Minimal volume entropy and fiber growth
MMINIMAL VOLUME ENTROPY AND FIBER GROWTH
IVAN BABENKO AND ST´EPHANE SABOURAU
Abstract.
This article deals with topological assumptions under which the minimal volumeentropy of a closed manifold M , and more generally of a finite simplicial complex X , vanishes oris positive. These topological conditions are expressed in terms of the growth of the fundamentalgroup of the fibers of maps from a given finite simplicial complex X to lower dimensionalsimplicial complexes P . We also give examples of finite simplicial complexes with zero simplicialvolume and arbitrarily large minimal volume entropy. Introduction
The notion of volume entropy has attracted a lot of attention since the early works of Efre-movich [21], ˇSvarc [55] and Milnor [44]. This Riemannian invariant describes the asymptoticgeometry of the universal cover of a Riemannian manifold and is related to the growth of itsfundamental 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 flowof 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 moregenerally of a finite simplicial complex), a topological invariant introduced by Gromov [29] re-lated to the simplicial volume. More precisely, we give topological conditions which ensure, inone case, that the minimal volume entropy of a finite simplicial complex is positive and, in theother 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 piecewiseRiemannian metric g is the exponential growth rate of the volume of balls in the universal coverof X . More precisely, it is defined asent( X, g ) = lim R →∞ R log(vol (cid:101) B ( R )) (1.1)where (cid:101) B ( R ) is a ball of radius R centered at any point in the universal cover of X . The limitexists and does not depend on the center of the ball. Observe that the volume entropy of a finitesimplicial complex with a piecewise Riemannian metric is positive if and only if its fundamentalgroup 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 ω ( X ) = inf g ent( X, g ) vol(
X, g ) m where g runs over the space of all piecewise Riemannian metrics on X . This topological invariantis known to be a homotopic invariant for closed manifolds M , see [3], and more generally, an Mathematics Subject Classification.
Primary 53C23; Secondary 57N65.
Key words and phrases.
Minimal volume entropy, collapsing, exponential and subexponential growth, fibergrowth, Urysohn width.Partially supported by the ANR project Min-Max (ANR-19-CE40-0014). a r X i v : . [ m a t h . G T ] F e b I. BABENKO AND S. SABOURAU 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 manifoldis only known in a few cases; see [37], [9], [52], [53], [17], [43]. For instance, the minimal volumeentropy of a closed m -manifold M which carries a hyperbolic metric is attained by the hyperbolicmetric and is equal to ( m −
1) vol( M, hyp) m ; see [37] for m = 2 and [9] for m ≥ simplicial volume of a connected closed orientable m -manifold M is defined as (cid:107) M (cid:107) ∆ = inf c (cid:88) s | r s | where the infimum is taken over all real singular m -cycles c = (cid:80) s r s σ s representing the funda-mental class of M in H m ( M ; Z ). Here, the maps σ s : ∆ m → M are singular m -simplices. Thedefinition extends to finite simplicial m -complexes X whose fundamental class is well-defined,that is, with H m ( X ; Z ) = Z .The following inequality of Gromov [29, p. 37] connects the minimal volume entropy of aconnected closed manifold to its simplicial volume (see also [8] for a presentation of this result).Namely, every connected closed orientable m -manifold M satisfies ω ( M ) m ≥ c m (cid:107) M (cid:107) ∆ (1.2)for some positive constant c m depending only on m . Thus, every closed manifold with positivesimplicial volume has positive minimal volume entropy. In particular, the minimal volumeentropy of a closed manifold which carries a negatively curved metric is positive; see [29]. Othertopological conditions ensuring the positivity of the minimal volume entropy have recently beenobtained in [51] and extended in [7, Section 4]; see [10] for a presentation of numerous examplesand cases where these conditions apply. These conditions are related to the topology of the loopspace of the manifold. In a different direction, the minimal volume entropy provides a lowerbound 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 withzero simplicial volume has zero minimal volume entropy. This is known to be true in dimen-sion two [37] and in dimension three [49] (see also [2] combined with Perelman’s resolution ofThurston’s geometrization conjecture), where the cube of the minimal volume entropy is pro-portional to the simplicial volume. In dimension four, the same is known to be true but onlyfor closed orientable geometrizable manifolds; see [54]. The techniques developed in this articleallow us to provide a negative answer for finite simplicial complexes; see Theorem 1.6. Thequestion for closed orientable manifolds remains open despite recent progress made with theintroduction of the volume entropy semi-norm; see [6]. This geometric semi-norm in homologymeasures the minimal volume entropy of a real homology class throughout a stabilization pro-cess. Namely, given a path-connected topological space X , it is defined for every a ∈ H m ( X ; Z )as (cid:107) a (cid:107) E = lim k →∞ ω ( k a ) m k (1.3)where ω ( a ) is the infimum of the minimal relative volume entropy of the maps f : M → X from an orientable connected closed m -pseudomanifold M to X such that f ∗ ([ M ]) = a ; see [6]for a more precise definition. The volume entropy semi-norm shares similar functorial featureswith the simplicial volume semi-norm. Moreover, the two semi-norms are equivalent in everydimension. That is, c m (cid:107) a (cid:107) ∆ ≤ (cid:107) a (cid:107) E ≤ C m (cid:107) a (cid:107) ∆ (1.4) INIMAL VOLUME ENTROPY AND FIBER GROWTH 3 for some positive constants c m and C m depending only on m . Thus, a closed manifold with zerosimplicial volume has zero volume entropy semi-norm, but its minimal volume entropy may benonzero a priori . See [6] for further details.More generally, one may ask for a topological characterization of closed manifolds or simplicialcomplexes with positive minimal volume entropy. Such a topological characterization holds forthe systolic volume, a topological invariant sharing similar properties with the minimal volumeentropy; see [3], [4], [5], [13]. Namely, a closed m -manifold or simplicial m -complex has positivesystolic volume if and only if it is essential ( i.e. , its classifying map cannot be homotoped intothe ( m − Definition 1.2.
Let G be a finitely generated group and S be a finite generating set of G .Denote by B S ( t ) ⊆ G the ball centered at the identity element of G and of radius t for the worddistance induced by S . The group G has exponential growth if the exponential growth rate ofthe number of elements in B S ( t ) defined asent( G, S ) = lim t →∞ t log | B S ( t ) | is nonzero for some (and so any) finite generating set S . It has uniform exponential growth atleast h > B S ( t ) is at least h forevery finite generating set S . That is, its algebraic entropy satisfiesent( G ) = inf S ent( G, S ) ≥ 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 ν ( G ) = lim sup t →∞ 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 existsa polynomial P such that | B S ( t ) | ≤ P ( t )for every t ≥
0. By [28], a finitely generated group has polynomial growth if and only if it isvirtually 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], answeringa question raised by Milnor. Still, it is an open problem whether finitely presented groups ofintermediate growth exist. Examples of finitely generated groups of exponential growth whichdo not have uniform exponential growth were first constructed by Wilson [56], answering aquestion asked in [27] and [31]. Still, it is an open question whether all finitely presented groupsof exponential growth have uniform exponential growth.For our topological conditions, we consider connected finite simplicial m -complexes X alongwith simplicial maps π : X → P onto simplicial complexes P of dimension at most k < m , I. BABENKO AND S. SABOURAU where m ≥
2. We denote by i ∗ : π ( F p ) → π ( X ) the homomorphism induced by the inclusionmap i : F p (cid:44) → X of a connected component F p of a fiber π − ( p ) of π .The first condition considered for X is the fiber π -growth collapsing assumption (or collapsingassumption for short). Fiber π -growth collapsing assumption. Suppose there exists a simplicial map π : X → P onto a simplicial complex P of dimension at most k < m such that for every connected compo-nent F p of every fiber π − ( p ) with p ∈ P , the finitely generated subgroup i ∗ [ π ( F p )] of π ( X )has subexponential growth. Recall that m is the dimension of X .The fiber π -growth collapsing assumption with polynomial growth rate is defined similarlywith the condition that all the finitely generated subgroup i ∗ [ π ( F p )] of π ( X ) have polynomialgrowth.Likewise, the fiber π -growth collapsing assumption with subexponential growth rate at most ν is defined similarly with the condition that the subexponential growth rate of all the finitelygenerated subgroup i ∗ [ π ( F p )] of π ( X ) is at most ν .In these definitions, it is enough to check the condition for every vertex p ∈ P .The following result shows that if the subexponential growth rate in the collapsing assumptionis 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 π -growthcollapsing assumption with subexponential growth rate at most ν onto a simplicial k -complex P .Suppose that ν < m − km . Then X has zero minimal volume entropy, that is, ω ( X ) = 0 . In Section 2.7, we give an example of a closed manifold satisfying the assumption of Theo-rem 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 finitesimplicial m -complexes X satisfying the fiber π -growth collapsing assumption without anycondition on the subexponential growth rate of the subgroups i ∗ [ π ( F p )] (cid:54) π ( X ). In Section 2.7,we construct an example showing that the natural method presented in the proof of Theorem 1.3consisting of shrinking the fibers of the simplicial map π : X → P involved in the definition ofthe fiber π -growth collapsing assumption does not readily apply.Since the subexponential growth rate of a group with polynomial growth is zero, we immedi-ately derive the following corollary. Corollary 1.4.
Every connected finite simplicial complex satisfying the fiber π -growth collaps-ing assumption with polynomial growth rate has zero minimal volume entropy. The second condition considered for X is the fiber π -growth non-collapsing assumption (ornon-collapsing assumption for short). Fiber π -growth non-collapsing assumption. Suppose that for every simplicial map π : X → P onto a simplicial complex P of dimension k < m , there exists a connected compo-nent F p of some fiber π − ( p ) with p ∈ P such that the finitely generated subgroup i ∗ [ π ( F p )]of π ( X ) has uniform exponential growth at least h for some h = h ( X ) > X . Recall that m is the dimension of X .This topological condition ensures that the minimal volume entropy of X does not vanish. INIMAL VOLUME ENTROPY AND FIBER GROWTH 5
Theorem 1.5.
