’s embedding theorem [35], the group $G$
can be embedded into a finitely presented group. A concrete realization of
such an embedding is given in [26]. The construction goes as follows

Consider the group $\bar{G}$ defined in [26] given by the following
presentation

$\bar{G}=\langle a,c,d,u\mid a^{2}=c^{2}=d^{2}=(ad)^{4}=(adacac)^{4}=e;\\\
u^{-1}au=aca,u^{-1}cu=dc,u^{-1}du=c\rangle.$ (2.23)

The group $\bar{G}$ contains the first Grigorchuk group $G$; see [26]. More
precisely, the group $\bar{G}$ is an HNN-extension of $G$:

$\bar{G}=\langle G,u\mid u^{-1}xu=\sigma(x)\mbox{ for every }x\in G\rangle.$

where $\sigma:G\to G$ is a monomorphism; see [26] for further details. The
subgroup $G\leqslant\bar{G}$ is generated by $a$, $c$ and $d$. Note that
$\bar{G}$ contains a free subsemigroup with two generators, and therefore has
exponential growth.

Let us construct an orientable closed $5$-dimensional manifold $M$ with
$\pi_{1}(M)=\bar{G}$ as follows. Define

$N=({\mathbb{R}}P^{5})_{a}\\#({\mathbb{R}}P^{5})_{c}\\#({\mathbb{R}}P^{5})_{d}\\#(S^{1}\times
S^{4})_{u}$ (2.24)

where the indices $a$, $c$, $d$ and $u$ correspond to the generators of $G$.
Take five loops $\gamma_{1},\dots,\gamma_{5}$ in the homotopy classes
$(ad)^{4}$, $(adacac)^{4}$, $u^{-1}auaca$, $u^{-1}cudc$ and $u^{-1}duc$ of
$\pi_{1}(N)={\mathbb{Z}}_{2}*{\mathbb{Z}}_{2}*{\mathbb{Z}}_{2}*{\mathbb{Z}}$.
Placing the curves in generic position, we can assume that the loops
$\gamma_{1},\dots,\gamma_{5}$ are smooth simple closed curves which do not
intersect each other. Denote by $M$ the orientable closed manifold obtained
from $N$ by spherical surgeries of type $(1,4)$ along
$\gamma_{1},\dots,\gamma_{5}$. Since spherical surgeries of type $(1,4)$
correspond to attaching index $2$ handles, the fundamental group of $M$ is
given by the presentation (2.23). That is, $\pi_{1}(M)=\bar{G}$.

Let us construct a piecewise linear map $\pi:M\to S^{1}$ with subexponential
growth fibers. Consider the natural map $N\to S^{1}$ which takes the terms
$({\mathbb{R}}P^{5})_{a}\\#({\mathbb{R}}P^{5})_{c}\\#({\mathbb{R}}P^{5})_{d}$
in the connected sum (2.24) to a point $p_{0}\in S^{1}$ and projects the last
term $(S^{1}\times S^{4})_{u}$ to the $S^{1}$-factor of the product. By the
expression of the relations of the presentation (2.23) of $\bar{G}$, the
images by $N\to S^{1}$ of the loops $\gamma_{1},\dots,\gamma_{5}$ are
contractible in $S^{1}$. Thus, the map $N\to S^{1}$ extends to the handles of
$M$, which yields a map $M\to S^{1}$. Deforming the map, if necessary, by
sending the complement of a tubular neighborhood of a regular fiber $F$ of
$M\to S^{1}$ to a point, we can assume that the map $M\to S^{1}$ is smooth
with a unique critical value $p_{0}\in S^{1}$ and that the inverse image
$\pi^{-1}(S^{1}\setminus\\{p_{0}\\})$ has a product structure $(0,1)\times F$
whose vertical slices coincide with the fibers of $M\to S^{1}$. We can further
deform $M\to S^{1}$ into a piecewise linear map $\pi:M\to S^{1}$ by taking
fine enough triangulations of $M$ and $S^{1}$, and by applying the simplical
approximation theorem, without changing the topology of the fibers above
$S^{1}\setminus\\{p_{0}\\}$.

Let us show that $\ker\pi_{*}=G$, where $\pi_{*}:\pi_{1}(M)\to\pi_{1}(S^{1})$
is the $\pi_{1}$-homomorphism induced by $\pi:M\to S^{1}$. Every element
$w\in\ker\pi_{*}$ can be represented by a word in the letters $a$, $b$, $d$
and $u$ with a minimal number of occurrences of $u^{\pm 1}$. By construction,
$\pi_{*}(a)=\pi_{*}(c)=\pi_{*}(d)=0$ and $\pi_{*}(u)$ is a generator of
$\pi_{1}(S^{1})$. Thus, the word $w$ has as many $u$’s as $u^{-1}$’s. If the
word $w$ contains a letter $u$ or $u^{-1}$, then it contains a subword
$uw^{\prime}u^{-1}$ or $u^{-1}w^{\prime}u$, where $w^{\prime}$ is a word in
$a$, $c$ and $d$ (without $u$). According to the presentation (2.23), these
subwords can be replaced with subwords in the letters $a$, $b$, $d$ (without
$u$) in the representation of $w$, which contradicts the choice of the word
representing $w$. Thus, $w$ lies in the subgroup of $\bar{G}$ generated by
$a$, $c$ and $d$. That is, $\ker\pi_{*}\leqslant G$. The reverse inclusion is
obvious. Hence, $\ker\pi_{*}=G$.

