Volume entropy for minimal presentations of surface groups in all ranks
VVOLUME ENTROPY FOR MINIMAL PRESENTATIONS OFSURFACE GROUPS IN ALL RANKS
LLU´IS ALSED `A, DAVID JUHER, J´ER ˆOME LOS AND FRANCESC MA ˜NOSAS
Abstract.
We study the volume entropy of a class of presentations(including the classical ones) for all surface groups, called minimal geo-metric presentations . We rediscover a formula first obtained by Cannonand Wagreich [6] with the computation in a non published manuscriptby Cannon [5]. The result is surprising: an explicit polynomial of de-gree n , the rank of the group, encodes the volume entropy of all classicalpresentations of surface groups. The approach we use is completely dif-ferent. It is based on a dynamical system construction following an ideadue to Bowen and Series [3] and extended to all geometric presenta-tions in [15]. The result is an explicit formula for the volume entropyof minimal presentations for all surface groups, showing a polynomialdependence in the rank n >
2. We prove that for a surface group G n ofrank n with a classical presentation P n the volume entropy is log( λ n ),where λ n is the unique real root larger than one of the polynomial x n − n − n − (cid:88) j =1 x j + 1 . Introduction
In the beginning of the 80’s several breakthroughs occurred in group the-ory. The main one was the development of large scale geometry for groups,largely due to M. Gromov with, for instance, the classification of groups withpolynomial growth function [14] or the introduction of the now standard no-tion of hyperbolic groups [13]. At about the same time R. Grigorchuck [12]found a class of groups with intermediate growth function . In all these classesof groups the growth function plays a central role. The growth function de-pends on the generating set X or on the presentation P = (cid:104) X/R (cid:105) of thegroup G . It is defined as the map N (cid:55)→ N such that n (cid:55)→ f G,P ( n ) = Card { g ∈ G : length X ( g ) ≤ n } . From the growth function f G,P several asymptotic functions are defined suchas the volume entropy or the growth series also called the
Poincar´e series . Date : June 12, 2014.1991
Mathematics Subject Classification.
Primary: 57M07, 57M05. Secondary: 37E10,37B40, 37B10.
Key words and phrases.
Surface groups, Bowen-Series Markov maps, topological en-tropy, volume entropy.The authors have been partially supported by MINECO grant numbers MTM2008-01486 and MTM2011-26995-C02-01. This work has been carried out thanks to the supportof the ARCHIMEDE Labex (ANR-11-LABX- 0033). a r X i v : . [ m a t h . D S ] J u l LLU´IS ALSED `A, DAVID JUHER, J´ER ˆOME LOS AND FRANCESC MA ˜NOSAS
The computational issues appeared also at about the same period. Anidea due to J. Cannon [7] allows an inductive way to describe geodesics inthe Cayley graph Cay ( G, P ) via the notion of cone types . This notion hasbeen intensively used later on by Epstein, Cannon, Levy, Holt, Patterson,Thurston [9] with the introduction of a very large class of groups, called au-tomatic , that contains the hyperbolic groups of Gromov. The computationof the growth function or the growth series becomes possible in principlefrom a geodesic automatic structure, when it exists. This is the case forhyperbolic groups. This computation, as it was noticed in [6], can also beobtained using the Floyd-Plotnick method [10].In practice, finding an explicit geodesic automatic structure from thepresentation is not so simple. For free groups with the free presentation allthe computations are easy and, for instance, the volume entropy is simplylog(2 n − n (see for instance [8]). The nextsimple case is the class of surface groups. For the classical presentationsof surface groups, the growth series appeared in a paper by Cannon andWagreich [6] without the explicit computation, leading to those series thatwere earlier obtained in a non published manuscript of Cannon [5]. Forhyperbolic groups, the existence of a geodesic automatic structure for eachpresentation implies that the growth series is a rational function (see [9, 7]).In this case the volume entropy (sometimes called the critical exponent) isrelated to the largest pole of the growth series, i.e. the largest root of thedenominator of the growth series (see for instance [4]). The result of Cannonand Wagreich for the classical presentations of surface groups shows that thedenominator Q n of the growth series is an explicit polynomial depending onthe rank n ≥ Q n ( x ) := x n − n − n − (cid:88) j =1 x j + 1 . The fact that a single, explicit polynomial could encode the volume en-tropy for all surface groups is mysterious a priori, specially since the originalcomputations of Cannon did not appear in published form.In this paper we rediscover the polynomial Q n ( x ) from a completely dif-ferent point of view and we hope that a part of the mystery will disappear.In our approach we compute the volume entropy of the group presentationsfrom a dynamical system argument based on an idea due to R. Bowen andC. Series [3] and generalized in [15].The original idea of Bowen and Series was to associate a specific mapΦ B − S : S −→ S to a particular action of the group G = π ( S ) on thehyperbolic plane H , where S is seen as the space at infinity of H . In[15], S is considered as the Gromov boundary of the group ∂G and a mapΦ P : S −→ S is constructed for each presentation P in a class, called geo-metric , characterized by the fact that the two dimensional Cayley complexCay ( G, P ) is planar. The maps Φ P are called Bowen-Series-Like and theysatisfy several interesting properties, in particular the volume entropy of thepresentation P equals the topological entropy of the map Φ P . In additionthe map Φ P admits a finite Markov partition and the computation of thetopological entropy for such maps is standard. OLUME ENTROPY FOR SURFACE GROUPS 3
For any surface S , the classical presentation of the corresponding surfacegroup Γ = π ( S ) is geometric. These classical presentations are given bythe minimal number of generators n and one relation of length 2 n . Fororientable surfaces, n is even and equals 2 g , where g is the genus of thesurface. In this case, the classical relation is a product of g commutators.In the non-orientable case, there is no restriction on the parity of n andthe relation is given by the product of the squares of all generators (seefor instance [18]). A presentation with the minimal number of generatorsis called minimal . The rank 2 cases (torus and Klein bottle) are, as usual,special: they are not hyperbolic, the growth function is quadratic and thusthe volume entropy is 0. For n > Q n ( x ) , r ≥
3. Wewill see that Q n ( x ) has a unique real root larger than one denoted λ n . Moreprecisely we prove: Theorem 1.1.
For n > , let Γ be a surface group of rank n with a minimalgeometric presentation P . Then, the volume entropy of Γ with respect to thepresentation P is log( λ n ) . Moreover, for n ≥ , λ n satisfies: n − − n − n − < λ n < n − . The above inequalities show that the difference between the volume en-tropy for the surface group and for the free group of the same rank is ex-plicitly very small.The genesis of this rediscovery is interesting. The dynamical system ap-proach discussed above allows to compute the volume entropy of any geomet-ric presentation P from an explicit Bowen-Series-Like map: Φ P : S −→ S .We developed an algorithm to compute the entropy of such maps, for theclassical presentations of orientable surfaces, via the well known kneadinginvariant technique of Milnor and Thurston [16]. The polynomial Q n ( x )appears that way in the computation for all orientable surfaces of genus g ≤
43. To obtain the theorem we needed to compute the determinant ofa matrix whose size grows either linearly in n with polynomial entries us-ing the Milnor-Thurston method, or quadratically in n with integer entriesusing the Markov matrix method. The computation leading to the proof ofthe theorem became possible by a succession of two surprises. First, by aparticular choice of a minimal presentation P + n the corresponding BSL mapΦ P + n admits an explicit symmetry of order 2 n . By a quotient process, theMarkov matrix is reduced to an integer matrix whose size grows linearlyin n . Then a method, developed in [2] under the nice name of the “Rometechnique”, was directly applicable to our case and reduced the computationto a 2 × P in some particular geometricpresentations. The map is then given explicitly for the particular minimalgeometric presentations with a symmetry property, together with its Markovpartition. In Section 3 we obtain a first formula for the volume entropy in LLU´IS ALSED `A, DAVID JUHER, J´ER ˆOME LOS AND FRANCESC MA ˜NOSAS the orientable case, in terms of the Markov matrix of the Bowen-Series-Likemap. We exploit the symmetry of the presentation to obtain a formula forthe volume entropy in terms of the spectral radius of a simpler matrix calledthe compacted matrix of rank n . In Section 4 we extend these results to thenon-orientable case by showing that the volume entropy in this case is alsothe logarithm of the spectral radius of the compacted matrix of rank n .The computation of the spectral radius of this new matrix is still somewhatdifficult. In Section 5 we obtain a simpler matrix with the same spectralradius. Finally, in Section 6, the “Rome method” is explained and appliedto compute this spectral radius, and proving Theorem 1.1.2. Bowen-Series-Like maps for geometric presentations
In this section we review the necessary ingredients for the construction ofthe Bowen-Series-Like maps defined in [15].2.1.