Every connected finite simplicial m -complex X satisfying the fiber π -growthnon-collapsing assumption has positive minimal volume entropy, that is, ω ( X ) > . As an example, we show in Proposition 3.7 that closed aspherical manifolds whose fundamentalgroup 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 simplicialcomplex is small then its volume entropy is large. The same conclusion holds true withoutassuming that the volume of the whole simplicial complex is small. More specifically, under thesame topological conditions, if the volume of every ball of small radius is small, then the volumeentropy is large; see Theorem 3.6 for a precise statement.Note that the fibers of the simplicial map π : X → P in the definition of the collapsing andnon-collapsing conditions can always be assumed to be connected; see Proposition 2.1. We willalso give alternative formulations of both the collapsing and non-collapsing assumptions in termsof 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 betweenthe minimal volume entropy and the simplicial volume of simplicial complexes whose funda-mental class is well-defined. In view of the lower and upper bounds (1.4), one can ask whetherthere is a complementary inequality to the bound (1.2). Namely, does there exist a positiveconstant C m such that ω ( M ) m ≤ C m (cid:107) M (cid:107) ∆ for every connected closed orientable m -manifold M ? The question also makes sense for everyconnected finite simplicial m -complex X whose fundamental class is well-defined. Our nextresult 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 volumeentropy of X n tends to infinity. We emphasize that both Theorem 1.3 and Theorem 1.5 hold for the class of finite simpli-cial complexes (including compact CAT(0) simplicial or cubical complexes) and not solely forclosed manifolds. This contrasts with all previous works, which focus on closed manifolds. Inparticular, 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 theinequality (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 De-partment of Mathematics at the University of Toronto for their hospitality while this work wascompleted. We express our gratitude to Rostislav Grigorchuk for multiple stimulating discus-sions. Finally, we thank Corey Bregman and Matt Clay who pointed out a mistake in a previousversion 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 satis-fying the fiber π -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 assumptionvanishes. Several examples and counter-examples are presented throughout this section. I. BABENKO AND S. SABOURAU
Connected and non-connected fibers.
The following result shows that we can assume that the fibers of the simplicial map π : X → P in the definition of the collapsing and non-collapsing conditions are connected. Proposition 2.1.
Let π : X → P be a simplicial map between two finite simplicial complexes.Denote by k the dimension of P . Then there exists a surjective simplicial map ¯ π : X → ¯ P to afinite simplicial complex ¯ P of dimension at most k such that the fibers of ¯ π : X → ¯ P agree withthe connected components of the fibers of π : X → P .Proof. Without loss of generality, we can assume that the simplicial map π : X → P is onto.Define ¯ P = X/ ∼ as the quotient space of X , where x ∼ y if x and y lie in the same connectedcomponent of a fiber of π : X → P . Since the map π : X → P is simplicial, the quotient space ¯ P is a simplicial complex of the same dimension as P . By construction, the map π : X → P factorsout through a simplicial map ¯ π : X → ¯ P whose fibers agree with the connected components ofthe fibers of π . (cid:3) Construction of a family of piecewise flat metrics.
Let π : X → P be simplicial map from a connected finite simplicial m -complex X to asimplicial k -complex P with k < m . We will assume that the map π : X → P is onto and thatits 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 whichcollapses onto P . The construction relies on some simplicial embeddings of X and P into anEuclidean space E of large dimension.Let ∆ s = ∆ s ( p , . . . , p s ) be the abstract s -simplex with the same vertices p , . . . , p s as P . Fixan ( s + 1)-dimensional Euclidean space H with an orthonormal basis e , . . . , e s . Identify theabstract s -simplex ∆ s with the regular s -simplex of H with vertices √ e , . . . , √ e s . Define thesubcomplex R i = π − ( p i ) ⊆ X. As previously, let ∆( R i ) be the abstract simplex with the same vertices as R i . Denote by m i the dimension of ∆( R i ). Fix an ( m i + 1)-dimensional Euclidean space H i with an orthonormalbasis e i , . . . , e im i . Identify the abstract m i -simplex ∆( R i ) with the regular m i -simplex of H i with vertices √ e i , . . . , √ e im i .Consider the orthogonal sum E = H ⊕ H ⊕ · · · ⊕ H s . (2.1)Denote by g E the scalar product on E . There is a natural simplicial embedding χ : X (cid:44) → E taking every vertex v ∈ X , identified with some element √ e ij with 0 ≤ i ≤ s and 0 ≤ j ≤ m i ,to χ ( v ) = √ e i + √ e ij . (Here, a simplicial embedding means an embedding whose restriction to each simplex is anaffine map.) Thus, the whole space R i is sent under χ : X (cid:44) → E to a point lying in thesubspace H (cid:48) i = √ e i + H i orthogonal to H , parallel to H i and passing through √ e i . Byour choices of identification, the composition of χ : X (cid:44) → E with the orthogonal projection p H : E → H onto H coincides with the simplicial map π : X → P , that is, π = p H ◦ χ. The piecewise flat metric on X induced by the simplicial embedding χ : X (cid:44) → E can bedeformed as follows. Let h t : E → E be the endomorphism of E preserving each factor of the INIMAL VOLUME ENTROPY AND FIBER GROWTH 7 decomposition (2.1) whose restriction to H is the identity map and restriction to each H i is thehomothety with coefficient t . For every t ∈ (0 , χ t : X (cid:44) → E defined as χ t = h t ◦ χ is a simplicial embedding. Note that h t preserves the subspaces H (cid:48) i . By construction, we stillhave π = p H ◦ χ t . Endow X with the piecewise flat metric g t induced by the simplicial embedding χ t : X (cid:44) → E defined as g t = χ ∗ t ( g E ) . Endow also P with the natural piecewise flat metric where all its simplices are isometric to thestandard Euclidean simplex induced by the simplicial embedding P ⊆ E . By construction, themap π : X → P is 1-Lipschitz. Observe also that the g t -length of every edge lying in somefiber π − ( p i ) ⊆ X over a vertex p i ∈ P is equal to t . Since P is a k -dimensional simplicialcomplex, we conclude that vol( X, g t ) = O ( t m − k ) (2.2)as t goes to zero. Note also that diam g t (∆ pX ) ≤ √ Construction of Lipschitz retractions around each fiber.
Using the same notations as in the previous section, we construct a Lipschitz retraction froma neighborhood of each fiber of π : X → P onto the fiber itself. This is an important technicalresult which will be used in Section 2.4 to deform paths of X into the 1-skeleton of X withoutincreasing their g t -length too much (uniformly in t ).Let ∆ q = ∆ qP be a q -simplex of P . Recall that ∆ q lies in H . Fix a vertex v of ∆ q , say v = p .Denote by ∆ q − v the ( q − q opposite to v . Consider a p -simplex ∆ pX of X mappedonto ∆ qP under π : X → P . Denote by δ r = π − ( v ) ∩ ∆ pX the r -simplex of ∆ pX sent to v by π : X → P . Construct a retraction¯ (cid:37) t : ∆ pX \ π − (∆ q − v ) → δ r onto δ r as follows. First, embed ∆ pX into the Euclidean space E through χ t : X (cid:44) → E . Underthis identification, the image h t ( δ r ) of δ r lies in the subspace H (cid:48) orthogonal to H , parallel to H and passing through v . Then, take the orthogonal projection to H ⊕ H . Note that the imageof ∆ pX under the composition of these maps agrees with the convex hull Conv( h t ( δ r ) ∪ ∆ q − v ).Thus, every point x ∈ ∆ pX \ π − (∆ q − v ) is sent to a point ¯ x ∈ Conv( h t ( δ r ) ∪ ∆ q − v ). Then, forevery ¯ x ∈ Conv( h t ( δ r ) ∪ ∆ q − v ) \ ∆ q − v not lying in h t ( δ r ), take the orthogonal projection ¯ x (cid:48) ∈ ∆ q of ¯ x to ∆ q , send ¯ x (cid:48) to the point ¯ x (cid:48)(cid:48) ∈ ∆ q − v where the ray arising from v and passing through ¯ x (cid:48) meets ∆ q − v , and map ¯ x to the point y (cid:48) ∈ h t ( δ r ) where the ray arising from ¯ x (cid:48)(cid:48) and passingthrough ¯ x intersects h t ( δ r ). The map taking ¯ x to y (cid:48) extends by continuity into the identity mapon h t ( δ r ). Finally, take the image y ∈ δ r of y (cid:48) under the inverse map χ − t : h t ( δ r ) → δ r . Theresulting map ¯ (cid:37) t : ∆ pX \ π − (∆ q − v ) → δ r sending x to y is a retraction onto δ r .The map ¯ (cid:37) t : ∆ pX \ π − (∆ q − v ) → δ r is not Lipschitz as the Lipschitz constant at a point goesto infinity when the point moves to ∆ pX ∩ π − (∆ q − v ). For the map to be Lipschitz, we needto restrict it to a domain away from π − (∆ q − v ) ∩ ∆ pX . In order to use the map as a buildingblock to construct further maps on simplicial complexes, we also need to take domains that arecoherent in terms of face inclusion. Extend ∆ q into a regular Euclidean m -simplex ∆ m ⊆ H ,where ∆ q is a face of ∆ m . The perpendicular bisector hyperplane of the segment joining the I. BABENKO AND S. SABOURAU barycenters of ∆ m and ∆ mv intersects ∆ q along a subspace H . Let τ q,m = d ( v, H ) be the distancefrom v to H in ∆ q . Observe that the sequence τ q,m is decreasing in q . In particular, τ q,m ≥ τ m := τ m,m . (2.3)Note also that τ m ≥ . See Figure 1 below. τ q,m ∆ q H ∆ m H (cid:48) v Figure 1.