Now, since $i_{*}[\pi_{1}(F_{p_{0}})]$ is a subgroup of $\ker\pi_{*}$
containing the generators $a$, $c$ and $d$ of $G$, we derive that
$i_{*}[\pi_{1}(F_{p_{0}})]=\ker\pi_{*}=G$. All the other fibers $F_{p}\simeq
F$ with $p\in S^{1}$ different from $p_{0}$ can be deformed into $F_{p_{0}}$,
which implies that $i_{*}[\pi_{1}(F_{p})]$ is a subgroup of
$i_{*}[\pi_{1}(F_{p_{0}})]=G$. Since $G$ has subexponential growth, the image
$i_{*}[\pi_{1}(F_{p})]$ of the fundamental group of every fiber $F_{p}$ of
$\pi:M\to S^{1}$ has also subexponential growth, where $p\in S^{1}$.

Since $\nu(G)<\frac{m-k}{m}=\frac{4}{5}$ (with $m=5$ and $k=1$), the
orientable closed $5$-dimensional manifold $M$ satisfies the collapsing
assumption of Theorem 2.6.

######  Remark 2.9.

The group $\bar{G}$ is amenable; see [26]. Thus, by Gromov’s vanishing
simplicial volume theorem, see Theorem 2.17, every manifold with fundamental
group $\bar{G}$ has zero simplicial volume.

######  Remark 2.10.

One can show that the manifold $M$ is essential. (This is not direct and
requires some work.) An easier way to obtain an essential manifold
$M^{\prime}$ is to modify our construction by taking the connected sum of $M$
with a nilmanifold, say ${\mathbb{T}}^{m}$. In this case, we collapse
$M^{\prime}={\mathbb{T}}^{m}\\#M$ to the graph
$P=[0,1]\cup_{\\{1\\}=p_{1}}S^{1}$ so that the preimage of $p_{1}\neq p_{0}$
is the attaching sphere of the connected sum, the torus
${\mathbb{T}}^{m}\setminus B^{m}$ with a ball removed is sent to $[0,1]$ and
the term $M\setminus B^{m}$ is sent to $S^{1}$ as before. The manifold
$M^{\prime}$ still satisfies the collapsing assumption of Theorem 2.6 with the
map $\pi:M^{\prime}\to P$, and the image $i_{*}[\pi_{1}(F^{\prime}_{p_{0}})]$
of the fundamental group of the fiber $F^{\prime}_{p_{0}}$ of
$\pi:M^{\prime}\to P$ still agrees with the group $G$ of intermediate growth.

In dimension $4$, the construction does not go through as the loop
$\gamma_{4}$ representing $u^{-1}cudc$ is orientation-reversing. Still, we can
define $N$ as the connected sum $\\#_{i=1}^{4}(S^{1}\times S^{4})_{i}$.
Performing spherical surgeries along eight loops $\gamma_{1},\dots,\gamma_{8}$
representing the eight relations of the presentation (2.23) of $\bar{G}$ as
previously, we obtain an orientable closed $4$-manifold $M$ and a piecewise
linear map $\pi:M\to S^{1}$ with $i_{*}[\pi_{1}(F_{p_{0}})]=G$ and
$i_{*}[\pi_{1}(F_{p})]\leqslant G$ for every $p\in S^{1}$. In this case ($m=4$
and $k=1$), we have $\nu(G)>\frac{m-k}{m}=\frac{3}{4}$. Thus, the manifold $M$
does not satisfy the collapsing assumption of Theorem 2.6 with the simplicial
map $\pi:M\to S^{1}$.

We show in this case that the natural method presented in the proof of Theorem
2.6 consisting of shrinking the fibers of the map $\pi:M\to S^{1}$ does not
readily apply, even though the images of the fundamental groups of the fibers
of $\pi:M\to S^{1}$ have subexponential growth. Endow $M$ with a piecewise
flat metric where all simplices of $N$ are isometric to the standard Euclidean
simplex with the same dimension. Since the subexponential growth rate of
$i_{*}[\pi_{1}(F_{p_{0}})]=G$ is greater than $\frac{3}{4}$, the number of
homotopy classes of $F_{p_{0}}$ which can be represented by a loop based at a
vertex $\star\in F_{p_{0}}$ of length at most $T$ is at least
$C\,\exp(T^{\frac{3}{4}})$ for some positive constant $C$. Fix a loop $\beta$
of $M$ based at $\star$ representing $u\in\pi_{1}(M)=\bar{G}$ and denote by
$\ell$ its length. Modify the metric on $M$ by shrinking the length of the
fibers of $\pi:M\to S^{1}$ by a factor $t$ as in the proof of Theorem 2.6 and
denote by $g_{t}$ the metric so-obtained. Note that ${\rm vol}(M,g_{t})\geq
C^{\prime}\,t$ for some positive constant $C^{\prime}$. Observe also that the
number of homotopy classes of the fiber $F_{p_{0}}$ which can be represented
by loops based at $\star$ of $g_{t}$-length at most $T$ is at least
$C\,\exp\left((\frac{T}{t})^{\frac{3}{4}}\right)$. Fix $T=n(\ell+1)$. Consider
all the loops $\gamma=\alpha_{1}\beta\cdots\alpha_{n}\beta$ of $M$ based at
$\star$ of $g_{t}$-length at most $T$ which can be constructed by taking a
loop $\alpha_{1}$ in $F_{p_{0}}$ of $g_{t}$-length at most $1$, followed by
the loop $\beta$ of $g_{t}$-length $\ell$, then by a loop $\alpha_{2}$ in
$F_{p_{0}}$ of $g_{t}$-length at most $1$, then by $\beta$ and so on. We
obtain at least
$C^{n}\,\exp\left(\frac{n}{t^{\frac{3}{4}}}\right)=C^{\frac{T}{\ell+1}}\,\exp\left(\frac{T}{(\ell+1)t^{\frac{3}{4}}}\right)$
homotopy classes. Thus, ${\rm ent}(M,g_{t})\gtrsim\frac{1}{t^{\frac{3}{4}}}$.
Hence, ${\rm ent}(M,g_{t})^{4}\,{\rm vol}(M,g_{t})\gtrsim\frac{1}{t^{2}}$
which tends to infinity as $t$ goes to zero.