Geometric presentations.
Let P = (cid:104) X/R (cid:105) = (cid:10) x ± , x ± , . . . , x ± n /R , . . . , R k (cid:11) be a presentation of agroup Γ. Recall that the Cayley graph Cay (Γ , P ) is a metric space andlet B m be the ball of radius m centred at the identity. We denote thecardinality of any finite set A by | A | . The volume entropy of Γ with respectto the presentation P is denoted by h vol (Γ , P ) and defined as:lim m →∞ m log | B m | . A presentation P of a surface group Γ = π ( S ) is called geometric if theCayley 2-complex Cay (Γ , P ) is a plane. In particular the Cayley graphCay (Γ , P ) is a planar graph. A geometric presentation P is called minimal if the number of generators is minimal. For a group of an orientable surfaceof genus g it is well known that the minimal number of generators is 2 g (see [18] for instance) and, in this case, there is a presentation with a singlerelation of length 4 g . The standard classical presentation in this case is thefollowing: (cid:42) x ± , y ± , x ± , y ± , . . . , x ± g , y ± g / g (cid:89) i =1 [ x i , y i ] (cid:43) , where [ x i , y i ] = x i · y i · x − i · y − i is a commutator.For a rank n group of a non-orientable surface there is also a classicalpresentation with a single relation of length 2 n : (cid:42) x ± , x ± , . . . , x ± n / n (cid:89) i =1 x i (cid:43) . It is easy to check that such classical presentations are geometric (seebelow).Geometric presentations satisfy very simple combinatorial properties:
Lemma 2.1 (Floyd and Plotnick [10]) . If P = (cid:10) x ± , . . . , x ± n /R , . . . , R k (cid:11) isa geometric presentation of a surface group Γ then P satisfies the followingproperties: OLUME ENTROPY FOR SURFACE GROUPS 5 (a) The set { x ± , . . . , x ± n } admits a cyclic ordering that is preserved bythe Γ -action.(b) Each generator appears exactly twice (with plus or minus exponent)in the set R = { R , . . . , R k } of relations.(c) Each pair of adjacent generators, according to the cyclic ordering (a),appears exactly once in R and defines uniquely a relation R i ∈ R . The following statement is the main ingredient to compute the volumeentropy of a geometric presentation. The statement also contains the mainresult about minimal geometric presentations. In what follows S will denotea (topological) circle. Recall that any surface group Γ is Gromov-hyperbolic[13] and its boundary is: ∂ Γ (cid:39) S .Let us introduce the notion of a Markov partition. Let W be a finite set of S . An interval of S will be called W -basic if it is the closure of a connectedcomponent of S \ W . Observe that two different W -basic intervals havepairwise disjoint interiors. Let φ : S −→ S and let W ⊂ S be finite. Wesay that W is a Markov partition of φ if W is φ − invariant (i.e., φ ( W ) ⊂ W )and the image by φ of every basic interval is a union of basic intervals. Theorem 2.2 (Los [15]) . Let P be a geometric presentation of a surfacegroup Γ . Then there exists a map Φ P : ∂ Γ = S −→ ∂ Γ = S with the fol-lowing properties:(a) The map Φ P is Markov, i.e. it admits a finite Markov partition.(b) The topological entropy of Φ P , h top (Φ P ) , is equal to the volume en-tropy h vol (Γ , P ) .In addition, the volume entropy is minimal, among geometric presentations,for all minimal geometric presentations. The map Φ P satisfies more properties that are not needed here. Prop-erty (a) is specially interesting for computations since, for Markov maps,it is classical that the topological entropy is nothing but the logarithm ofthe spectral radius of a finite integer matrix, the Markov transition matrix(see [17] or [1] for instance). The goal of the next sections is to make sucha Markov partition explicit in the particular cases of minimal geometricpresentations.2.2. Construction of the Bowen-Series-Like map.
In this subsection we review the definition and the necessary propertiesof the BSL maps, in the particular case of minimal geometric presentations.2.2.1.
Bigons.
As we have seen, a presentation P = (cid:104) X/R (cid:105) defines the Cay-ley graph Cay (Γ , P ) and the Cayley 2-complex Cay (Γ , P ). A bigon inCay (Γ , P ) is a pair of distinct geodesics { γ , γ } connecting two vertices { v, v (cid:48) } ∈ Cay (Γ , P ). We denote by B v ( x, y ) the set of bigons { γ , γ } whose initial vertex is v and so that the geodesic γ starts at v by the edgelabelled x and γ starts at v by the edge labelled y , with x (cid:54) = y . By theΓ-action we can fix the initial vertex v to be the identity and we denote B id ( x, y ) by B ( x, y ).For geometric presentations of surface groups the set of bigons is partic-ularly simple. LLU´IS ALSED `A, DAVID JUHER, J´ER ˆOME LOS AND FRANCESC MA ˜NOSAS
Lemma 2.3. If P = (cid:104) X/R (cid:105) is a geometric presentation of a surface group Γ then the set of bigons B ( x, y ) is non empty if and only if ( x, y ) is an adjacentpair of generators, according to the cyclic ordering of Lemma 2.1(a). Inaddition, if ( x, y ) is an adjacent pair of generators there is a unique bigon β ( x, y ) ∈ B ( x, y ) of finite minimal length, called minimal bigon . This lemma is proved in [15, Lemmas 2.6 and 2.12]. Observe that eachminimal bigon is particularly simple for a geometric presentation where allrelations have even length. Indeed, each pair of adjacent generators ( x, y )defines a unique relation by Lemma 2.1(c). The relation can be written,up to cyclic permutation and inversion ( R i → ( R i ) − ), as: R i = γ · ( γ ) − ,where γ is a word (or a path) starting by the letter x , γ starts by the letter y and l ( γ ) = l ( γ ). The observation is that for geometric presentations thetwo paths γ and γ start at the identity and end at the same vertex (since R i is a relation) and are geodesics. In other words the pair { γ , γ } is abigon. It is minimal and unique by Lemma 2.1(c).2.2.2. Bigon-Rays.