Construction of H Consider the domain ∆ q ( v ) of ∆ q containing v delimited by H . The restriction (cid:37) t : π − (∆ q ( v )) ∩ ∆ pX → δ r of ¯ (cid:37) t is κ m -Lipschitz for some constant κ m ≥ m . Note that this constructionis coherent. That is, if ∆ P and ∆ (cid:48) P are two simplices of P containing v , and ∆ X and ∆ (cid:48) X aretwo simplices of X mapped onto ∆ P and ∆ (cid:48) P under π : X → P , then the retractions (cid:37) t definedon π − (∆ P ( v )) ∩ ∆ X and π − (∆ (cid:48) P ( v )) ∩ ∆ (cid:48) X coincide with the intersection of their domains ofdefinition. This will allows us to put together the retractions (cid:37) t .Given a point z of ∆ pX lying in H , let z − be a vertex of ∆ pX lying in δ r and z + be a vertexof ∆ pX not lying in δ r at minimal g t -distance from z . Recall that ∆ pX collapses onto ∆ qP in E as t goes to zero. By our choice of H , there exist ε m , σ m ∈ (0 ,
1) depending only on m such thatfor t small enough d ( z, z + ) ≤ d ( z, z − ) − ε m (2.4)and d ( z, z + ) + σ m ≤ τ m . (2.5)Now, define P v = ∪ ∆ qP ( v ) ⊆ P (2.6)as the union over all the domains ∆ qP ( v ) ⊆ ∆ qP , where ∆ qP is a simplex of P of any dimension q containing v . Denote also X v = π − ( P v ) ⊆ X. (2.7)Putting together the retractions (cid:37) t : π − (∆ q ( v )) ∩ ∆ pX → δ r where ∆ pX is a simplex of X v projecting to a simplex ∆ qP of P containing v and δ r = π − ( v ) ∩ ∆ pX , we obtain a κ m -Lipschitzretraction of X v onto π − ( v ), still denoted by (cid:37) t : X v → π − ( v ) . (2.8) INIMAL VOLUME ENTROPY AND FIBER GROWTH 9
Deforming arcs into edge-arcs.
Considering the family of piecewise flat metrics g t on X previously defined, we show thefollowing result about the deformation of arcs of X into its 1-skeleton. This result will allow usto apply combinatorial techniques to count homotopy classes in Section 2.5. Proposition 2.2.
Every arc γ of X joining two vertices can be deformed into an arc γ e lyingin the -skeleton of X (i.e., γ e is an edge-arc), while keeping its endpoints fixed, with length g t ( γ e ) ≤ C m length g t ( γ ) (2.9) for every t ∈ (0 , , 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 ∆ m withendpoints on ∂ ∆ m can be deformed into an arc of ∂ ∆ m with the same endpoints at the cost ofmultiplying its length by a factor bounded by a constant λ m depending only on m . Applyingthis observation successively on the simplices of the skeleta of X , we deduce by induction thatthe inequality (2.9) holds with C m = λ (cid:48) m for t = 1, where λ (cid:48) m = (cid:81) mi =2 λ 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 whereall simplices are isometric to the standard Euclidean simplex of the same dimension. Denoteby v the image of the starting point of γ by π : X → P . Note that v is a vertex of P . Considerthe domains P v and X v introduced in (2.6) and (2.7). For every q -simplex ∆ q ⊆ P v containing v ,the distance between v and its opposite side in ∆ q ( v ) is at least τ m . Since the map π : X v → P v is 1-Lipschitz, we deduce that if γ leaves X v then its g t -length is greater than τ m .Let us argue by induction on the integer n ≥ nτ m ≤ length g t ( γ ) < ( n + 1) τ m where τ m is defined in (2.3). The value of C m in (2.9) can be taken to be equal to C m = 12 λ (cid:48) m κ m σ m ,where κ m is the Lipschitz constant of the map (cid:37) t defined in (2.8), σ m is defined in (2.5), and λ (cid:48) m is defined above.Suppose that γ lies in X v . (This is the case for instance if length g t ( γ ) < τ m , i.e. , n = 0.) Theimage γ (cid:48) of γ under the κ m -Lipschitz retraction (cid:37) t : X v → π − ( v ) satisfieslength g t ( γ (cid:48) ) ≤ κ m length g t ( γ ) . By construction, the fiber π − ( v ) is a simplicial complex of dimension at most m composed ofregular Euclidean simplices of size t . As observed at the beginning of the proof, the arc γ (cid:48) lyingin π − ( v ) can be deformed into an arc γ e lying in the 1-skeleton of π − ( v ), and so of X , withthe same endpoints multiplying its length by a factor bounded by at most λ (cid:48) m . This concludesthe proof of the proposition in this case with C m = κ m λ (cid:48) m .Suppose that γ leaves X v . Denote by z the first point where γ leaves X v . The point z splits γ into two subarcs, γ (cid:48) and γ (cid:48)(cid:48) , with γ (cid:48) ⊆ X v . Let ∆ pX be the simplex of X containing z . (Thissimplex is not necessarily unique.) Pick a vertex z − of ∆ pX lying in π − ( v ) and a vertex z + of ∆ pX not lying in π − ( v ) at minimal distance from z . By (2.4), we have d ( z, z + ) ≤ d ( z, z − ) − ε m ≤ length g t ( γ (cid:48) ) − ε m . (2.10)Since z and z ± lie in the same simplex ∆ pX , the arc γ is homotopic to α (cid:48) ∪ [ z − , z + ] ∪ α (cid:48)(cid:48) , wherethe two arcs α (cid:48) = γ (cid:48) ∪ [ z, z − ] and α (cid:48)(cid:48) = [ z + , z ] ∪ γ (cid:48)(cid:48) start and end at vertices of P . As previously observed, we have length g t ( γ (cid:48) ) ≥ τ m . Recall alsothat diam g t (∆ pX ) ≤ √
2. Thus,length g t ( α (cid:48) ) ≤ length g t ( γ (cid:48) ) + √ ≤ (cid:16) √ τ m (cid:17) length g t ( γ (cid:48) )for t small enough. The arc α (cid:48) lies in X v and is sent to an arc of π − ( v ) with the same endpointsunder the κ m -Lipschitz retraction (cid:37) t : X v → π − ( v ). In turn, this arc can be deformed into anarc α (cid:48) e lying in the 1-skeleton of X with the same endpoints withlength g t ( α (cid:48) e ) ≤ λ (cid:48) m κ m length g t ( α (cid:48) ) ≤ λ (cid:48) m κ m (cid:16) √ τ m (cid:17) length g t ( γ (cid:48) ) . (2.11)Now, by (2.10), we have length g t ( α (cid:48)(cid:48) ) ≤ length g t ( γ (cid:48)(cid:48) ) + d ( z, z + ) ≤ length g t ( γ ) − ε m . By induction, the arc α (cid:48)(cid:48) can be deformed into an edge-arc α (cid:48)(cid:48) e with the same endpoints withlength g t ( α (cid:48)(cid:48) e ) ≤ C m length g t ( α (cid:48)(cid:48) ) ≤ C m length g t ( γ (cid:48)(cid:48) ) + C m d ( z, z + ) . (2.12)As a result of (2.11) and (2.12), the arc γ can be deformed into the edge-arc γ e = α (cid:48) e ∪ [ z − , z + ] ∪ α (cid:48)(cid:48) e ,where length g t ( γ e ) ≤ λ (cid:48) m κ m (cid:16) √ τ m (cid:17) length g t ( γ (cid:48) ) + √ C m length g t ( γ (cid:48)(cid:48) ) + C m d ( z, z + ) . In order to have length g t ( γ e ) ≤ C m length g t ( γ ), it is enough to have A m length g t ( γ (cid:48) ) + √ C m d ( z, z + ) ≤ C m length g t ( γ (cid:48) )where A m = λ (cid:48) m κ m (cid:16) √ τ m (cid:17) ≤ λ (cid:48) m κ m (recall that τ m ≥ ). That is, C m d ( z, z + ) + √ C m − A m ≤ length g t ( γ (cid:48) ) . Recall that d ( z, z + ) + σ m ≤ τ m ; see (2.5). Thus, for C m large enough ( e.g. , C m ≥ λ (cid:48) m κ m σ m ≥ (1+ √ σ m ) A m σ m ), we have C m d ( z, z + ) + √ C m − A m ≤ d ( z, z + ) + σ m ≤ τ m ≤ length g t ( γ (cid:48) )as desired. (cid:3) Edge-loop entropy.
In this section, we introduce the edge-loop entropy – a discrete substitute for the volumeentropy – and show that the growth of the edge-loop entropy of (
X, g t ) is controlled as t goesto zero. Definition 2.3.