######  Remark 2.11.

Note that the orientable closed $4$-manifold $M$ constructed above satisfies
the collapsing assumption of Theorem 2.6 with $k=2$. Indeed, one can show that
there exists a map $M\to K$ from $M$ to a $2$-dimensional polyhedron $K$ which
induces a $\pi_{1}$-isomorphism whose fundamental groups of the fibers have a
trivial image in $\pi_{1}(M)$. Therefore, the minimal volume entropy of $M$
vanishes.

###  2.8. Collapsing assumption and open covers

In this section, we give a characterization of the collapsing assumption in
terms of open covers. This characterization is similar to the description of
the Urysohn width in terms of either fiber diameters or open covers; see
Proposition 3.2.

######  Definition 2.12.

A path-connected open subset $U$ of a path-connected topological space $X$ has
_subexponential $\pi_{1}$-growth in $X$_ if the subgroup
$\Gamma_{U}:=i_{*}[\pi_{1}(U)]$ of $\pi_{1}(X)$ has subexponential growth,
where $i:U\hookrightarrow X$ is the inclusion map. In this case, the
subexponential $\pi_{1}$-growth rate of $U$ in $X$ is defined as the
subexponential growth rate of $\Gamma_{U}$.

######  Proposition 2.13.

A connected finite simplicial $m$-complex $X$ satisfies the collapsing
assumption (with subexponential growth rate at most $\nu$) onto a simplicial
$k$-complex $P$ if and only if it admits a cover of multiplicity $k+1$ by open
subsets of subexponential $\pi_{1}$-growth in $X$ (with subexponential growth
rate at most $\nu$).

###### Proof.

Suppose that $X$ satisfies the collapsing assumption. Then there exists a
simplicial map $\pi:X\to P$ onto a simplicial $k$-complex $P$ such that for
every connected component $F_{p}$ of every fiber $\pi^{-1}(p)$, where $p$ is a
vertex of $P$, the subgroup $i_{*}[\pi_{1}(F_{p})]$ of $\pi_{1}(X)$ has
subexponential growth. Since $P$ is a finite simplicial complex of dimension
$k$, the open cover formed by the open stars $\textrm{st}(p)\subseteq P$ of
the vertices $p$ of $P$ has multiplicity $k+1$. The connected components of
the preimages $\pi^{-1}(\textrm{st}(p))\subseteq X$ of these open stars form
an open cover of $X$ with the same multiplicity $k+1$ as the previous cover of
$P$. Furthermore, the open subsets of this open cover of $X$ strongly
deformation retract onto the connected components $F_{p}$ of the fibers
$\pi^{-1}(p)$. In particular, they have subexponential $\pi_{1}$-growth in $X$
(with the same subexponential growth rate as the subgroups induced by the
fibers). This prove the first implication.

For the converse implication, let $\\{U_{i}\\}_{i=0,\dots,s}$ be a cover of
$X$ of multiplicity $k+1$ by open subsets of subexponential $\pi_{1}$-growth
in $X$. Take a partition of unity $\\{\phi_{i}\\}$ of $X$, where each function
$\phi_{i}:X\to[0,1]$ has its support in $U_{i}$. Consider the map
$\Phi:X\to\Delta^{s}$ defined by

$\Phi(x)=(\phi_{0}(x),\dots,\phi_{s}(x))$

in the barycentric coordinates of $\Delta^{s}$. The nerve $P$ of the cover
$\\{U_{i}\\}$ is a simplicial complex with one vertex $v_{i}$ for each open
set $U_{i}$, where $v_{i_{0}},\dots,v_{i_{n}}$ span an $n$-simplex of $P$ if
and only if the intersection $\cap_{j=1}^{n}U_{i_{j}}$ is nonempty. By
construction, the dimension of the nerve $P$ is one less than the multiplicity
of the cover $\\{U_{i}\\}$. That is, $\dim P=k$. We identify in a natural way
the vertices $\\{v_{i}\\}$ of $P$ with the vertices of $\Delta^{s}$. With this
identification, the nerve $P$ of $X$ lies in $\Delta^{s}$. Furthermore, the
image of $\Phi$ lies in $P$. By [34, §2.C], subdividing $X$ and $P$ if
necessary, we can approximate $\Phi:X\to P$ by a simplicial map $\pi:X\to P$
close to $\Phi$ for the $C^{0}$-topology, whose normalized barycentric
coordinates $\pi_{i}:X\to[0,1]$ have their support in $U_{i}$. Thus, every
fiber $\pi^{-1}(p)$ lies in one of the open subsets $U_{i}$. Therefore, for
every connected component $F_{p}$ of $\pi^{-1}(p)$, the subgroup
$i_{*}[\pi_{1}(F_{p})]$ lies in some subgroup $i_{*}[\pi_{1}(U_{i})]$. Since
the open subsets $U_{i}$ have subexponential $\pi_{1}$-growth in $X$, the
subgroups $i_{*}[\pi_{1}(F_{p})]$ have subexponential growth (with a
subexponential growth rate bounded by the one of the subsets of the open
cover) and the simplicial complex $X$ satisfies the collapsing assumption as
required. ∎

An illustration of the characterization of the collapsing assumption in terms
of open covers is given by the following example.