We describe a canonical way to define a point on theboundary ∂ Γ associated to an adjacent pair of generators ( x, y ). Recall thata surface group is hyperbolic in the sense of Gromov [13] and its boundary ∂ Γ is the circle S . By definition of ∂ Γ, a point ξ ∈ ∂ Γ is the limit of geodesicrays, for instance starting at the identity, modulo the equivalence relationamong rays that two rays are equivalent if they stay at a uniform boundeddistance from each others (c.f. [13]). If ξ ∈ ∂ Γ is a point on the boundarywe denote by { ξ } a geodesic ray starting at identity and converging to ξ .In what follows, given two integers k and l we will denote k (mod l ) by[ k ] l . Also, we choose 1 , , . . . , l as the representatives of the classes modulo l ; that is, [0] l = [ l ] l = l. However, unless necessary we omit the modulo partin the notations.
Notation 2.4.
In what follows we denote the n generators (and their in-verses) by y , y , . . . , y n in such a way that y [ i ± n are the elements adjacentto y i with respect to the cyclic ordering from Lemma 2.1(a). We denote anadjacent pair by ( y i , y [ i +1] n ) where, by convention, the edges denoted y i and y [ i +1] n are adjacent and oriented from the vertex. We also adopt theconvention that y i is on the left of y [ i +1] n (see Figure 1). This conventiondefines an orientation of the plane Cay (Γ , P ).The parity of the number of adjacent pairs at each vertex implies that( y i , y [ i +1] n ) defines an opposite pair, with respect to the cyclic ordering ofLemma 2.1(a), defined by:( y i , y [ i +1] n ) opp := ( y [ i + n ] n , y [ i + n +1] n )(see Figure 2).We construct a unique infinite sequence of adjacent pairs, bigons andvertices from any given pair ( y i , y [ i +1] n ) by the following process: Step
1. Each adjacent pair, at the identity, defines a unique minimal bigon β ( y i , y i +1 ) by Lemma 2.3. The bigon β ( y i , y i +1 ) is a pair of geo-desics { γ l , γ r } , where the indices l, r stand for left and right, withrespect to an orientation of the plane Cay (Γ , P ). The geodesics { γ l , γ r } connect the identity to a vertex v = v [ β ( y i , y i +1 )] . OLUME ENTROPY FOR SURFACE GROUPS 7 y y y n ·· ·· ·· ·· ·· ·· ·· y n + y n + y n Figure 1.
The labelling of the generators (and the cyclicordering) fixed in Notation 2.4.
Step
2. The two geodesics { γ l , γ r } end at v by two generators that areadjacent by Lemma 2.3. Therefore the bigon β ( y i , y i +1 ) defines aunique adjacent pair at v , called a top pair of β ( y i , y i +1 ), whichis denoted: topp[ β ( y i , y i +1 )], based at v = v [ β ( y i , y i +1 )] and isuniquely defined by ( y i , y i +1 ) . Step
3. The pair topp[ β ( y i , y i +1 )] defines an opposite pair at v , denotedby: (cid:0) topp[ β ( y i , y i +1 )] (cid:1) ) opp := ( y i , y i +1 ) (1) . Step
4. We consider then the unique minimal bigon, at v , defined by thepair ( y i , y i +1 ) (1) by Lemma 2.3: β (1) ( y i , y i +1 ) := β v [( y i , y i +1 ) (1) ] . Step
5. The bigon β (1) ( y i , y i +1 ) defines a new top pair topp[ β (1) ( y i , y i +1 )],at the vertex v .The Steps 1–5 define, by induction, a unique infinite sequence of verticesand bigons (see Figure 2):id , v , v , · · · β ( y i , y i +1 ) , β (1) ( y i , y i +1 ) , β (2) ( y i , y i +1 ) · · · . (2)Each bigon in the infinite sequence (cid:8) β ( k ) ( y i , y i +1 ) : k ∈ N (cid:9) is a pair of geodesics (cid:110) γ ( k ) l , γ ( k ) r (cid:111) with k ∈ N connecting the vertices v k and v k +1 .By definition, the terminal vertex v k +1 of β ( k ) is the initial vertex of thenext bigon β ( k +1) in the sequence. Therefore a finite concatenation of bigons β (0) ( y i , y i +1 ) β (1) ( y i , y i +1 ) · · · β ( k ) ( y i , y i +1 ) makes sense. It is defined by thefinite collection of paths: (cid:110) γ (0) (cid:15) (0) · γ (1) (cid:15) (1) · · · γ ( k ) (cid:15) ( k ) : (cid:15) ( j ) ∈ { l, r } , j ∈ { , , . . . , k } (cid:111) . We denote the infinite concatenation of all these paths as: β ∞ ( y i , y i +1 ) := lim k →∞ β (0) ( y i , y i +1 ) β (1) ( y i , y i +1 ) · · · β ( k ) ( y i , y i +1 ) . Lemma 2.5 (Los [15, Lemma 3.1]) . With the above notation the followingstatements hold.
LLU´IS ALSED `A, DAVID JUHER, J´ER ˆOME LOS AND FRANCESC MA ˜NOSAS ( a, b ) ( a, b ) opp v v x yIdβ ( x, y ) β (1) g ( x, y ) (1) top pair at v Figure 2.
Opposite pair and bigon rays. (a) Each path in the collection: β (0) ( y i , y i +1 ) β (1) ( y i , y i +1 ) · · · β ( k ) ( y i , y i +1 ) is a geodesic segment, for all k ∈ N .(b) Two geodesic segments in (a) stay at a uniform distance from eachother for any k ∈ N . In consequence, the infinite concatenation β ∞ ( y i , y i +1 ) defines infinitelymany geodesic rays with a unique limit point in ∂ Γ . It will be denoted by( y i , y i +1 ) ∞ .2.2.3. Cylinders, definition of the BSL map.
We define the cylinder of lengthone as the subset of the boundary: C x := { ξ ∈ ∂ Γ : there is a geodesic ray { ξ } starting at id by x ∈ X } . Lemma 2.6.
Let P = (cid:104) X/R (cid:105) be a geometric presentation of Γ . The bound-ary ∂ Γ = S is covered by the cylinder sets C x , x ∈ X and:(a) Two cylinders have non-empty intersection: C x (cid:84) C y (cid:54) = ∅ if and onlyif ( x, y ) is an adjacent pair of generators.(b) Each cylinder C x , x ∈ X is a non trivial connected interval of ∂ Γ . This lemma is proved in [15, Lemmas 2.13 and 2.14]. Observe that thepoint ( y i , y i +1 ) ∞ of Lemma 2.5 belongs, by definition, to the intersection C y i (cid:84) C y i +1 .In what follows we consider the points in the circle ordered clockwise .That is, if r, s, t are pairwise different points of S we will write r < s < t if s belongs to the clockwise arc starting at r and ending at t . The notation r ≤ s ≤ t will also be used in the natural way. Then the interval [ r, t ] isdefined as the set (cid:8) s ∈ S : r ≤ s ≤ t (cid:9) . Also, if
I, J, K are closed connectedsubsets of S with pairwise disjoint interiors we will write I < J < K whenever r ≤ s ≤ t for every r ∈ I, s ∈ J and t ∈ K . Definition 2.7. If P = (cid:104) X/R (cid:105) is a geometric presentation of a hyperbolicsurface group Γ, then we denote by I y i the interval [( y i − , y i ) ∞ , ( y i , y i +1 ) ∞ ] . Clearly I y i is a subset of C y i for every y i ∈ X . OLUME ENTROPY FOR SURFACE GROUPS 9
We define the Bowen-Series-Like map Φ P : ∂ Γ −→ ∂ Γ byΦ P ( ξ ) = x − ( ξ ) if ξ ∈ I x ,where x − ( ξ ) is the action, by homeomorphism, on ∂ Γ by the group element x − .The map Φ P satisfies the following elementary properties:(i) It depends explicitly on the presentation P (the exact dependence willbe explained below).(ii) Since I x ⊂ C x , each ξ ∈ I x has a writing, as a limit of a ray, as { ξ } = x · ω. The image under Φ P is given by: { Φ P ( ξ ) } = { x − ( x · ω ) } = { ω } . In other words, the map Φ P is a shift map, on this particular writingas a ray.2.3. Markov partition for minimal geometric presentations.