Let X be a connected finite simplicial complex with a piecewise Riemannianmetric g . The volume entropy of ( X, g ), see (1.1), can also be defined as the exponential growthrate of the number of homotopy classes induced by loops of length at most T . Namely,ent( X, g ) = lim T →∞ T log N ( X, g ; T ) (2.13) INIMAL VOLUME ENTROPY AND FIBER GROWTH 11 where N ( X, g ; T ) = card { [ γ ] ∈ π ( X, (cid:63) ) | γ loop of g -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 entropyof ( X, g ) as ent e ( X, g ) = lim T →∞ T log N e ( X, g ; T )where N e ( X, g ; T ) = card { [ γ ] ∈ π ( X, (cid:63) ) | γ edge-loop of g -length at most T } . Clearly, one hasent e ( X, g ) ≤ ent( X, g ).Let A be a subcomplex of X with basepoint a . We also define N ( A ⊆ X ; T ) = card { [ γ ] ∈ π ( X, a ) | γ ⊆ A and length( γ ) ≤ T } 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 ∗ [ π ( F p )] of π ( X ) is at most ν , where F p = π − ( p ) is a (connected) fiber of π : X → P and i : F p (cid:44) → X is the inclusion map. Then ent e ( X, g t ) = O (cid:0) t ν (cid:1) . (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 anedge of P by the simplicial map π : X → P . It is said to be short otherwise (in which case, itis sent to a vertex of P ). By construction, every long edge of X is of length √ t and everyshort edge of X is of length t . Denote also by n e the number of edges of X .Observe that g t = t g on every (connected) fiber F p = π − ( p ) of π : X → P . Hence,diam( F p , g t ) = t · diam( F p , g ) −−→ t → . 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 diam( F p , g t ) < for every vertex p ∈ P .Let us estimate the number of homotopy classes of edge-loops in X of g t -length at most T .Every edge-loop γ in X of g t -length at most T decomposes as γ = α ∪ β ∪ α ∪ · · · ∪ β N (2.15)where α i is a long edge of X and β i is a possibly constant edge-path lying in a (connected) fiber F i = π − ( p i ) of π : X → P over a vertex p i ∈ P , which joins the terminal endpoint of α i to theinitial endpoint of α i +1 .Fix a basepoint a i ∈ F i . Denote by (cid:96) i the g t -length of β i . Let ¯ β i be the loop of F i based at a i obtained by connecting the endpoints x i and y i of β i to the basepoint a i along two paths of F i of g t -length at most diam( F i , g t ) < . The number N ex i ,y i ( F i ⊆ ( X, g t ); (cid:96) 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 -lengthat most (cid:96) 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 (cid:96) i + 2 diam( F i , g t ). Thus, N ex i ,y i ( F i ⊆ ( X, g t ); (cid:96) i ) ≤ N ( F i ⊆ ( X, g t ); (cid:96) i + 2 diam( F i , g t )) ≤ N (cid:16) F i ⊆ ( X, g ); (cid:96) i +1 t (cid:17) (2.16)since g t = t g on the fiber F i , where we refer to Definition 2.3 for the definition of N . By assumption, the subgroups i ∗ [ π ( F p )] (cid:54) π ( X ) have a subexponential growth at most ν and the same holds for N ( F p ⊆ ( X, g ); T ); see [42]. More specifically, there exists a function Q ( T ) = A exp( T ν ) with A > N ( F p ⊆ ( X, g ); T ) ≤ Q ( T ) (2.17)for every vertex p ∈ P and every T ≥ X induced by thedifferent possibilities for the edge-path β i of length (cid:96) i is at most N ex i ,y i ( F i ⊆ ( X, g t ); (cid:96) i ) ≤ Q (cid:16) (cid:96) i +1 t (cid:17) where (cid:96) i is the g t -length of β i .Now, there are at most n e possible choices for each long edge α i . (Recall that n e is the numberof 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 β i of g t -length (cid:96) i ≤ θ i , where θ i = (cid:100) (cid:96) i (cid:101) , is bounded by n Ne N (cid:89) i =1 Q (cid:16) θ i +1 t (cid:17) . Since every edge α i is of g t -length at least 1, we have N ≤ T and (cid:80) Ni =1 (cid:96) i ≤ T − N . Since θ i = (cid:100) (cid:96) i (cid:101) , we also have (cid:80) Ni =1 θ i ≤ T . Therefore, the number N e ( X, g t ; T ) of homotopy classes ofedge-loops in X of g t -length at most T is bounded by N e ( X, g t ; T ) ≤ (cid:88) N ≤(cid:98) T (cid:99) (cid:88) ( θ i ) N ≤(cid:98) T (cid:99) n Ne N (cid:89) i =1 Q (cid:16) θ i +1 t (cid:17) (2.18)where the second sum is over all N -tuples ( τ , . . . , τ N ) of positive integers such that (cid:80) Ni =1 θ i ≤ (cid:98) T (cid:99) .The double sum (2.18) has at most T T terms (the first sum has (cid:98) T (cid:99) terms and the secondsum has 2 (cid:98) T (cid:99)− terms given by the distinct decomposition of the integer (cid:98) T (cid:99) ). Consider thelargest term of (2.18) attained by some N ≤ T and ( θ i ) N ≤ T . We have N e ( X, g t ; T ) ≤ T T n Te N (cid:89) i =1 Q (cid:16) θ i +1 t (cid:17) (2.19) ≤ T T n Te A T exp (cid:32) t ν N (cid:88) i =1 ( θ i + 1) ν (cid:33) . Applying H¨older’s inequality to the sum (cid:80) Ni =1 ( θ i + 1) ν with p = − ν and q = ν , we obtain N (cid:88) i =1 ( θ i + 1) ν ≤ (cid:32) N (cid:88) i =1 p (cid:33) /p · (cid:32) N (cid:88) i =1 ( θ i + 1) (cid:33) /q ≤ T − ν · ν T ν ≤ T since νq = 1, N ≤ T and (cid:80) Ni =1 ( θ i + 1) ≤ (cid:80) Ni =1 θ i + N ≤ T . Hence, N e ( X, g t ; T ) ≤ T T n Te A T exp (cid:18) Tt ν (cid:19) . This implies that ent e ( X, g t ) ≤ log(2 n e A ) + 2 t ν . (cid:3) INIMAL VOLUME ENTROPY AND FIBER GROWTH 13
Remark 2.5. If X satisfies the collapsing assumption with polynomial growth rate, we canderive a stronger bound than (2.14). Namely, the edge-loop entropy of ( X, g t ) has a logarithmicgrowth when t goes to zero, that is,ent e ( X, g t ) = O (cid:0) log (cid:0) t (cid:1)(cid:1) . 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, usingthe expression of Q , the concavity of the nondecreasing function log(1 + · ), and the inequalities N ≤ T and (cid:80) Ni =1 ( θ i + 1) ≤ T , we obtainlog (cid:32) N (cid:89) i =1 Q (cid:16) θ i +1 t (cid:17)(cid:33) ≤ T log( a ) + d N (cid:88) i =1 log (cid:16) θ i +1 t (cid:17) ≤ T log( a ) + d N log (cid:0) TNt (cid:1) (2.20)Introduce f t ( x ) = x log(1 + xt ) with x ∈ [0 , t ≤ e − , we have f (cid:48) t ( x ) = log(1 + xt ) − xt +1 ≥ log(1 + t ) − ≥ . Thus, for x = N T and t small enough, we deduce that · NT log (cid:0) TNt (cid:1) = f t ( N T ) ≤ f t (1) = log (cid:0) t (cid:1) . (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 ent e ( X, g t ) = O (cid:0) log (cid:0) t (cid:1)(cid:1) 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 simpli-cial map π : X → P to a simplicial k -complex P with k < m such that for every connected com-ponent F p of every fiber π − ( p ) with p ∈ P , the finitely generated subgroup i ∗ [ π ( F p )] of π ( X ) has subexponential growth rate at most ν . Suppose that ν < m − km . Then X has zero minimalvolume entropy.Proof. By Proposition 2.1, we can assume that the simplicial map π : X → P in Theorem 2.6is 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 ent e ( X, g t ) ≤ ent( X, g t ); see Definition 2.3. By Propo-sition 2.2, a reverse inequality holds true. Namely, there exists C m > X, g t ) ≤ C m ent e ( X, g t ) (2.22)for every t ∈ (0 , X, g t ) vol( X, g t ) m = O (cid:16) t m − km − ν (cid:17) . Since ν < m − km , we conclude that ent( X, g t ) vol( X, g t ) m converges to zero as t goes to zero. (cid:3) We conclude with an application. Let us recall the definition of an F -structure, first introducedby 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 opencover { U i } of M , finite normal coverings π i : ˜ U i → U i and effective smooth actions of tori T k i on ˜ U i which extend the action of the deck transformation group such that if U i and U j intersecteach other, then π − i ( U i ∩ U j ) and π − j ( U i ∩ U j ) have a common covering space on which thelifting actions of T k i and 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 dimensionof the orbits.As an application of Corollary 1.4, we derive the following result, which is also a consequenceof Paternain and Petean’s work on the connection between the topological entropy of the geodesicflow and F -structures; see [48, Theorem A]. Corollary 2.8.
Every closed manifold admitting an F -structure (of possibly zero rank) has zerominimal volume entropy.Proof. By the Slice Theorem and its consequences, see [32, Appendix B], we derive the followingproperties. The orbits of the F -structure of a closed m -manifold M partition the manifold intoclosed submanifolds covered by tori; see also [15] and [48]. The trivial orbits form a submanifoldof codimension at least one (at least two if the manifold is orientable) and the orbit space isan orbifold of dimension at most m −
1. Since the fibers of the natural projection from M tothe orbit space have almost abelian fundamental groups, the manifold M satisfies the collapsingassumption with polynomial growth rate and has zero minimal volume entropy by Corollary 1.4. (cid:3) Examples of manifolds satisfying the collapsing assumption.