######  Example 2.14.

For $i=1,2$, let $M_{i}$ be a connected closed manifold of dimension $m\geq 3$
with fundamental group $\pi_{1}(M_{i})$ of subexponential growth rate at most
$\nu<\frac{m-1}{m}$. Let $N$ be a connected closed $n$-manifold embedded both
in $M_{1}$ and $M_{2}$ with $n\leq m-3$. Suppose that the embedding
$N\subseteq M_{i}$ induces a $\pi_{1}$-monomorphism and that its normal fiber
bundle $\mathcal{N}_{i}(N)\subseteq TM_{i}$ is trivial for $i=1,2$. Define the
$m$-manifold

$X=(M_{1}\setminus U_{1}(N))\mathop{\cup}_{N\times S^{m-n-1}}(M_{2}\setminus
U_{2}(U))$

where $U_{i}(N)$ is a small tubular neighborhood of $N$ in $M_{i}$. By van
Kampen’s theorem, $\pi_{1}(M_{i}\setminus U_{i}(N))$ is isomorphic to
$\pi_{1}(M_{i})$, and thus has subexponential growth rate at most $\nu$. Take
a small tubular neighborhood $U_{i}$ of $M_{i}\setminus U_{i}(N)$ in $X$ for
$i=1,2$. Since $U_{i}$ strongly deformation retracts onto $M_{i}\setminus
U_{i}(N)$, its fundamental group $\pi_{1}(U_{i})$ is isomorphic to
$\pi_{1}(M_{i}\setminus U_{i}(N))$. This yields a cover of $X$ of multiplicity
two by open subsets $U_{1}$ and $U_{2}$ with subexponential $\pi_{1}$-growth
at most $\nu$ in $X$. According to Proposition 2.13, the closed $m$-manifold
$X$ satisfies the collapsing assumption. Note however that the fundamental
group of $X$ has exponential growth in general. This construction provides
numerous examples of closed essential manifolds with a fundamental group of
exponential growth and zero minimal volume entropy. For instance, when $N$ is
reduced to a singleton, the manifold $X$ is the connected sum $M_{1}\\#M_{2}$
of $M_{1}$ and $M_{2}$. This special case can also be recovered from [6,
Theorem 2.8].

Combining Theorem 2.6 and Proposition 2.13, we immediately derive the
following result.

######  Corollary 2.15.

Every connected finite simplicial $m$-complex $X$ which admits a cover of
multiplicity $k+1$ by open subsets of subexponential $\pi_{1}$-growth in $X$
with subexponential growth rate at most $\nu<\frac{m-k}{m}$ has zero minimal
volume entropy.

###  2.9. Collapsing assumption and zero simplicial volume

Drawing a parallel with the simplicial volume through Gromov’s vanishing
simplicial volume theorem, we show that a manifold satisfying the collapsing
assumption has zero simplicial volume

######  Definition 2.16.

A group $G$ is _amenable_ if it admits a finitely-additive left-invariant
probability measure. A path-connected open subset $U$ of a path-connected
topological space $X$ is _amenable in $X$_ if $i_{*}[\pi_{1}(U)]$ is an
amenable subgroup of $\pi_{1}(X)$, where $i:U\hookrightarrow X$ is the
inclusion map.

Gromov’s vanishing simplicial volume theorem can be stated as follows.

######  Theorem 2.17 ([29], see also [36]).

Let $M$ be a connected closed $m$-manifold. Suppose that $M$ admits a cover by
amenable open subsets of multiplicity at most $m$. Then

$\|M\|_{{\Delta}}=0.$

In particular, the simplicial volume of a connected closed manifold with
amenable fundamental group is zero.

The characterization of the collapsing assumption in terms of covers allows us
to derive the following result about the effect of the collapsing assumption
on the simplicial volume.

######  Proposition 2.18.

Every closed $m$-manifold $M$ satisfying the collapsing assumption has zero
simplicial volume.

###### Proof.

Recall that every finitely generated group with subexponential growth is
amenable; see [1] or [14, Theorem 6.11.12] for instance. Thus, every open
subset $U\subseteq M$ with subexponential $\pi_{1}$-growth in $M$, see
Definition 2.12, is amenable in $M$. By Proposition 2.13, the manifold $M$
admits a cover of multiplicity at most $m$ by open subsets of subexponential
$\pi_{1}$-growth in $M$, and so by amenable open subsets. It follows from
Theorem 2.17 that $M$ has zero simplicial volume. ∎

##  3\. Simplicial complexes with positive minimal volume entropy

By relying on the notion of Urysohn width, we show that the minimal volume
entropy of a connected finite simplicial complex satisfying the fiber
$\pi_{1}$-growth non-collapsing assumption is positive. We also give a
characterization of the non-collapsing assumption in terms of open covers.
Finally, we construct finite simplicial complexes with zero simplicial volume
and arbitrarily large minimal volume entropy.

###  3.1. Urysohn width and volume

Let us go over the notion of Urysohn width in metric geometry.

######  Definition 3.1.

The _Urysohn $q$-width_ of a compact metric space $X$, denoted by ${\rm
UW}_{q}(X)$, is defined as the least real $\delta>0$ such that there is a
continuous map $\pi:X\to P$ from $X$ to a simplicial $q$-complex $P$, where
all the fibers $\pi^{-1}(p)$ have diameter at most $\delta$ in $X$. That is,

${\rm UW}_{q}(X)=\adjustlimits{\inf}_{\pi:X\to P}{\sup}_{p\in P}\,\mathop{\rm
diam}\nolimits_{X}[\pi^{-1}(p)]$ (3.1)

where $\pi:X\to P$ runs over all continuous map from $X$ to a simplicial
$q$-complex $P$. Note that the simplicial complex $P$ may vary with $\pi$. For
a simplicial $m$-complex $X$, we will simply write ${\rm UW}(X)$ for ${\rm
UW}_{m-1}(X)$.

The Urysohn width can also be interpreted in terms of open covers; see [33,
Lemma 0.8] for instance.

######  Proposition 3.2.

A compact metric space $X$ has Urysohn $q$-width less than $w$ if and only if
there is a finite cover of $X$ of multiplicity at most $q+1$ by open subsets
of diameter less than $w$.

In the case of simplicial complexes, we can further require extra structural
properties on the map $\pi:X\to P$ in the previous definition.