Theorem 2.2 states that the map Φ P admits a Markov partition. In thissubsection we will define a particular presentation, which will be called sym-metric , and we will make the Markov partition explicit for this presentation.The first step is to define subdivision points in each interval I x , x ∈ X .Let us recall that the extreme points ( y, x ) ∞ and ( x, z ) ∞ of the intervals I x are limit points of bigon rays β ∞ ( y, x ) and β ∞ ( x, z ). Let us focus on ( y, x ) ∞ .Let β ∞ v ( y, x ) be the bigon ray starting at the vertex v ∈ Cay (Γ , P ). Observethat with this definition we can write:(3) β ∞ ( y, x ) = β ( y, x ) · β ∞ v [( y, x ) (1) ] , with the notations of Subsection 2.2.2.The particular property of a minimal geometric presentation that is usefulhere is that there is only one relation R of even length 2 n , when Γ is a surfacegroup of rank n . In this case, any bigon β ( y, x ) has the form { γ l , γ r } with γ l · ( γ r ) − being one of the words representing the relation R , up to cyclicpermutation and inversion. This word starts with the letter y and terminateswith the letter x − .Since the relation R has length 2 n , let us write the two paths { γ l , γ r } as:(4) { y · x (cid:48) i · · · x (cid:48) i n , x · x i · · · x i n } . We focus on the “ x ” side of Equations (3),(4), i.e. on the infinite collectionof rays:(5) x · x i · · · x i n · β ( ∞ ) v [( y, x ) (1) ] , where v is the group element written: v = x · x i · · · x i n . The vertices v = x and v j = x · x i · · · x i j , for j = 2 , , . . . , n − (Γ , P ) belong to γ r andare ordered along γ r (this notation is consistent with v = v n ).The following pairs of consecutive edges:(6) (cid:8) ( x, x i ) , ( x i , x i ) , . . . , ( x i n − , x i n ) (cid:9) at the vertices (cid:8) v , . . . , v n − (cid:9) , are crossed by the path γ r , where the notation x i j means the edge x i j with the opposite orientation. Observe that each pairof consecutive letters along the paths γ r are adjacent generators. Lemma 2.8.
If the relation defining β ( y, x ) has even length n then thecollection: (7) R xL := (cid:110) x · x i · · · x i j · β ( ∞ ) v j [( x i j , x i j +1 ) opp ] : j = 1 , . . . , n − (cid:111) , (see Figure 3) is called the left (with respect to x ) subdivision rays . Theysatisfy the following properties:(a) Each path in the infinite collection R xL is a ray starting at the iden-tity.(b) For a given j ∈ { , , . . . , n − } , all the rays in R ( x,j ) L = x · x i · · · x i j · β ( ∞ ) v j [( x i j , x i j +1 ) opp ] converge to the same point λ jx ∈ ∂ Γ .(c) For any j (cid:54) = p , the rays in R ( x,j ) L and in R ( x,p ) L have a common begin-ning: x · x i · · · x i ν where ν := min { j, p } and are otherwise disjoint.(d) Each λ jx , j ∈ { , , . . . , n − } belongs to the interior of the interval I x of Definition 2.7.(e) The limit points λ jx are inversely ordered with respect to the index j ∈ { , , . . . , n − } along ∂ Γ (that is, λ n − x < λ n − x < · · · < λ x < λ x ). This lemma is proved in [15, Lemma 4.1]. xIdy ( y, x ) ∞ subdivision raysadjacent pairsalong the path x i x i x i Figure 3.
Subdivision rays.We denote L x = { λ x , . . . , λ n − x } this set of left (with respect to x ) limitpoints. By the same analysis the adjacent pair ( x, z ) defines the set of right (with respect to x ) limit points R x = { ρ x , . . . , ρ n − x } , which are ordered withrespect to the superindex. Observe that we use here the fact that a minimalgeometric presentation has only one relation (of length 2 n ). Consider nowthe set of all such points:(8) S = (cid:91) x ∈ X ( R x ∪ L x ∪ ∂I x ) , called the subdivision points . OLUME ENTROPY FOR SURFACE GROUPS 11
Lemma 2.9. If P is a geometric presentation of a hyperbolic surface group Γ so that all relations have even length, then the set of subdivision points S is invariant under the map Φ P of Definition 2.7 and defines a finite Markovpartition of ∂ Γ . This statement is a particular case of [15, Theorem 4.3]. For a minimalgeometric presentation there is only one relation of length 2 n for a surfacegroup of rank n . In this case the partition of each interval I x above is givenby the points R x ∪ L x ∪ ∂I x which are ordered in the following way: λ nx := ( y, x ) ∞ < λ n − x < · · · < λ x < ρ x < λ x < ρ x < · · · < ρ nx := ( x, z ) ∞ . We also observe here that the intervals I x are ordered, along S , by the(cyclic) ordering of the generators at the identity. Then, we can define apartition of each of the intervals I x consisting on the following subintervals: L jx = (cid:2) λ jx , λ j − x (cid:3) and R jx = (cid:2) ρ j − x , ρ jx (cid:3) , for j ∈ { , , . . . , n } , C Lx = (cid:2) λ x , ρ x (cid:3) and C Rx = (cid:2) λ x , ρ x (cid:3) , and C x = (cid:2) ρ x , λ x (cid:3) . (9)Recall that a subdivision point (left or right) has the following writing: { λ jx } = x · x i · · · x i j · β ∞ v j [( x i j , x i j +1 ) opp ] , for j ∈ { , , . . . , n } .Since the map Φ P acts, on each interval I x , on the ray writing as a shiftmap we obtain: { Φ P ( λ x ) } = β ∞ [( x, x i ) opp ] , and { Φ P ( λ jx ) } = x i · · · x i j · β ∞ x i ··· x ij [( x i j , x i j +1 ) opp ] for j ∈ { , , . . . , n } (10)and there is a similar writing for the points ρ jx . Lemma 2.10. If P is a geometric presentation of a surface group with allrelations of even length then the image of the central interval C x = [ ρ x , λ x ] under Φ P is a single interval I u , u ∈ X , where u is the generator that isopposite to x − for the cyclic ordering of Lemma 2.1(a) at the vertex x .Proof. By (10) we observe that { Φ P ( λ x ) } = β ∞ [( x, x i ) opp ] and similarly { Φ P ( ρ x ) } = β ∞ [( x (cid:48) i , x ) opp ] . Since the two adjacent pairs ( x, x i ) and ( x (cid:48) i , x )are adjacent at the vertex v = x then the two opposite pairs ( x, x i ) opp and( x (cid:48) i , x ) opp are also adjacent. That means that they share one edge u . Thisedge is just the one that is opposite to x − at the vertex x (see Figure 4). (cid:3) Next we define a particular presentation, which we call symmetric , for therank n group π ( S ), where S is an orientable surface. Recall that n = 2 g ,where g is the genus of S . Definition 2.11.