In this section, we construct a closed orientable manifold with fundamental group of exponen-tial growth satisfying the collapsing assumption with fibers of subexponential growth which donot have polynomial growth. Furthermore, this example satisfies the condition on the subexpo-nential growth rate of the subgroups i ∗ [ π ( F p )] of Theorem 2.6 (which implies that their minimalvolume entropy is zero). We also show through another example which satisfies the collapsingassumption but does not satisfy the condition of Theorem 2.6 that the natural method presentedin 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 generatedgroup 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 computedin [22]. It is roughly equal to ν ( G ) (cid:39) . ∈ [ , ] . The group G is a finitely generated recursively presented group – a description of its presen-tation can be found in [40] – but it is not finitely presented. It is an open question whetherfinitely 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 anembedding is given in [26]. The construction goes as followsConsider the group ¯ G defined in [26] given by the following presentation¯ G = (cid:104) a, c, d, u | a = c = d = ( ad ) = ( adacac ) = e ; u − au = aca, u − cu = dc, u − du = c (cid:105) . (2.23)The group ¯ G contains the first Grigorchuk group G ; see [26]. More precisely, the group ¯ G is anHNN-extension of G : ¯ G = (cid:104) G, u | u − xu = σ ( x ) for every x ∈ G (cid:105) . INIMAL VOLUME ENTROPY AND FIBER GROWTH 15 where σ : G → G is a monomorphism; see [26] for further details. The subgroup G (cid:54) ¯ G isgenerated by a , c and d . Note that ¯ G contains a free subsemigroup with two generators, andtherefore has exponential growth.Let us construct an orientable closed 5-dimensional manifold M with π ( M ) = ¯ G as follows.Define N = ( R P ) a R P ) c R P ) d S × S ) u (2.24)where the indices a , c , d and u correspond to the generators of G . Take five loops γ , . . . , γ in thehomotopy classes ( ad ) , ( adacac ) , u − auaca , u − cudc and u − duc of π ( N ) = Z ∗ Z ∗ Z ∗ Z .Placing the curves in generic position, we can assume that the loops γ , . . . , γ are smooth simpleclosed curves which do not intersect each other. Denote by M the orientable closed manifoldobtained from N by spherical surgeries of type (1 ,
4) along γ , . . . , γ . Since spherical surgeriesof type (1 ,
4) correspond to attaching index 2 handles, the fundamental group of M is given bythe presentation (2.23). That is, π ( M ) = ¯ G .Let us construct a piecewise linear map π : M → S with subexponential growth fibers.Consider the natural map N → S which takes the terms ( R P ) a R P ) c R P ) d in theconnected sum (2.24) to a point p ∈ S and projects the last term ( S × S ) u to the S -factorof the product. By the expression of the relations of the presentation (2.23) of ¯ G , the images by N → S of the loops γ , . . . , γ are contractible in S . Thus, the map N → S extends to thehandles of M , which yields a map M → S . Deforming the map, if necessary, by sending thecomplement of a tubular neighborhood of a regular fiber F of M → S to a point, we can assumethat the map M → S is smooth with a unique critical value p ∈ S and that the inverse image π − ( S \ { p } ) has a product structure (0 , × F whose vertical slices coincide with the fibersof M → S . We can further deform M → S into a piecewise linear map π : M → S by takingfine enough triangulations of M and S , and by applying the simplical approximation theorem,without changing the topology of the fibers above S \ { p } .Let us show that ker π ∗ = G , where π ∗ : π ( M ) → π ( S ) is the π -homomorphism inducedby π : M → S . Every element w ∈ ker π ∗ can be represented by a word in the letters a , b , d and u with a minimal number of occurrences of u ± . By construction, π ∗ ( a ) = π ∗ ( c ) = π ∗ ( d ) = 0and π ∗ ( u ) is a generator of π ( S ). Thus, the word w has as many u ’s as u − ’s. If the word w contains a letter u or u − , then it contains a subword uw (cid:48) u − or u − w (cid:48) u , where w (cid:48) is a wordin a , c and d (without u ). According to the presentation (2.23), these subwords can be replacedwith subwords in the letters a , b , d (without u ) in the representation of w , which contradicts thechoice of the word representing w . Thus, w lies in the subgroup of ¯ G generated by a , c and d .That is, ker π ∗ (cid:54) G . The reverse inclusion is obvious. Hence, ker π ∗ = G .Now, since i ∗ [ π ( F p )] is a subgroup of ker π ∗ containing the generators a , c and d of G , wederive that i ∗ [ π ( F p )] = ker π ∗ = G . All the other fibers F p (cid:39) F with p ∈ S different from p can be deformed into F p , which implies that i ∗ [ π ( F p )] is a subgroup of i ∗ [ π ( F p )] = G . Since G has subexponential growth, the image i ∗ [ π ( F p )] of the fundamental group of every fiber F p of π : M → S has also subexponential growth, where p ∈ S .Since ν ( G ) < m − km = (with m = 5 and k = 1), the orientable closed 5-dimensional mani-fold M satisfies the collapsing assumption of Theorem 2.6. Remark 2.9.
The group ¯ G is amenable; see [26]. Thus, by Gromov’s vanishing simplicialvolume theorem, see Theorem 2.17, every manifold with fundamental group ¯ G has zero simplicialvolume. Remark 2.10.
One can show that the manifold M is essential. (This is not direct and requiressome work.) An easier way to obtain an essential manifold M (cid:48) is to modify our constructionby taking the connected sum of M with a nilmanifold, say T m . In this case, we collapse M (cid:48) = T m M to the graph P = [0 , ∪ { } = p S so that the preimage of p (cid:54) = p is the attaching sphere of the connected sum, the torus T m \ B m with a ball removed is sent to [0 ,
1] and theterm M \ B m is sent to S as before. The manifold M (cid:48) still satisfies the collapsing assumptionof Theorem 2.6 with the map π : M (cid:48) → P , and the image i ∗ [ π ( F (cid:48) p )] of the fundamental groupof the fiber F (cid:48) p of π : M (cid:48) → P still agrees with the group G of intermediate growth.In dimension 4, the construction does not go through as the loop γ representing u − cudc isorientation-reversing. Still, we can define N as the connected sum i =1 ( S × S ) i . Performingspherical surgeries along eight loops γ , . . . , γ representing the eight relations of the presenta-tion (2.23) of ¯ G as previously, we obtain an orientable closed 4-manifold M and a piecewiselinear map π : M → S with i ∗ [ π ( F p )] = G and i ∗ [ π ( F p )] (cid:54) G for every p ∈ S . In this case( m = 4 and k = 1), we have ν ( G ) > m − km = . Thus, the manifold M does not satisfy thecollapsing assumption of Theorem 2.6 with the simplicial map π : M → S .We show in this case that the natural method presented in the proof of Theorem 2.6 consistingof shrinking the fibers of the map π : M → S does not readily apply, even though the imagesof the fundamental groups of the fibers of π : M → S have subexponential growth. Endow M with a piecewise flat metric where all simplices of N are isometric to the standard Euclideansimplex with the same dimension. Since the subexponential growth rate of i ∗ [ π ( F p )] = G isgreater than , the number of homotopy classes of F p which can be represented by a loop basedat a vertex (cid:63) ∈ F p of length at most T is at least C exp( T ) for some positive constant C . Fixa loop β of M based at (cid:63) representing u ∈ π ( M ) = ¯ G and denote by (cid:96) its length. Modify themetric on M by shrinking the length of the fibers of π : M → S by a factor t as in the proofof Theorem 2.6 and denote by g t the metric so-obtained. Note that vol( M, g t ) ≥ C (cid:48) t for somepositive constant C (cid:48) . Observe also that the number of homotopy classes of the fiber F p whichcan be represented by loops based at (cid:63) of g t -length at most T is at least C exp (cid:16) ( Tt ) (cid:17) . Fix T = n ( (cid:96) + 1). Consider all the loops γ = α β · · · α n β of M based at (cid:63) of g t -length at most T which can be constructed by taking a loop α in F p of g t -length at most 1, followed by theloop β of g t -length (cid:96) , then by a loop α in F p of g t -length at most 1, then by β and so on. Weobtain at least C n exp (cid:16) nt (cid:17) = C T(cid:96) +1 exp (cid:18) T ( (cid:96) +1) t (cid:19) homotopy classes. Thus, ent( M, g t ) (cid:38) t .Hence, ent( M, g t ) vol( M, g t ) (cid:38) t which tends to infinity as t goes to zero. Remark 2.11.
Note that the orientable closed 4-manifold M constructed above satisfies thecollapsing assumption of Theorem 2.6 with k = 2. Indeed, one can show that there exists amap M → K from M to a 2-dimensional polyhedron K which induces a π -isomorphism whosefundamental groups of the fibers have a trivial image in π ( M ). Therefore, the minimal volumeentropy 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 fiberdiameters 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 π -growth in X if the subgroup Γ U := i ∗ [ π ( U )] of π ( X ) has subexponentialgrowth, where i : U (cid:44) → X is the inclusion map. In this case, the subexponential π -growth rateof U in X is defined as the subexponential growth rate of Γ U . Proposition 2.13.
A connected finite simplicial m -complex X satisfies the collapsing assump-tion (with subexponential growth rate at most ν ) onto a simplicial k -complex P if and only ifit admits a cover of multiplicity k + 1 by open subsets of subexponential π -growth in X (withsubexponential growth rate at most ν ). INIMAL VOLUME ENTROPY AND FIBER GROWTH 17
Proof.