######  Proposition 3.3.

Let $X$ be a finite simplicial complex with a piecewise Riemannian metric.
Subdividing $X$ if necessary, we can assume that the maps $\pi:X\to P$ in the
definition of the Urysohn width are simplicial and that their fibers are
connected.

###### Proof.

Suppose ${\rm UW}_{q}(X)<\delta$. By Proposition 3.2, there is a finite open
cover $\mathcal{U}=\\{U_{i}\\}_{i=1,\cdots,s}$ of $X$ of multiplicity $q+1$
and diameter less than $\delta$. Consider the natural map $\Phi:X\to
P\subseteq\Delta^{s-1}$ to the nerve $P$ of $\mathcal{U}$ given by a partition
of unity of the cover. As in the proof of Proposition 2.13, subdividing $X$
and $P$, we can approximate $\Phi:X\to P$ by a simplicial map $\pi:X\to P$
close to $\Phi$ for the $C^{0}$-topology, whose normalized barycentric
coordinates $\pi_{i}:X\to[0,1]$ have their support in $U_{i}$; see [34, §2.C].
Thus, every fiber $\pi^{-1}(p)$ lies in one of the open sets $U_{i}$.
Therefore, $\mathop{\rm diam}\nolimits_{X}\pi^{-1}(p)<\delta$. As a result, we
can assume that the map $\pi:X\to P$ is simplicial in the definition of the
Urysohn width; see (3.1). Now, by Proposition 2.1, we can replace $\pi:X\to P$
with a simplicial map $\bar{\pi}:X\to\bar{P}$ to a simplicial complex
$\bar{P}$ of dimension at most $q$, whose fibers are connected and of diameter
less than $\delta$. ∎

We will need the following recent result of Liokumovich-Lishak-Nabutovsky-
Rotman [38], extending a theorem of L. Guth [33]. The proof of this result was
later on simplified by P. Papasoglu [47]; see also [46].

######  Theorem 3.4 ([33], [38], [47], [46]).

Let $X$ be a finite simplicial $m$-complex with a piecewise Riemannian metric.
Then

${\rm vol}(X)\geq C_{m}\,{\rm UW}(X)^{m}$

where $C_{m}$ is an explicit positive constant depending only on $m$. More
precisely (by contraposition), if for some $R>0$ every ball $B(R)\subseteq X$
of radius $R$ has volume at most $C_{m}\,R^{m}$ then

${\rm UW}(X)\leq R.$

A more general statement involving the lower dimensional widths and the
Hausdorff content of balls holds true; see [38], [47], [46].

###  3.2. Diameter and uniform group growth

Let us present the following classical result relating the diameter and the
volume entropy of a space, similar in spirit to the Švarc-Milnor lemma; see
[31, §5.16].

######  Proposition 3.5.

Let $U$ be a connected open subset in a connected finite simplicial complex
$X$ with a piecewise Riemannian metric. Suppose that the subgroup
$\Gamma_{U}:=i_{*}[\pi_{1}(U)]$ of $\pi_{1}(X)$ defined as the image of
$\pi_{1}(U)$ under the group homomorphism induced by the inclusion map
$i:U\hookrightarrow X$ is finitely generated. Then

$\mathop{\rm diam}\nolimits_{X}(U)\cdot{\rm ent}(X)\geq\frac{1}{2}\,{\rm
ent}(\Gamma_{U}).$

###### Proof.

The proof of this result is classical. The construction in the proof of [31,
Proposition 3.22] shows that there exist finitely many loops
$\gamma_{i}\subseteq X$ based at some fixed basepoint $x_{0}\in X$ whose
homotopy classes in $X$ form a generating set $S$ of
$\Gamma_{U}=i_{*}[\pi_{1}(U)]$ with

${\rm length}(\gamma_{i})<2\,\mathop{\rm diam}\nolimits_{X}(U)+\varepsilon$

for some arbitrarily small $\varepsilon>0$. Clearly, every homotopy class
$\alpha\in\Gamma_{U}$ can be represented by a loop $\gamma\subseteq X$ based
at $x_{0}$ of length at most

$(2\,\mathop{\rm diam}\nolimits_{X}(U)+\varepsilon)\cdot d_{S}(e,\alpha)$

where $d_{S}$ is the word distance on $\Gamma_{U}$ induced by $S$. Thus, the
number $\mathcal{N}(X;T)$ of homotopy classes represented by loops based at
$x_{0}$ of length at most $T$, see Definition 2.3, satisfies

$\mathcal{N}(X;T)\geq{\rm card}\left\\{\alpha\in\Gamma_{U}\mid
d_{S}(e,\alpha)\leq\frac{T}{2\,\mathop{\rm
diam}\nolimits_{X}(U)+\varepsilon}\right\\}$

It follows from (2.13) that

${\rm ent}(X)\geq\frac{1}{2\,\mathop{\rm
diam}\nolimits_{X}(U)+\varepsilon}\,{\rm ent}(\Gamma_{U},S)$

for every $\varepsilon>0$. Hence the result. ∎

###  3.3. Non-collapsing assumption and minimal volume entropy

We can now prove the following result complementing Theorem 2.6.

######  Theorem 3.6.

Every connected finite simplicial $m$-complex $X$ satisfying the non-
collapsing assumption has positive minimal volume entropy. More generally (by
contraposition), given a piecewise Riemannian metric $g$ on $X$, if for some
$R>0$ every ball $B(R)\subseteq X$ of radius $R$ has volume at most
$C_{m}\,R^{m}$ then

${\rm ent}(X,g)\geq\frac{h(X)}{2R}$

where $C_{m}$ is an explicit positive constant depending only on $m$.