Given a surface group π ( S g ) of rank n = 2 g , where S g isorientable of genus g, the presentation (cid:10) x ± , x ± , . . . , x ± n (cid:14) x x · · · x n x − x − · · · x − n (cid:11) will be called symmetric and denoted by P + n . Proposition 2.12.
The symmetric presentation P + n is minimal and geo-metric. Idxy x i x i u Figure 4.
Central Interval.
Proof.
Consider the polygon ∆ n with 2 n sides, labelled by the elements of { x ± , x ± , . . . , x ± n } in the ordering: x · x · · · x n · x − · x − · · · x − n . The identification of the side labelled x i with the one labelled x − i defines anorientable surface of genus g . The identification is an equivalence relation ∼ and ∆ n / ∼ is the surface of genus g . The presentation P + n is minimalsince it has n generators and it is geometric because the universal cover ofthe surface ∆ n / ∼ is nothing but the Cayley 2-complex Cay (Γ , P + n ) that isa plane. (cid:3) Lemma 2.1 says that for geometric presentations the generators have acyclic ordering at each vertex. For the presentation P + n the cyclic orderingis x < x − < x < x − < · · · < x n − < x − n < x − < x < · · · < x − n − < x n . The topological entropy of the map Φ P + n The aim of this section is to start the computation of the topologicalentropy of the Bowen-Series-Like map Φ P + n for the symmetric presentation P + n = (cid:104) X/R (cid:105) of the orientable surface group of rank n .Since the surface is orientable and the presentation is geometric and min-imal, then all generators x ∈ X act on ∂ Γ as an orientation preservinghomeomorphism. By Definition 2.7(ii), Φ P (cid:12)(cid:12) I x is an orientation preservinghomeomorphism for every x ∈ X and from Lemma 2.9 the set S defines aMarkov partition of Φ P . Since ∂I x i ⊂ S we also have that Φ P is a homeo-morphism on every S− basic interval.In this situation the topological entropy can be easily computed as thelogarithm of the spectral radius of the associated Markov matrix . Let usrecall such result.Let W be a Markov partition of a map φ : S −→ S , and let U , U , . . . ,U | W | be a labelling of the W − basic intervals. The Markov matrix of W is OLUME ENTROPY FOR SURFACE GROUPS 13 x x x x n ·· ·· ·· ·· ·· ·· ·· x x n Figure 5.
The cyclic ordering of the symmetric presentation.defined as the | W | × | W | (0 , − matrix M = ( m ij ) | W | i,j =1 such that m ij = 1 ifand only if φ ( U i ) ⊃ U j .For any square matrix M , we will denote its spectral radius by ρ ( M ).It is well known (see for instance [2] or [1, Theorem 4.4.5]), that if φ ismonotone on each basic interval then(11) h top ( φ ) = log max { ρ ( M ) , } . We will use (11) to compute h top (Φ P + n ) . To this end we first have tocompute the Markov matrix of S that, in what follows, will be denotedby M + n . As we will see, a direct computation of ρ (cid:0) M + n (cid:1) is infeasible at apractical level because the size of the matrix grows quadratically with n .So, the computation of ρ (cid:0) M + n (cid:1) will be done in two steps by using spectralradius preserving transformations of the matrix M + n . In this section wewill compute the Markov matrix M + n for a symmetric presentation in theorientable case.To do this, we need to specify completely the map Φ P + n and then computeits Markov matrix. Recall that, for the symmetric presentation P + n (seecomments after Proposition 2.12), the cyclic ordering of Lemma 2.1 at anyvertex is given by: x < x − < · · · < x n − < x − n < x − < x < · · · < x − n − < x n < x (see Figure 5). The main property of this cyclic ordering which makesthe symmetric presentation very special and useful is that the edge that isopposite to x at any vertex is simply the edge x − .The above cyclic ordering of the generators induces the following orderingof the intervals I x along the boundary ∂ Γ = S : I x < I x − < · · · < I x − n < I x − < I x < · · · < I x n < I x . The fact that the symmetric presentation has associated the above cyclicordering gives the following immediate corollary of Lemma 2.10:
Corollary 3.1.
Let P + n be the symmetric presentation of an orientable sur-face group of rank n . Then, Φ P + n ( C x ) = I x for each generator x . For notational reasons we denote the ordered generators as y < y < · · · < y n < y
14 LLU´IS ALSED `A, DAVID JUHER, J´ER ˆOME LOS AND FRANCESC MA ˜NOSAS and the corresponding intervals as I y < I y < · · · < I y n < I y , where y i = x ( − i +1 i for 1 ≤ i ≤ n , and y i = x ( − i i − n for n + 1 ≤ i ≤ n. Also,the fact that the edge that is opposite to x at any vertex is the edge x − now gives(12) y − i = y [ i + n ] n . Observe from (9) that each of the 2 n intervals I y i is divided into 2 n − L ny i < · · · < L y i < C Ly i < C y i < C Ry i < R y i < · · · < R ny i . Hence, |S| = 2 n (2 n −
1) and thus, the matrix M + n is 2 n (2 n − × n (2 n − . Equations (10), (12) and Corollary 3.1 give the following images of thepartition intervals defined in (9) (see Figure 6):Φ P + n ( L jy i ) = L j − y [ i + n +1]2 n for j ∈ { , , . . . , n } , Φ P + n ( L y i ) = C Ly [ i + n +1]2 n ∪ C y [ i + n +1]2 n , Φ P + n ( C Ly i ) = C Ry [ i + n +1]2 n ∪ n (cid:91) j =2 R jy [ i + n +1]2 n ∪ [ i − n (cid:91) k =[ i + n +2] n I y k , Φ P + n ( C y i ) = I y i , Φ P + n ( C Ry i ) = C Ly [ i + n − n ∪ n (cid:91) j =2 L jy [ i + n − n ∪ [ i + n − n (cid:91) k =[ i +1] n I y k , Φ P + n ( R y i ) = C y [ i + n − n ∪ C Ry [ i + n − n , Φ P + n ( R jy i ) = R j − y [ i + n − n for j ∈ { , , . . . , n } . (14)From the formulae (14) it follows that the Markov matrix M + n has astructure in blocks, all of size (2 n − × (2 n − . So, it is convenient towrite the matrix M + n as(15) M M . . . M , n M M . . . M , n . . . . . . . . . . . . . . . . . . . . . . . . . . .M n M n . . . M n, n . . . . . . . . . . . . . . . . . . . . . . . . . . .M n, M n, . . . M n, n where each of the matrices M lt = ( m ltij ) n − i,j =1 is of size (2 n − × (2 n − . Accordingly, we will label the basic intervals contained in I y i as U ij in sucha way that they preserve the ordering given in (13). So, for i = 1 , , . . . , n OLUME ENTROPY FOR SURFACE GROUPS 15 I y [ i + n + ] n I y − i = I y [ i + n ]2 n I y [ i + n − ] n L j y i C L y i C y i C R y i R j y i I y i Figure 6.
The intervals I y i in the circle together with the in-terior intervals. The outer curve is the image Φ P + n (cid:12)(cid:12) I yi (whichis order preserving). The intervals L jy i , R jy i and their imagesare drawn with a continuous black line, L y i , R y i and theirimages are drawn with a dotted line, C Ly i , C Ry i and their im-ages are drawn with a continuous thick black line and finally, C y i and its image are drawn with a continuous thick greyline.and j = 1 , , . . . , n − U ij = L ( n +1) − jy i for j = 1 , , . . . , n − ,C Ly i for j = n − ,C y i for j = n,C Ry i for j = n + 1 ,R j − ( n − y i for j = n + 2 , n + 3 , . . . , n − . With this labelling we define the matrix M + n so that m ltij = 1 if and only if φ ( U li ) ⊃ U tj .The next theorem is a first reduction in the effective computation of h top (Φ P + n ). Figure 7.