Suppose that X satisfies the collapsing assumption. Then there exists a simplicial map π : X → P onto a simplicial k -complex P such that for every connected component F p of everyfiber π − ( p ), where p is a vertex of P , the subgroup i ∗ [ π ( F p )] of π ( X ) has subexponentialgrowth. Since P is a finite simplicial complex of dimension k , the open cover formed by theopen stars st( p ) ⊆ P of the vertices p of P has multiplicity k + 1. The connected componentsof the preimages π − (st( p )) ⊆ X of these open stars form an open cover of X with the samemultiplicity k + 1 as the previous cover of P . Furthermore, the open subsets of this open coverof X strongly deformation retract onto the connected components F p of the fibers π − ( p ). Inparticular, they have subexponential π -growth in X (with the same subexponential growth rateas the subgroups induced by the fibers). This prove the first implication.For the converse implication, let { U i } i =0 ,...,s be a cover of X of multiplicity k + 1 by opensubsets of subexponential π -growth in X . Take a partition of unity { φ i } of X , where eachfunction φ i : X → [0 ,
1] has its support in U i . Consider the map Φ : X → ∆ s defined byΦ( x ) = ( φ ( x ) , . . . , φ s ( x ))in the barycentric coordinates of ∆ s . The nerve P of the cover { U i } is a simplicial complexwith one vertex v i for each open set U i , where v i , . . . , v i n span an n -simplex of P if and onlyif the intersection ∩ nj =1 U i j is nonempty. By construction, the dimension of the nerve P is oneless than the multiplicity of the cover { U i } . That is, dim P = k . We identify in a naturalway the vertices { v i } of P with the vertices of ∆ s . With this identification, the nerve P of X lies in ∆ s . Furthermore, the image of Φ lies in P . By [34, § X and P ifnecessary, we can approximate Φ : X → P by a simplicial map π : X → P close to Φ forthe C -topology, whose normalized barycentric coordinates π i : X → [0 ,
1] have their supportin U i . Thus, every fiber π − ( p ) lies in one of the open subsets U i . Therefore, for every connectedcomponent F p of π − ( p ), the subgroup i ∗ [ π ( F p )] lies in some subgroup i ∗ [ π ( U i )]. Since the opensubsets U i have subexponential π -growth in X , the subgroups i ∗ [ π ( F p )] have subexponentialgrowth (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. (cid:3) An illustration of the characterization of the collapsing assumption in terms of open covers isgiven by the following example.
Example 2.14.
For i = 1 ,
2, let M i be a connected closed manifold of dimension m ≥ π ( M i ) of subexponential growth rate at most ν < m − m . Let N be aconnected closed n -manifold embedded both in M and M with n ≤ m −
3. Suppose that theembedding N ⊆ M i induces a π -monomorphism and that its normal fiber bundle N i ( N ) ⊆ T M i is trivial for i = 1 ,
2. Define the m -manifold X = ( M \ U ( N )) ∪ N × S m − n − ( M \ U ( U ))where U i ( N ) is a small tubular neighborhood of N in M i . By van Kampen’s theorem, π ( M i \ U i ( N )) is isomorphic to π ( M i ), and thus has subexponential growth rate at most ν . Take asmall tubular neighborhood U i of M i \ U i ( N ) in X for i = 1 ,
2. Since U i strongly deformationretracts onto M i \ U i ( N ), its fundamental group π ( U i ) is isomorphic to π ( M i \ U i ( N )). Thisyields a cover of X of multiplicity two by open subsets U and U with subexponential π -growthat most ν in X . According to Proposition 2.13, the closed m -manifold X satisfies the collapsingassumption. Note however that the fundamental group of X has exponential growth in general.This construction provides numerous examples of closed essential manifolds with a fundamentalgroup of exponential growth and zero minimal volume entropy. For instance, when N is reducedto a singleton, the manifold X is the connected sum M M of M and M . This special casecan 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 mul-tiplicity k + 1 by open subsets of subexponential π -growth in X with subexponential growth rateat most ν < m − km has zero minimal volume entropy. Collapsing assumption and zero simplicial volume.
Drawing a parallel with the simplicial volume through Gromov’s vanishing simplicial volumetheorem, 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 probabilitymeasure. A path-connected open subset U of a path-connected topological space X is amenablein X if i ∗ [ π ( U )] is an amenable subgroup of π ( X ), where i : U (cid:44) → 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 (cid:107) M (cid:107) ∆ = 0 . In particular, the simplicial volume of a connected closed manifold with amenable fundamentalgroup is zero.
The characterization of the collapsing assumption in terms of covers allows us to derive thefollowing 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 zerosimplicial 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 ⊆ M with subexponential π -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 π -growth in M ,and so by amenable open subsets. It follows from Theorem 2.17 that M has zero simplicialvolume. (cid:3) Simplicial complexes with positive minimal volume entropy
By relying on the notion of Urysohn width, we show that the minimal volume entropy of aconnected finite simplicial complex satisfying the fiber π -growth non-collapsing assumption ispositive. We also give a characterization of the non-collapsing assumption in terms of open cov-ers. Finally, we construct finite simplicial complexes with zero simplicial volume and arbitrarilylarge 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 UW q ( X ), isdefined as the least real δ > π : X → P from X to asimplicial q -complex P , where all the fibers π − ( p ) have diameter at most δ in X . That is,UW q ( X ) = inf π : X → P sup p ∈ P diam X [ π − ( p )] (3.1)where π : X → P runs over all continuous map from X to a simplicial q -complex P . Note thatthe simplicial complex P may vary with π . For a simplicial m -complex X , we will simply writeUW( X ) for UW m − ( X ). INIMAL VOLUME ENTROPY AND FIBER GROWTH 19
The Urysohn width can also be interpreted in terms of open covers; see [33, Lemma 0.8] forinstance.
Proposition 3.2.
A compact metric space X has Urysohn q -width less than w if and only ifthere 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 themap π : X → 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 π : X → P in the definition of theUrysohn width are simplicial and that their fibers are connected.Proof. Suppose UW q ( X ) < δ . By Proposition 3.2, there is a finite open cover U = { U i } i =1 , ··· ,s of X of multiplicity q +1 and diameter less than δ . Consider the natural map Φ : X → P ⊆ ∆ s − to the nerve P of 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 Φ : X → P by a simplicial map π : X → P closeto Φ for the C -topology, whose normalized barycentric coordinates π i : X → [0 ,
1] have theirsupport in U i ; see [34, § π − ( p ) lies in one of the open sets U i . Therefore,diam X π − ( p ) < δ . As a result, we can assume that the map π : X → P is simplicial in thedefinition of the Urysohn width; see (3.1). Now, by Proposition 2.1, we can replace π : X → P with a simplicial map ¯ π : X → ¯ P to a simplicial complex ¯ P of dimension at most q , whose fibersare connected and of diameter less than δ . (cid:3) 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. Pa-pasoglu [47]; see also [46].
Theorem 3.4 ([33], [38], [47], [46]) . Let X be a finite simplicial m -complex with a piecewiseRiemannian metric. Then vol( X ) ≥ C m UW( X ) m where C m is an explicit positive constant depending only on m . More precisely (by contraposi-tion), if for some R > every ball B ( R ) ⊆ X of radius R has volume at most C m R m then UW( X ) ≤ R. A more general statement involving the lower dimensional widths and the Hausdorff contentof 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 ofa space, similar in spirit to the ˇSvarc-Milnor lemma; see [31, § 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 Γ U := i ∗ [ π ( U )] of π ( X ) definedas the image of π ( U ) under the group homomorphism induced by the inclusion map i : U (cid:44) → X is finitely generated. Then diam X ( U ) · ent( X ) ≥
12 ent(Γ 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 γ i ⊆ X based at some fixed basepoint x ∈ X whosehomotopy classes in X form a generating set S of Γ U = i ∗ [ π ( U )] withlength( γ i ) < X ( U ) + ε for some arbitrarily small ε >
0. Clearly, every homotopy class α ∈ Γ U can be represented by aloop γ ⊆ X based at x of length at most(2 diam X ( U ) + ε ) · d S ( e, α )where d S is the word distance on Γ U induced by S . Thus, the number N ( X ; T ) of homotopyclasses represented by loops based at x of length at most T , see Definition 2.3, satisfies N ( X ; T ) ≥ card (cid:26) α ∈ Γ U | d S ( e, α ) ≤ T X ( U ) + ε (cid:27) It follows from (2.13) that ent( X ) ≥
12 diam X ( U ) + ε ent(Γ U , S )for every ε >
0. Hence the result. (cid: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-collapsingassumption has positive minimal volume entropy. More generally (by contraposition), given apiecewise Riemannian metric g on X , if for some R > every ball B ( R ) ⊆ X of radius R hasvolume at most C m R m then ent( X, g ) ≥ h ( X )2 R 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, thereexists a constant C (cid:48) m > π : X → P to some simplicial ( m − P ,where every fiber F p = π − ( p ) is connected, such thatdiam X ( F p ) < C (cid:48) m vol( X, g ) m . (3.2)The constant C (cid:48) m can be taken arbitrarily close to C − m m , where C m is the constant in The-orem 3.4. By the non-collapsing assumption, one of the subgroups i ∗ [ π ( F p )] has uniformexponential growth at least h ( X ). The point p ∈ P has a small enough open neighborhood B p ⊆ P whose preimage U p = π − ( B p ) is homotopy equivalent to the fiber F p and hasdiameter close to the one of F p so thatdiam X ( U p ) < C (cid:48) m vol( X, g ) m . (3.3)The group Γ p = i ∗ [ π ( U p )], which is isomorphic to the finitely generated subgroup i ∗ [ π ( F p )],has uniform exponential growth at least h ( X ). It follows from Proposition 3.5 that12 h ( X ) ≤
12 ent(Γ p ) ≤ diam X ( U ) · ent( X, g ) ≤ C (cid:48) m ent( X, g ) vol(
X, g ) m . (3.4)Hence, the minimal volume entropy of X is positive and satisfies ω ( X ) ≥ C (cid:48) m h ( X ) > . Suppose that every ball B ( R ) of radius R has volume at most C m R m , where C m is the constantin 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 thedesired lower bound for the volume entropy of X . (cid:3) INIMAL VOLUME ENTROPY AND FIBER GROWTH 21
The following result provides examples of simplicial complexes satisfying the non-collapsingassumption.
Proposition 3.7.