###### Proof.

Let $g$ be a piecewise Riemannian metric on $X$. By Theorem 3.4 and
Proposition 3.3, there exists a constant $C^{\prime}_{m}>0$ and a simplicial
map $\pi:X\to P$ to some simplicial $(m-1)$-complex $P$, where every fiber
$F_{p}=\pi^{-1}(p)$ is connected, such that

$\mathop{\rm diam}\nolimits_{X}(F_{p})<C^{\prime}_{m}\,{\rm
vol}(X,g)^{\frac{1}{m}}.$ (3.2)

The constant $C^{\prime}_{m}$ can be taken arbitrarily close to
$C_{m}^{-\frac{1}{m}}$, where $C_{m}$ is the constant in Theorem 3.4. By the
non-collapsing assumption, one of the subgroups $i_{*}[\pi_{1}(F_{p_{0}})]$
has uniform exponential growth at least $h(X)$. The point $p_{0}\in P$ has a
small enough open neighborhood $B_{p_{0}}\subseteq P$ whose preimage
$U_{p_{0}}=\pi^{-1}(B_{p_{0}})$ is homotopy equivalent to the fiber
$F_{p_{0}}$ and has diameter close to the one of $F_{p_{0}}$ so that

$\mathop{\rm diam}\nolimits_{X}(U_{p_{0}})<C^{\prime}_{m}\,{\rm
vol}(X,g)^{\frac{1}{m}}.$ (3.3)

The group $\Gamma_{p_{0}}=i_{*}[\pi_{1}(U_{p_{0}})]$, which is isomorphic to
the finitely generated subgroup $i_{*}[\pi_{1}(F_{p_{0}})]$, has uniform
exponential growth at least $h(X)$. It follows from Proposition 3.5 that

$\frac{1}{2}\,h(X)\leq\frac{1}{2}\,{\rm ent}(\Gamma_{p_{0}})\leq\mathop{\rm
diam}\nolimits_{X}(U)\cdot{\rm ent}(X,g)\leq C^{\prime}_{m}\,{\rm
ent}(X,g)\,{\rm vol}(X,g)^{\frac{1}{m}}.$ (3.4)

Hence, the minimal volume entropy of $X$ is positive and satisfies

$\omega(X)\geq\frac{1}{2C^{\prime}_{m}}\,h(X)>0.$

Suppose that every ball $B(R)$ of radius $R$ has volume at most
$C_{m}\,R^{m}$, where $C_{m}$ is the constant in Theorem 3.4. By Theorem 3.4,
we can replace the right-hand sides in the inequalities (3.2) and (3.3) by
$R$. By making the appropriate change in the inequality chain (3.4), we derive
the desired lower bound for the volume entropy of $X$. ∎

The following result provides examples of simplicial complexes satisfying the
non-collapsing assumption.

######  Proposition 3.7.

Let $X$ be a finite aspherical simplicial $m$-complex with
$H_{m}(X;{\mathbb{R}})$ nontrivial, where $m\geq 2$. Suppose the fundamental
group of $X$ is a non-elementary word hyperbolic group. Then $X$ satisfies the
non-collapsing assumption.

In particular, every closed aspherical manifold whose fundamental group is a
non-elementary word hyperbolic group satisfies the non-collapsing assumption.

###### Proof.

First observe that since $X$ is aspherical, its fundamental group $\pi_{1}(X)$
is torsion-free, otherwise there would exist a finite-dimensional aspherical
space with a finite fundamental group, which is impossible; see [34,
Proposition 2.45]. Let $\pi:X\to P$ be a simplicial map to a finite simplicial
$(m-1)$-complex $P$. Suppose that all the subgroups
$H=i_{*}[\pi_{1}(F_{p})]\leqslant\pi_{1}(X)$, where $F_{p}$ is a connected
component of a fiber $\pi^{-1}(p)$ and $i:F_{p}\hookrightarrow X$ is the
inclusion map, have subexponential growth (and so are amenable). That is, the
simplicial complex $X$ satisfies the collapsing assumption. According to the
generalization given by [36, Theorem 9.2] (also proved via different
approaches in [23] and [39]) of Gromov’s vanishing simplicial volume theorem,
see Theorem 2.17, the canonical homomorphism
$\widehat{H}^{m}(X;{\mathbb{R}})\to H^{m}(X;{\mathbb{R}})$ between bounded
cohomology and singular cohomology vanishes. By [45], the canonical
homomorphism $\widehat{H}^{m}(X;{\mathbb{R}})\to H^{m}(X;{\mathbb{R}})$ is
also surjective. Hence, $H^{m}(X;{\mathbb{R}})$ is trivial, which leads to a
contradiction. Indeed, by assumption, $H_{m}(X;{\mathbb{R}})$ is nontrivial,
and by the universal coefficient theorem for cohomology,
$H^{m}(X;{\mathbb{R}})={\rm Hom}(H_{m}(X;{\mathbb{R}}),{\mathbb{R}})$ is also
nontrivial. Therefore, one of the subgroups
$H_{0}=i_{*}[\pi_{1}(F_{p_{0}})]\leqslant\pi_{1}(X)$ has exponential growth
and thus is not abelian. Now, by [18], given a torsion-free non-elementary
hyperbolic group $G$, there exists an integer $n_{G}$ such that for every
$x,y\in G$ which do not commute and for every $n\geq n_{G}$, the subgroup
$\langle x^{n},y^{n}\rangle$ of $G$ generated by $x^{n}$ and $y^{n}$ is free
of rank $2$. In particular, if a finite generating set $S$ of $G$ has two
elements $x,y$ which do not commute then $B_{S_{0}}(\frac{t}{n_{G}})\subseteq
B_{S}(t)$, where $S_{0}=\\{x^{\pm n_{G}},y^{\pm n_{G}}\\}$. Here, $B_{S}(t)$
is formed of the elements of $G$ at distance at most $t$ induced by $S$ from
the identity element with respect to the word distance. In this case, the
algebraic entropy of $G$ is bounded from below by the algebraic entropy of the
free group of rank $2$ generated by $S_{0}$. Thus, every finitely generated
subgroup $H\leqslant G$ is either abelian or has uniform exponential growth at
least $h(G)=\frac{1}{n_{G}}\log(3)$. This implies that the non-abelian
subgroup $H_{0}\leqslant\pi_{1}(X)$ has uniform exponential growth at least
$h$, where $h=h(X)$ is a positive constant depending only on $X$. Hence the
result. ∎

######  Question 3.8.

Does every closed orientable manifold $M$ satisfying the non-collapsing
assumption have positive simplicial volume? Otherwise, find examples of closed
orientable manifolds with zero simplicial volume satisfying the non-collapsing
assumption.