The first three (of the total of eight) block rowsof the Markov matrix M P +4 corresponding to the symmetricpresentation of an orientable surface group of rank 4. Theorem 3.2. h top (Φ P + n ) = log max (cid:110) ρ (cid:0) M + n (cid:1) , (cid:111) = log max (cid:40) ρ (cid:32) n (cid:88) k =1 M k (cid:33) , (cid:41) . An ( r, s ) − block circulant matrix is a matrix of the form A A A . . . A r A r A A . . . A r − A r − A r A . . . A r − . . . . . . . . . . . . . . . . . . . . . . . . . .A A A . . . A where each A i is an s × s matrix. Notice that a circulant matrix is completelydetermined by its first block row ( A A A . . . A r ).The next lemma will be crucial in effectively computing the spectral radiusof M + n (see Figure 7 for an example in the case of rank 4). Lemma 3.3.
The Markov matrix M + n is a (2 n, n − − block circulantmatrix.Proof. From the formulae (14) it follows that Φ P + n ( U li ) ⊃ U tj if and only ifΦ P + n ( U [ l +1] n i ) ⊃ U [ t +1] n j . In terms of the Markov matrix this amounts to m ltij = m [ l +1] n , [ t +1] n ij for every l, t ∈ { , , . . . , n } and i, j ∈ { , , . . . , n − } . This implies that M lt = M [ l +1] n , [ t +1] n . (cid:3) The next technical lemma provides a nice and useful result about thespectral radius of block circulant matrices.
OLUME ENTROPY FOR SURFACE GROUPS 17
Lemma 3.4.
Let A = A A A . . . A r A r A A . . . A r − A r − A r A . . . A r − . . . . . . . . . . . . . . . . . . . . . . . . . .A A A . . . A be a non-negative block circulant matrix. Then ρ ( A ) = ρ (cid:32) r (cid:88) i =1 A i (cid:33) . Proof.
Since A is a block matrix, for every m ≥ A m is a block matrix A ( m )11 A ( m )12 . . . A ( m )1 r A ( m )21 A ( m )22 . . . A ( m )2 r . . . . . . . . . . . . . . . . . . . . . . .A ( m ) r A ( m ) r . . . A ( m ) rr where each block has size s × s and is a sum of r m − non-commutativeproducts of m matrices among the blocks A , A , A , . . . , A r . That is, each A ( m ) ij is the sum of r m − matricial products of the form A r A r · · · A r m with r , r , . . . , r m ∈ { , , . . . , r } . Moreover, since every block A i appears exactly once in every block rowand every block column it can be proved by induction that every productof the form A r A r · · · A r m appears exactly once in every block row andevery block column of A m . Therefore, for every q ∈ { , , . . . , r } , (cid:80) ri =1 A ( m ) qi and (cid:80) ri =1 A ( m ) iq are the sum of r m matricial products, and every product A r A r · · · A r m appears exactly once in each of these expressions. Hence, r (cid:88) i =1 A ( m ) qi = r (cid:88) i =1 A ( m ) iq = (cid:32) r (cid:88) i =1 A i (cid:33) m . Consequently, the classical matrix norms satisfy: (cid:107) A m (cid:107) ∞ = (cid:107) ( (cid:80) ri =1 A i ) m (cid:107) ∞ and (cid:107) A m (cid:107) = (cid:107) ( (cid:80) ri =1 A i ) m (cid:107) . Since the spectral radius is defined as: ρ ( A ) = lim m →∞ (cid:107) A m (cid:107) /m for anymatrix norm, it follows that ρ ( A ) = ρ ( (cid:80) ri =1 A i ) . (cid:3) Remark 3.5.
In fact the above lemma holds for every matrix for which agiven block appears exactly once in every block row and every block column.Now we are endowed with the necessary ingredients to prove Theorem 3.2.
Proof of Theorem 3.2.
It follows from (11) and Lemmas 3.3 and 3.4. (cid:3)
Next, to complete the first reduction in the computation of h top (Φ P + n ),we will give an explicit formula for the matrix (cid:80) nk =1 M k . The compacted matrix of rank n is the (2 n − × (2 n −
1) matrix C n = ( c ij )defined by(17) c ij = j = i + 1 and i ∈ { , , . . . , n − } , j ∈ { n − , n } and i = n − ,n − j ∈ { , , . . . , n } and i = n − ,n − j ∈ { n + 1 , n + 2 , . . . , n − } and i = n − , i = n,n − j ∈ { , , . . . , n − } and i = n + 1 ,n − j ∈ { n, n + 1 , . . . , n − } and i = n + 1 , j ∈ { n, n + 1 } and i = n + 2 , j = i − i ∈ { n + 3 , n + 4 , . . . , n − } , and0 otherwise.In matrix form, C n is · · · · · · · · · · · · ... ... ... . . . · · · ... ... ... ... ... ... ... ...... ... ... · · · . . . ... ... ... ... ... ... ... ... · · · · · · n − n − n − · · · n − n − n − n − n − · · · n − n − n −
11 1 1 · · · · · · n − n − n − · · · n − n − n − n − n − · · · n − n − n −
20 0 0 · · · · · · ... ... ... ... ... ... ... ... . . . · · · ... ... ...... ... ... ... ... ... ... ... · · · . . . ... ... ... · · · · · · · · · · · ·
Lemma 3.6. n (cid:88) k =1 M k = C n . To prove this lemma we need to explicitly describe the matrix M + n . Tothis end we introduce the following notation. The zero matrix of size k × k will be denoted by k , and J k will denote the k × k (0 , − matrix with onesin the anti-diagonal. Also, if i ∈ { , , . . . , k } , U ik will denote the k × k matrix such that all entries in the i − th row are 1 and all other entries are0. Finally, for k ≥ T k = ( t ij ) is the k × k (0 , − matrix such that t ij = 1 if and only if (see Figure 8 for examples): • j = i + 1 and i ∈ { , , . . . , (cid:101) k − } , or • j ∈ { (cid:101) k − , (cid:101) k } and i = (cid:101) k − • (cid:101) k + 1 ≤ j ≤ k and i = (cid:101) k − , where (cid:101) k = k +12 . Observe that (see again Figure 8) J k T k J k is the matrixobtained from T k by a symmetry with respect to the central coordinate t (cid:101) k, (cid:101) k . Proof of Lemma 3.6.
From formulae (14) and taking into account the la-belling of the basic intervals (16) it follows that m ltij = 1 if and only if (see OLUME ENTROPY FOR SURFACE GROUPS 19 U = T = J = J T J = Figure 8.