Let X be a finite aspherical simplicial m -complex with H m ( X ; R ) nontrivial,where m ≥ . 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-elementaryword hyperbolic group satisfies the non-collapsing assumption.Proof. First observe that since X is aspherical, its fundamental group π ( X ) is torsion-free, oth-erwise there would exist a finite-dimensional aspherical space with a finite fundamental group,which is impossible; see [34, Proposition 2.45]. Let π : X → P be a simplicial map to a fi-nite simplicial ( m − P . Suppose that all the subgroups H = i ∗ [ π ( F p )] (cid:54) π ( X ),where F p is a connected component of a fiber π − ( p ) and i : F p (cid:44) → X is the inclusion map,have subexponential growth (and so are amenable). That is, the simplicial complex X sat-isfies 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 volumetheorem, see Theorem 2.17, the canonical homomorphism (cid:98) H m ( X ; R ) → H m ( X ; R ) betweenbounded cohomology and singular cohomology vanishes. By [45], the canonical homomorphism (cid:98) H m ( X ; R ) → H m ( X ; R ) is also surjective. Hence, H m ( X ; R ) is trivial, which leads to a con-tradiction. Indeed, by assumption, H m ( X ; R ) is nontrivial, and by the universal coefficienttheorem for cohomology, H m ( X ; R ) = Hom( H m ( X ; R ) , R ) is also nontrivial. Therefore, one ofthe subgroups H = i ∗ [ π ( F p )] (cid:54) π ( 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 suchthat for every x, y ∈ G which do not commute and for every n ≥ n G , the subgroup (cid:104) x n , y n (cid:105) 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 ( tn G ) ⊆ B S ( t ), where S = { x ± n G , y ± n G } .Here, B S ( t ) is formed of the elements of G at distance at most t induced by S from the identityelement with respect to the word distance. In this case, the algebraic entropy of G is boundedfrom below by the algebraic entropy of the free group of rank 2 generated by S . Thus, ev-ery finitely generated subgroup H (cid:54) G is either abelian or has uniform exponential growth atleast h ( G ) = n G log(3). This implies that the non-abelian subgroup H (cid:54) π ( X ) has uniformexponential growth at least h , where h = h ( X ) is a positive constant depending only on X .Hence the result. (cid:3) Question 3.8.
Does every closed orientable manifold M satisfying the non-collapsing assump-tion have positive simplicial volume? Otherwise, find examples of closed orientable manifoldswith 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 opencovers in the same vein as in the collapsing case; see Section 2.8.
Definition 3.9.
A cover U = { U i } of a path-connected topological space X by path-connectedopen subsets has uniform exponential π -growth at least h if the subgroup Γ U := i ∗ [ π ( U )]of π ( X ) has uniform exponential growth at least h for at least one open subset U of U , where i : U (cid:44) → X is the inclusion map.The following proposition mirrors Proposition 2.13 with an analoguous proof, which is left tothe reader. Proposition 3.10.
A connected finite simplicial m -complex X satisfies the non-collapsing as-sumption if and only if every open cover of multiplicity at most m has uniform exponential π -growth at least h , where h = h ( X ) > 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 multi-plicity at most m have uniform exponential π -growth at least h = h ( X ) has positive minimalvolume entropy. Simplicial volume and minimal volume entropy.
In this section, we construct a sequence of connected finite simplicial complexes X n withvanishing simplicial volume and minimal volume entropy going to infinity, proving Theorem 1.6.Let Σ be a surface of genus h ≥ d ≥
3. Consider thesimplicial complex X obtained by attaching Σ to S along a map ∂ Σ → S of degree d . Thefundamental group of X has the following presentation π ( X ) = (cid:104) x , y , . . . , x h , y h , z | h (cid:89) i =1 [ x i , y i ] z d = e (cid:105) . (3.5)The space X admits a CAT( −
1) metric and therefore is aspherical; see [12]. (This is also aconsequence of the fact that X is the cellular 2-complex associated to the one-relator grouppresentation (3.5) whose relator is not a proper power; see [20].) However, one cannot directlyapply Proposition 3.7 to X . Indeed, consider the cell decomposition of X with a single 2-cellattached to a bouquet of 2 h + 1 circles according to the relation (3.5). A cellular homologycomputation of the boundary map on the cellular chains of the cell decomposition of X showsthat H ( X ; Z ) = 0 . Thus, H ( X ; R ) is trivial and so is H ( X ; 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 π : X → P to a one-dimensional simplicial complex ( i.e. , a graph).The space X admits a d -sheeted cover q : (cid:98) X → X with (cid:98) X = S ∪ ψ Σ · · · ∪ ψ d Σ d where the d copies Σ i of Σ with i = 1 , . . . , d are attached to S along the maps ψ i : ∂ Σ i −→ S defined as follows. Choose a homeomorphism ψ : ∂ Σ → S and define ψ i : ∂ Σ i → S as ψ i = ρ i − ◦ ψ , where ρ is the rotation of S of angle πd . By definition, X = (cid:98) X/ Z q , where thecyclic group Z d freely acts on (cid:98) X by cyclic permutations of the copies Σ i of Σ.The cover (cid:98) X contains the π -embedded genus 2 h surface M = Σ ∪ Σ . Consider the compositemap f = π ◦ q : M → P . Proposition 3.7 ensures that for some p ∈ P , the subgroup i ∗ [ π ( F p )] (cid:54) π ( M ) has uniform exponential growth at least h ( π ( M )) = . Thus, since π ( M ) is asubgroup of π ( (cid:98) X ), and so of π ( X ), the space X satisfies the non-collapsing assumption.By Theorem 3.6, the minimal volume entropy of X is positive. Namely, ω ( X ) ≥ C (cid:48) log(3) > C (cid:48) 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). INIMAL VOLUME ENTROPY AND FIBER GROWTH 23
Consider the simplicial complex X n defined as the bouquet X n = T ∨ (cid:32) n (cid:95) i =1 X (cid:33) of the torus with n copies of X . By construction, we have H ( X n ; Z ) = H ( T ; Z ) = Z and (cid:107) X n (cid:107) ∆ = 0 . By [6, Theorem 2.6], we obtain ω ( X n ) = n ω ( X ) . Since ω ( X ) is positive, see (3.6), we conclude that ω ( X n ) goes to infinity, which proves Theo-rem 1.6. Remark 3.12.
Applying the “graph of groups” construction, see [34, § m = 2 k with positive simplicial volume. Theresulting space Σ is a manifold with boundary ∂ Σ (cid:39) S k − . Fix an integer d ≥
3. Denoteby Y the quotient of Σ by the natural free action of Z d on S k − given by rotation of the Hopffibration. Observe that π ( Y ) (cid:39) π (Σ) ∗ Z d and H m ( Y ; Z ) = 0. Define the simplicial m -complex X n = ni =1 Y i by taking the connected sum of n copies of Y . Note that H m ( X n ; Z ) = 0.The space X n admits a d -sheeted cyclic cover which can be described as follows. The con-nected sum ni =1 Σ i of n copies of Σ is a manifold whose boundary identifies with the disjointunion (cid:116) S k − i of n spheres. Let (cid:98) X n be the space obtained by gluing d copies of ni =1 Σ i alongthis disjoint union (cid:98) X n = ( (cid:116) S k − i ) ∪ ψ ( ni =1 Σ i ) · · · ∪ ψ d ( ni =1 Σ i )where the attaching maps ψ j are given by the action of α j on the boundary components of ni =1 Σ i (for a fixed generator α of Z d ). The cover (cid:98) X n → X n is the natural map sending the d copies ni =1 Σ i to X n . By the comparison principle, see [13, Lemma 4.1], we have ω ( (cid:98) X n ) ≤ d m ω ( X n ) . (3.7)Now, take two copies ni =1 Σ i and ni =1 ¯Σ i in (cid:98) X n . By construction, the boundaries ∂ Σ i and ∂ ¯Σ i agree and the union M n = ( ni =1 Σ i ) ∪ ( ni =1 ¯Σ i )is a closed m -manifold homeomorphic to M n (cid:39) ni =1 (Σ i i ) ni =1 ( S × S k − ) . Since the simplicial volume is additive under connected sums in dimension at least three, see [29],we obtain (cid:107) M n (cid:107) ∆ = 2 n (cid:107) Σ (cid:107) ∆ > . Thus, by (1.2), the minimal volume entropy ω ( M n ) of M n goes to infinity when n or (cid:107) Σ (cid:107) ∆ tendto infinity.To conclude, consider the simplicial m -complex Z n defined as the connected sum Z n = X n T m . Observe that Z n is a cellular m -complex with a single m -cell. Clearly, H m ( Z n ; Z ) = Z and (cid:107) Z n (cid:107) ∆ = 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 , here X n , is a cellular m -complex with a single m -cell),we have ω ( Z n ) ≥ ω ( X n ). Since π ( M n ) is a subgroup of π ( (cid:98) X n ) and the manifold M n containedin (cid:98) X n has the same dimension m as (cid:98) X n , we deduce that ω ( (cid:98) X n ) ≥ ω ( M n ). Thus, by (3.7), theminimal volume entropy ω ( Z n ) of Z n goes to infinity. Remark 3.13.
Similar examples exist in odd dimensions but their construction is more tech-nical.
References [1] Adel’son-Vel’ski˘ı, G. M.; Shre˘ıder, Yu. A.: The Banach mean on groups (in Russian).
Usp. Mat. Nauk
Cited on page 18 [2] Anderson, J.; Paternain, G.: The minimal entropy problem for 3-manifolds with zero simplicial volume. In:Geometric methods in dynamics I,
Ast´erisque
286 (2003) 63–79.
Cited on page 2 [3] Babenko, I.: Asymptotic invariants of smooth manifolds.
Russian Acad. Sci. Izv. Math.
41 (1993), no. 1,1–38.
Cited on page 1, 3 [4] Babenko, I.: Topologie des systoles unidimensionnelles.
Enseign. Math. (2)
52 (2006), no. 1-2, 109–142.