###  3.4. Non-collapsing assumption and open covers

In this section, we give a characterization of the non-collapsing assumption
in terms of open covers in the same vein as in the collapsing case; see
Section 2.8.

######  Definition 3.9.

A cover $\mathcal{U}=\\{U_{i}\\}$ of a path-connected topological space $X$ by
path-connected open subsets has _uniform exponential $\pi_{1}$-growth_ at
least $h$ if the subgroup $\Gamma_{U}:=i_{*}[\pi_{1}(U)]$ of $\pi_{1}(X)$ has
uniform exponential growth at least $h$ for at least one open subset $U$ of
$\mathcal{U}$, where $i:U\hookrightarrow X$ is the inclusion map.

The following proposition mirrors Proposition 2.13 with an analoguous proof,
which is left to the reader.

######  Proposition 3.10.

A connected finite simplicial $m$-complex $X$ satisfies the non-collapsing
assumption if and only if every open cover of multiplicity at most $m$ has
uniform exponential $\pi_{1}$-growth at least $h$, where $h=h(X)>0$ depends
only on $X$ and not on the open cover.

The following result is the counterpart of Corollary 2.15.

######  Corollary 3.11.

Every connected finite simplicial $m$-complex $X$ whose open covers of
multiplicity at most $m$ have uniform exponential $\pi_{1}$-growth at least
$h=h(X)$ has positive minimal volume entropy.

###  3.5. Simplicial volume and minimal volume entropy

In this section, we construct a sequence of connected finite simplicial
complexes $X_{n}$ with vanishing simplicial volume and minimal volume entropy
going to infinity, proving Theorem 1.6.

Let $\Sigma$ be a surface of genus $h\geq 2$ with a disk removed. Fix an
integer $d\geq 3$. Consider the simplicial complex $X$ obtained by attaching
$\Sigma$ to $S^{1}$ along a map $\partial\Sigma\to S^{1}$ of degree $d$. The
fundamental group of $X$ has the following presentation

$\pi_{1}(X)=\langle
x_{1},y_{1},\dots,x_{h},y_{h},z|\mathop{\prod}\limits_{i=1}^{h}[x_{i},y_{i}]z^{d}=e\rangle.$
(3.5)

The space $X$ admits a ${\rm CAT}(-1)$ metric and therefore is aspherical; see
[12]. (This is also a consequence of the fact that $X$ is the cellular
$2$-complex associated to the one-relator group presentation (3.5) whose
relator is not a proper power; see [20].) However, one cannot directly apply
Proposition 3.7 to $X$. Indeed, consider the cell decomposition of $X$ with a
single $2$-cell attached to a bouquet of $2h+1$ circles according to the
relation (3.5). A cellular homology computation of the boundary map on the
cellular chains of the cell decomposition of $X$ shows that

$H_{2}(X;{\mathbb{Z}})=0.$

Thus, $H_{2}(X;{\mathbb{R}})$ is trivial and so is $H^{2}(X;{\mathbb{R}})$ by
the universal coefficient theorem for cohomology.

To show that $X$ satisfies the non-collapsing assumption, one needs further
topological insight. Consider a simplicial map $\pi:X\to P$ to a one-
dimensional simplicial complex (i.e., a graph). The space $X$ admits a
$d$-sheeted cover $q:\widehat{X}\to X$ with

$\widehat{X}=S^{1}\mathop{\cup}\limits_{\psi_{1}}\Sigma_{1}\cdots\mathop{\cup}\limits_{\psi_{d}}\Sigma_{d}$

where the $d$ copies $\Sigma_{i}$ of $\Sigma$ with $i=1,\dots,d$ are attached
to $S^{1}$ along the maps $\psi_{i}:\partial\Sigma_{i}\longrightarrow S^{1}$
defined as follows. Choose a homeomorphism $\psi_{1}:\partial\Sigma_{1}\to
S^{1}$ and define $\psi_{i}:\partial\Sigma_{i}\to S^{1}$ as
$\psi_{i}=\rho^{i-1}\circ\psi_{1}$, where $\rho$ is the rotation of $S^{1}$ of
angle $\frac{2\pi}{d}$. By definition, $X=\widehat{X}/{\mathbb{Z}}_{q}$, where
the cyclic group ${\mathbb{Z}}_{d}$ freely acts on $\widehat{X}$ by cyclic
permutations of the copies $\Sigma_{i}$ of $\Sigma$.

The cover $\widehat{X}$ contains the $\pi_{1}$-embedded genus $2h$ surface
$M=\Sigma_{1}\cup\Sigma_{2}$. Consider the composite map $f=\pi\circ q:M\to
P$. Proposition 3.7 ensures that for some $p\in P$, the subgroup
$i_{*}[\pi_{1}(F_{p})]\leqslant\pi_{1}(M)$ has uniform exponential growth at
least $h(\pi_{1}(M))=\frac{1}{\log(3)}$. Thus, since $\pi_{1}(M)$ is a
subgroup of $\pi_{1}(\widehat{X})$, and so of $\pi_{1}(X)$, the space $X$
satisfies the non-collapsing assumption.

By Theorem 3.6, the minimal volume entropy of $X$ is positive. Namely,

$\omega(X)\geq\frac{1}{2C^{\prime}_{2}\log(3)}>0$ (3.6)

where $C^{\prime}_{2}$ is the constant involved in (3.2). The positivity of
the minimal volume entropy of $X$ can also be deduced from simplicial volume
considerations (see below).