Examples of the matrices U ik , T k , J k and J k T k J k with k = 7.Figure 7 for an example in the case of rank 4): j = i + 1 , t = [ l + n + 1] n for i = 1 , , . . . , n − ,j ∈ { n − , n } , t = [ l + n + 1] n for i = n − ,n +1 ≤ j ≤ n − , t =[ l + n +1] n , and j ∈{ , ,..., n − } , [ l + n +2] n ≤ t ≤ [ l − n (cid:27) for i = n − ,j ∈ { , , . . . , n − } , t = l for i = n,j ∈{ , ,...,n − } , t =[ l + n − n , and j ∈{ , ,..., n − } , [ l +1] n ≤ t ≤ [ l + n − n (cid:27) for i = n + 1 ,j ∈ { n, n + 1 } , t = [ l + n − n for i = n + 2 ,j = i − , t = [ l + n − n for i = n + 3 , . . . , n − . In matrix block form the above formulae become (see again Figure 7): M l, [ l + n +1] n = T n − M lt = U n − n − for [ l + n + 2] n ≤ t ≤ [ l − n M ll = U n n − M lt = U n +12 n − for [ l + 1] n ≤ t ≤ [ l + n − n M l, [ l + n − n = J n − T n − J n − M l, [ l + n ] n = n − for l ∈ { , , . . . , n } . Consequently, n (cid:88) t =1 M t = T n − + J n − T n − J n − + U n n − + ( n − (cid:0) U n − n − + U n +12 n − (cid:1) = C n . (cid:3) The next corollary gives an explicit formula for the entropy in the ori-entable case in terms of the spectral radius of a (2 n − × (2 n −
1) matrixwhich is a “compacted” version of the Markov matrix M + n . Corollary 3.7. h top (Φ P + n ) = log max { ρ ( C n ) , } . Proof.
It follows from Theorem 3.2 and Lemma 3.6. (cid:3)
Remark 3.8.
Note that the map Φ P + n commutes with a rigid rotation R of period 2 n. The quotient space obtained by identifying each orbit of R toa point is a circle. The map induced by Φ P + n on this quotient space is alsoa Markov map. The matrix C n is nothing but the Markov matrix of thisinduced map (see (14) and Figure 6).4. The non-orientable case
We start this section by extending the definition of symmetric presentation (Definition 2.11) to non orientable surface groups.
Definition 4.1.
Given a surface group Γ = π ( S ) of rank n , where S is a nonorientable surface, the following presentation of Γ will be called symmetric and denoted by P − n . Its definition depends on the parity of n as follows. For n odd, we define P − n as (cid:10) x ± , x ± , . . . , x ± n / x x · · · x n x n − x n − · · · x x n (cid:11) while, for n even, P − n is defined as (cid:10) x ± , x ± , . . . , x ± n (cid:14) x x · · · x n x n − x n − · · · x x − n (cid:11) . Similar arguments to the ones used in the proof of Proposition 2.12 yieldthat the symmetric presentation P − n is minimal and geometric.As in the orientable case, the nomenclature symmetric for the presenta-tion P − n accounts for the fact that, at each vertex, the cyclic ordering of thegenerators (Lemma 2.1) exhibits the useful property that the edge oppositeto x at any vertex is simply the edge x − . Indeed, one can check that theordering of the generators at any vertex is x < x − < x < · · · < x n − < x − n < x − < x < x − < · · · < x − n − < x n when n is even, and x < x − < x < · · · < x − n − < x n < x − < x < x − < · · · < x n − < x − n when n is odd.The fact that the symmetric presentation has associated the above cyclicordering implies that Corollary 3.1 also holds for the non-orientable case: Corollary 4.2.
Let P − n be the symmetric presentation of a non-orientablesurface group of rank n. Then, Φ P − n ( C x ) = I x for each generator x . Notice that map Φ P + n and the Markov matrix M + n are only defined for n even since the group corresponds to an orientable surface. However, allassociated formulae extend to the case n odd. In this sense below we will OLUME ENTROPY FOR SURFACE GROUPS 21
Figure 9.
The first three (of the total of six) block rowsof the Markov matrix M P − corresponding to the symmetricpresentation of a non-orientable surface group of rank 3.compare the maps Φ P + n and Φ P − n and the associated Markov matrices M + n and M − n , independently on the parity of n. Using the notations introduced in the previous sections one can check thatthe Markov map Φ P − n behaves essentially as Φ P + n in all intervals I y i exceptwhen i ∈ { n, n } . In these two intervals the map reverses orientation. So,when i / ∈ { n, n } the equation (14) holds with Φ P − n instead of Φ P + n . When i = n, Φ P − n ( L jy n ) = R j − y n − for j ∈ { , , . . . , n } , Φ P − n ( L y n ) = C y n − ∪ C Ry n − , Φ P − n ( C Ly n ) = C Ly n − ∪ n (cid:91) j =2 L jy n − ∪ (cid:32) n − (cid:91) k = n +1 I y k (cid:33) , Φ P − n ( C y n ) = I y n , Φ P − n ( C Ry n ) = C Ry ∪ n (cid:91) j =2 R jy ∪ (cid:32) n − (cid:91) k =2 I y k (cid:33) , Φ P − n ( R y n ) = C Ly ∪ C y , Φ P − n ( R jy n ) = L j − y for j ∈ { , , . . . , n } , (18)and analogous formulae hold for the interval I y n . Hence, in a similar wayto the previous section it follows that the Markov matrix M − n of Φ P − n isof the form (15) with M i,j replaced by J n − M i,j for i ∈ { n, n } and j =1 , , . . . , n (see Figure 9).Next we want to prove Lemma 4.3 which is an analogue of Lemma 3.4for this case. This will allow us to simplify the computation of the spectralradius of M − n . As expected, a further consequence of Lemmas 4.3 and 3.4 willbe that the orientation-reversing character of Φ P − n in the intervals I y n and I y n has no effects in the entropy. To do this it is convenient to introduce the notion of disoriented block circulant matrix as follows. An ( r, s ) − disorientedblock circulant matrix is a matrix of the form A = A A . . . A r A A . . . A r . . . . . . . . . . . . . . . . . . . .A r A r . . . A rr where each A ij is an s × s matrix for which there exists an ( r, s ) − blockcirculant matrix (cid:101) A = A A . . . A r (cid:101) A (cid:101) A . . . (cid:101) A r . . . . . . . . . . . . . . . . . . . . (cid:101) A r (cid:101) A r . . . (cid:101) A rr such that given i ∈ { , . . . , r } , either A ij = (cid:101) A ij for every j = 1 , , . . . , r or A ij = J s (cid:101) A ij for every j = 1 , , . . . , r. That is, every block row of A coincideswith the corresponding block row of (cid:101) A or is obtained from the correspondingblock row of (cid:101) A by pre-multiplying each block by J s . Observe that this lastoperation permutes the individual rows of the block row symmetrically withrespect to the central horizontal axis. The matrix (cid:101) A will be called the parallelization of A . Observe that the assumption that the first block rowof A and (cid:101) A coincide implies that the parallelization of A is unique. Lemma 4.3.
Let A = A A . . . A r A A . . . A r . . . . . . . . . . . . . . . . . . . .A r A r . . . A rr be a non-negative disoriented ( r, s ) − block circulant matrix such that (19) r (cid:88) j =1 A j J s = J s r (cid:88) j =1 A j . Then ρ ( A ) = ρ r (cid:88) j =1 A j . Proof.