Cited on page 3 [5] Babenko, I.: Addenda `a l’article intitul´e “Topologie des systoles unidimensionnelles”
Enseign. Math. (2)
Cited on page 3 [6] Babenko, I.; Sabourau, S.: Volume entropy semi-norm. See arXiv:1909.10803.
Cited on page 2, 3, 17, 23, 24 [7] Babenko, I.; Sabourau, S.: Minimal volume entropy of simplicial complexes. See arXiv:2002.11069v1.
Citedon page 2 [8] Balacheff, F.; Karam, S.: Macroscopic Schoen conjecture for manifolds with nonzero simplicial volume.
Trans. Amer. Math. Soc.
372 (2019), no. 10, 7071–7086.
Cited on page 2 [9] Besson, G.; Courtois, G.; Gallot, S.: Volume et entropie minimale des espaces localement sym´etriques.
Invent. Math.
103 (1991), no. 2, 417–445.
Cited on page 2 [10] Besson, G.; Courtois, G.; Gallot, S.; Sambusetti, A.: Curvature-Free Margulis Lemma for Gromov-Hyperbolic Spaces. See arXiv:1712.08386v3.
Cited on page 2 [11] Bregman, C.; Clay, M.: Minimal volume entropy of free-by-cyclic groups and 2-dimensional right-angledArtin groups. See arXiv:2008.08504.
Cited on page 5 [12] Bridson, M.; Haefliger, A.: Metric spaces of non-positive curvature.
Grundlehren der Mathematischen Wis-senschaften , vol. 319, Springer-Verlag, 1999.
Cited on page 22 [13] Brunnbauer, M.: Homological invariance for asymptotic invariants and systolic inequalities.
Geom. Funct.Anal.
18 (2008), no. 4, 1087–1117.
Cited on page 2, 3, 23 [14] Ceccherini-Silberstein, T.; Coornaert, M.: Cellular automata and groups.
Springer Monographs in Mathe-matics , Springer-Verlag, 2010.
Cited on page 18 [15] Cheeger, J.; Gromov, M.: Collapsing Riemannian manifolds while keeping their curvature bounded. I.
J.Differential Geom.
23 (1986), no. 3, 309–346.
Cited on page 13, 14 [16] Cheeger, J.; Gromov, M.: Collapsing Riemannian manifolds while keeping their curvature bounded. II.
J.Differential Geom.
32 (1990), no. 1, 269–298.
Cited on page 13, 14 [17] Connell, C.; Farb, B.: Minimal entropy rigidity for lattices in products of rank one symmetric spaces.
Comm. Anal. Geom.
11 (2003), no. 5, 1001–1026.
Cited on page 2 [18] Delzant, T.: Sous-groupes distingu´es et quotients des groupes hyperboliques.
Duke Math. J.
83 (1996),no. 3, 661–682.
Cited on page 21 [19] Dinaburg, E. I.: A connection between various entropy characterizations of dynamical systems.
Izv. Akad.Nauk SSSR Ser. Mat.
35 (1971) 324–366.
Cited on page 1 [20] Dyer, E.; Vasquez, A. T.: Some small aspherical spaces. Collection of articles dedicated to the memory ofHanna Neumann, III.
J. Austral. Math. Soc.
16 (1973), 332–352.
Cited on page 22 [21] Efremovich, V. A: On proximity geometry of Riemannian manifolds.
Uspekhi Mat. Nauk.
8, 5(57) (1953),189–191.
Cited on page 1 [22] Erschler, A.; Zheng, T.: Growth of periodic Grigorchuk groups.
Invent. Math.
219 (2020), no. 3, 1069–1155.
Cited on page 14 [23] Frigerio, R.; Moraschini. M.: Gromov’s theory of multicomplexes with applications to bounded cohomologyand simplicial volume.
Mem. Amer. Math. Soc. , to appear. See arXiv:1808.07307.
Cited on page 21
INIMAL VOLUME ENTROPY AND FIBER GROWTH 25 [24] Grigorchuk, R.: On Burnside’s problem on periodic groups.
Functional Anal. Appl.
14 (1980), no. 1, 41–43.
Cited on page 14 [25] Grigorchuk, R.: Degrees of growth of finitely generated groups and the theory of invariant means. (Russian)
Izv. Akad. Nauk SSSR Ser. Mat.
48 (1984), no. 5, 939–985. English translation:
Math. USSR-Izv.
25 (1985),no. 2, 259–300.
Cited on page 3, 14 [26] Grigorchuk, R.: An example of a finitely presented amenable group not belonging to the class EG . Sb.Math.
189 (1998), no. 1-2, 75–95.
Cited on page 3, 14, 15 [27] Grigorchuk, R.; de la Harpe, P.: On problems related to growth, entropy, and spectrum in group theory.
J.Dynam. Control Systems
Cited on page 3 [28] Gromov, M.: Groups of polynomial growth and expanding maps.
Inst. Hautes ´Etudes Sci. Publ. Math.
Cited on page 3 [29] Gromov, M.: Volume and bounded cohomology.
Inst. Hautes ´Etudes Sci. Publ. Math.
56 (1982), 5–99.
Citedon page 1, 2, 18, 23 [30] Gromov, M.: Filling Riemannian manifolds.
J. Differential Geom.
18 (1983), no. 1, 1–147.
Cited on page 3 [31] Gromov, M. Metric structures for Riemannian and non-Riemannian spaces.
Progr. in Mathematics , vol. 152,Birkh¨auser, 1999.
Cited on page 3, 19 [32] Guillemin, V.; Ginzburg, V.; Karshon, Y.: Moment maps, cobordisms, and Hamiltonian group actions.Appendix J by M. Braverman.
Mathematical Surveys and Monographs , 98. Amer. Math. Soc., 2002.
Citedon page 14 [33] Guth, L.: Volumes of balls in Riemannian manifolds and Uryson width.
J. Topol. Anal.
Cited on page 19 [34] Hatcher, A.: Algebraic topology. Cambridge University Press, 2002.
Cited on page 17, 19, 21, 23 [35] Higman, G.: Subgroups of finitely presented groups.
Proc. Roy. Soc. London Ser. A
262 (1961) 455–475.
Cited on page 14 [36] Ivanov, N.: Notes on the bounded cohomology theory. See arXiv:1708.05150.
Cited on page 18, 21 [37] Katok, A.: Entropy and closed geodesics.
Ergodic Theory Dynam. Systems
Citedon page 2 [38] Liokumovich, Y.; Lishak, B.; Nabutovsky, A.; Rotman, R.: Filling metric spaces. See arXiv:1905.06522.
Cited on page 19 [39] L¨oh, C.; Sauer, R.: Bounded cohomology of amenable covers via classifying spaces.
Enseign. Math.
Cited on page 21 [40] Lys¨enok, I. G.: A set of defining relations for the Grigorchuk group.
Math. Notes
38 (1985), no. 3-4, 784–792.
Cited on page 14 [41] Manning, A.: Topological entropy for geodesic flows.
Ann. of Math. (2)
110 (1979), no. 3, 567–573.
Cited onpage 1 [42] Manning, A.: Relating exponential growth in a manifold and its fundamental group.
Proc. Amer. Math.Soc.
133 (2005), no. 4, 995–997.
Cited on page 12 [43] Merlin, L.: Minimal entropy for uniform lattices in product of hyperbolic planes.
Comment. Math. Helv.
Cited on page 2 [44] Milnor, J.: A note on curvature and fundamental group.
J. Differential Geometry
Cited onpage 1 [45] Mineyev, I.: Straightening and bounded cohomology of hyperbolic groups.
Geom. Funct. Anal.
11 (2001),no. 4, 807–839.
Cited on page 21 [46] Nabutovsky, A.: Linear bounds for constants in Gromov’s systolic inequality and related results. SeearXiv:1909.12225.
Cited on page 19 [47] Papasoglu, P.: Uryson width and volume.
Geom. Funct. Anal.
30 (2020), no. 2, 574–587.
Cited on page 19 [48] Paternain, G.; Petean, J.: Minimal entropy and collapsing with curvature bounded from below.
Invent.Math.
151 (2003), no. 2, 415–450.
Cited on page 14 [49] Pieroni, E.: Minimal entropy of 3-manifolds. See arXiv:1902.09190
Cited on page 2 [50] Sabourau, S.: Systolic volume and minimal entropy of aspherical manifolds.
J. Differential Geom.
74 (2006),no. 1, 155–176.
Cited on page 2, 11 [51] Sabourau, S.: Small volume of balls, large volume entropy and the Margulis constant.
Math. Ann.
Cited on page 2 [52] Sambusetti, A.: Minimal entropy and simplicial volume.
Manuscripta Math.
99 (1999), no. 4, 541–560.
Citedon page 2 [53] Sambusetti, A.: On minimal entropy and stability.
Geom. Dedicata
81 (2000), no. 1-3, 261–279.
Cited onpage 2 [54] Su´arez-Serrato, P.: Minimal entropy and geometric decompositions in dimension four.
Algebr. Geom. Topol.
Cited on page 2 [55] ˇSvarc, A. S.: A volume invariant of coverings.
Dokl. Akad. Nauk SSSR (N.S.)
105 (1955), 32–34.
Cited onpage 1 [56] Wilson, J.: On exponential growth and uniform exponential growth for groups.
Invent. Math.
155 (2004)287–303.
Cited on page 3
Universit´e Montpellier II, CNRS UMR 5149, Institut Montpelli´erain Alexander Grothendieck,Place Eug`ene Bataillon, Bˆat. 9, CC051, 34095 Montpellier CEDEX 5, France
Email address : [email protected] Univ Paris Est Creteil, CNRS, LAMA, F-94010 Creteil, FranceUniv Gustave Eiffel, LAMA, F-77447 Marne-la-Vall´ee, France
Email address ::