Consider the simplicial complex $X_{n}$ defined as the bouquet

$X_{n}={\mathbb{T}}^{2}\vee\left(\bigvee_{i=1}^{n}X\right)$

of the torus with $n$ copies of $X$. By construction, we have

$H_{2}(X_{n};{\mathbb{Z}})=H_{2}({\mathbb{T}}^{2};{\mathbb{Z}})={\mathbb{Z}}$

and

$\|X_{n}\|_{\Delta}=0.$

By [6, Theorem 2.6], we obtain

$\omega(X_{n})^{2}=n\,\omega(X)^{2}.$

Since $\omega(X)$ is positive, see (3.6), we conclude that $\omega(X_{n})$
goes to infinity, which proves Theorem 1.6.

######  Remark 3.12.

Applying the “graph of groups” construction, see [34, §1.B], we can construct
further examples of aspherical simplicial complexes satisfying the non-
collapsing assumption.

The previous example extends in higher dimension as follows using different
arguments. Remove a ball from a closed manifold of dimension $m=2k$ with
positive simplicial volume. The resulting space $\Sigma$ is a manifold with
boundary $\partial\Sigma\simeq S^{2k-1}$. Fix an integer $d\geq 3$. Denote by
$Y$ the quotient of $\Sigma$ by the natural free action of ${\mathbb{Z}}_{d}$
on $S^{2k-1}$ given by rotation of the Hopf fibration. Observe that
$\pi_{1}(Y)\simeq\pi_{1}(\Sigma)\ast{\mathbb{Z}}_{d}$ and
$H_{m}(Y;{\mathbb{Z}})=0$. Define the simplicial $m$-complex

$X_{n}=\\#_{i=1}^{n}Y_{i}$

by taking the connected sum of $n$ copies of $Y$. Note that
$H_{m}(X_{n};{\mathbb{Z}})=0$.

The space $X_{n}$ admits a $d$-sheeted cyclic cover which can be described as
follows. The connected sum $\\#_{i=1}^{n}\Sigma_{i}$ of $n$ copies of $\Sigma$
is a manifold whose boundary identifies with the disjoint union $\sqcup
S_{i}^{2k-1}$ of $n$ spheres. Let $\widehat{X}_{n}$ be the space obtained by
gluing $d$ copies of $\\#_{i=1}^{n}\Sigma_{i}$ along this disjoint union

$\widehat{X}_{n}=(\sqcup
S_{i}^{2k-1})\cup_{\psi_{1}}(\\#_{i=1}^{n}\Sigma_{i})\cdots\cup_{\psi_{d}}(\\#_{i=1}^{n}\Sigma_{i})$

where the attaching maps $\psi_{j}$ are given by the action of $\alpha^{j}$ on
the boundary components of $\\#_{i=1}^{n}\Sigma_{i}$ (for a fixed generator
$\alpha$ of ${\mathbb{Z}}_{d}$). The cover $\widehat{X}_{n}\to X_{n}$ is the
natural map sending the $d$ copies $\\#_{i=1}^{n}\Sigma_{i}$ to $X_{n}$. By
the comparison principle, see [13, Lemma 4.1], we have

$\omega(\widehat{X}_{n})\leq d^{\frac{1}{m}}\,\omega(X_{n}).$ (3.7)

Now, take two copies $\\#_{i=1}^{n}\Sigma_{i}$ and
$\\#_{i=1}^{n}\bar{\Sigma}_{i}$ in $\widehat{X}_{n}$. By construction, the
boundaries $\partial\Sigma_{i}$ and $\partial\bar{\Sigma}_{i}$ agree and the
union

$M_{n}=(\\#_{i=1}^{n}\Sigma_{i})\cup(\\#_{i=1}^{n}\bar{\Sigma}_{i})$

is a closed $m$-manifold homeomorphic to

$M_{n}\simeq\\#_{i=1}^{n}(\Sigma_{i}\\#\bar{\Sigma}_{i})\,\,\\#_{i=1}^{n}(S^{1}\times
S^{2k-1}).$

Since the simplicial volume is additive under connected sums in dimension at
least three, see [29], we obtain

$\|M_{n}\|_{\Delta}=2n\,\|\Sigma\|_{\Delta}>0.$

Thus, by (1.2), the minimal volume entropy $\omega(M_{n})$ of $M_{n}$ goes to
infinity when $n$ or $\|\Sigma\|_{\Delta}$ tend to infinity.

To conclude, consider the simplicial $m$-complex $Z_{n}$ defined as the
connected sum

$Z_{n}=X_{n}\\#{\mathbb{T}}^{m}.$

Observe that $Z_{n}$ is a cellular $m$-complex with a single $m$-cell.
Clearly, $H_{m}(Z_{n};{\mathbb{Z}})={\mathbb{Z}}$ and $\|Z_{n}\|_{\Delta}=0$.
Note also that $Z_{n}$ is not aspherical since its fundamental group has
torsion. By [6, Theorem 2.12] (which still holds when $M_{1}$, here $X_{n}$,
is a cellular $m$-complex with a single $m$-cell), we have
$\omega(Z_{n})\geq\omega(X_{n})$. Since $\pi_{1}(M_{n})$ is a subgroup of
$\pi_{1}(\widehat{X}_{n})$ and the manifold $M_{n}$ contained in
$\widehat{X}_{n}$ has the same dimension $m$ as $\widehat{X}_{n}$, we deduce
that $\omega(\widehat{X}_{n})\geq\omega(M_{n})$. Thus, by (3.7), the minimal
volume entropy $\omega(Z_{n})$ of $Z_{n}$ goes to infinity.

######  Remark 3.13.

Similar examples exist in odd dimensions but their construction is more
technical.

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