Let (cid:101) A = (cid:101) A (cid:101) A . . . (cid:101) A r (cid:101) A (cid:101) A . . . (cid:101) A r . . . . . . . . . . . . . . . . . . . . (cid:101) A r (cid:101) A r . . . (cid:101) A rr be the unique parallelization of A. We will prove that (cid:107) A m (cid:107) ∞ = (cid:107) (cid:101) A m (cid:107) ∞ forevery m ≥
0. Then, ρ ( A ) = lim m →∞ (cid:107) A m (cid:107) /m ∞ = lim m →∞ (cid:107) (cid:101) A m (cid:107) /m ∞ = ρ ( (cid:101) A ) , and the result follows from Lemma 3.4. OLUME ENTROPY FOR SURFACE GROUPS 23
For every m ∈ N we will write A m and (cid:101) A m as A ( m )11 A ( m )12 . . . A ( m )1 r A ( m )21 A ( m )22 . . . A ( m )2 r . . . . . . . . . . . . . . . . . . . . . . .A ( m ) r A ( m ) r . . . A ( m ) rr and (cid:101) A ( m )11 (cid:101) A ( m )12 . . . (cid:101) A ( m )1 r (cid:101) A ( m )21 (cid:101) A ( m )22 . . . (cid:101) A ( m )2 r . . . . . . . . . . . . . . . . . . . . . . . (cid:101) A ( m ) r (cid:101) A ( m ) r . . . (cid:101) A ( m ) rr , respectively. Then, (cid:107) A m (cid:107) ∞ = max i =1 , ,...,r (cid:13)(cid:13)(cid:13) r (cid:88) j =1 A ( m ) ij (cid:13)(cid:13)(cid:13) ∞ = max i =1 , ,...,r r (cid:88) j =1 A ( m ) ij u s and, similarly, (cid:107) (cid:101) A m (cid:107) ∞ = max i =1 , ,...,r r (cid:88) j =1 (cid:101) A ( m ) ij u s , where u s denotes the (column) vector of size s with all the entries equal to1. So, to prove the lemma it is enough to show that(20) r (cid:88) j =1 A ( m ) ij u s = r (cid:88) j =1 (cid:101) A ( m ) ij u s for every i = 1 , , . . . , r and m ∈ N . Before proving this claim we will prove two necessary technical resultson the block rows of the matrix A that are stronger versions of assumption(19). The first one is the following:(21) r (cid:88) j =1 A ij J s = J s r (cid:88) j =1 A ij and r (cid:88) j =1 A ij w s = r (cid:88) j =1 A rj w s for every non-negative vector w s of size s such that J s w s = w s and every i, r ∈ { , , . . . , r } . From the definitions of disoriented block circulant matrix, parallelizationand block circulant matrix we get:(22) r (cid:88) j =1 A j = r (cid:88) j =1 (cid:101) A j = r (cid:88) j =1 (cid:101) A ij and (cid:80) rj =1 A ij is either (cid:80) rj =1 (cid:101) A ij or J s (cid:16)(cid:80) rj =1 (cid:101) A ij (cid:17) . Thus, from (22) and(19) we get, respectively, r (cid:88) j =1 A ij = (cid:40) (cid:80) rj =1 (cid:101) A ij = (cid:80) rj =1 A j J s (cid:16)(cid:80) rj =1 (cid:101) A ij (cid:17) = J s (cid:16)(cid:80) rj =1 A j (cid:17) = (cid:16)(cid:80) rj =1 A j (cid:17) J s . Consequently, if w s is a non-negative vector of size s such that J s w s = w s , we have (cid:16)(cid:80) rj =1 A ij (cid:17) w s = (cid:16)(cid:80) rj =1 A j (cid:17) w s for every i ∈ { , , . . . , r } . Thisproves the second equality of (21). To prove the first one we use the aboveexpression for (cid:80) rj =1 A ij and (19). In the first case we have (cid:16)(cid:80) rj =1 A ij (cid:17) J s = (cid:16)(cid:80) rj =1 A j (cid:17) J s = J s (cid:16)(cid:80) rj =1 A j (cid:17) = J s (cid:16)(cid:80) rj =1 A ij (cid:17) , and in the second one, (cid:16)(cid:80) rj =1 A ij (cid:17) J s = J s (cid:16)(cid:80) rj =1 A j (cid:17) J s = J s J s (cid:16)(cid:80) rj =1 A j (cid:17) = J s (cid:16)(cid:80) rj =1 A ij (cid:17) . This ends the proof of (21).The second technical result that we need is a weaker version of (21) butthat holds for all powers of A . More precisely, for every i, r ∈ { , , . . . , r } and m ∈ N , (23) J s r (cid:88) j =1 A ( m ) ij u s = r (cid:88) j =1 A ( m ) rj u s . In fact this implies that (cid:16)(cid:80) rj =1 A ( m ) (cid:96)j (cid:17) u s is a vector independent on (cid:96), callit w ms , such that J s w ms = w ms .For m = 1 , (23) follows directly from (21) and the fact that J s u s = u s .Now we assume that (23) holds for m ≥ m + 1. Clearly,(24) r (cid:88) j =1 A ( m +1) ij = r (cid:88) j =1 (cid:32) r (cid:88) (cid:96) =1 A i(cid:96) A ( m ) (cid:96)j (cid:33) = r (cid:88) (cid:96) =1 A i(cid:96) r (cid:88) j =1 A ( m ) (cid:96)j for every i ∈ { , , . . . , r } . Thus, from the induction hypothesis and (21), J s (cid:16)(cid:80) rj =1 A ( m +1) ij (cid:17) u s = J s (cid:80) r(cid:96) =1 A i(cid:96) (cid:16)(cid:80) rj =1 A ( m ) (cid:96)j (cid:17) u s = J s (cid:0) (cid:80) r(cid:96) =1 A i(cid:96) (cid:1) w ms = (cid:0) (cid:80) r(cid:96) =1 A i(cid:96) (cid:1) J s w ms = (cid:0) (cid:80) r(cid:96) =1 A i(cid:96) (cid:1) w ms = (cid:0) (cid:80) r(cid:96) =1 A r(cid:96) (cid:1) w ms = (cid:80) r(cid:96) =1 A r(cid:96) (cid:16)(cid:80) rj =1 A ( m ) (cid:96)j (cid:17) u s = (cid:16)(cid:80) rj =1 A ( m +1) rj (cid:17) u s . This completes the induction step and, thus, (23) is proved.Now we will prove formula (20) by induction on m for a fixed but arbitrary i ∈ { , , . . . , r } . Assume that m = 1 . If A ij = (cid:101) A ij for j = 1 , , . . . , r then theequality is trivially true. Otherwise, A ij = J s (cid:101) A ij for j = 1 , , . . . , r. Hence,from (21), (cid:16)(cid:80) rj =1 A ij (cid:17) u s = (cid:16)(cid:80) rj =1 A ij (cid:17) ( J s u s ) = J s (cid:16)(cid:80) rj =1 A ij (cid:17) u s = J s (cid:16)(cid:80) rj =1 J s (cid:101) A ij (cid:17) u s = (cid:16)(cid:80) rj =1 (cid:101) A ij (cid:17) u s , because J s is an involution (i.e. J s is the identity of size s ).Assume that (20) holds for m ≥ . As above, we will consider two cases.If A i(cid:96) = (cid:101) A i(cid:96) for (cid:96) = 1 , , . . . , r, then from (24), the analogous formula for (cid:101) A and the induction assumption we get (cid:16)(cid:80) rj =1 A ( m +1) ij (cid:17) u s = (cid:80) r(cid:96) =1 A i(cid:96) (cid:16)(cid:80) rj =1 A ( m ) (cid:96)j (cid:17) u s = (cid:80) r(cid:96) =1 (cid:101) A i(cid:96) (cid:16)(cid:80) rj =1 (cid:101) A ( m ) (cid:96)j (cid:17) u s = (cid:16)(cid:80) rj =1 (cid:101) A ( m +1) ij (cid:17) u s . OLUME ENTROPY FOR SURFACE GROUPS 25
So, we are left with the case A i(cid:96) = J s (cid:101) A i(cid:96) for (cid:96) = 1 , , . . . , r. In a similar wayto the previous case we get (cid:16)(cid:80) rj =1 A ( m +1) ij (cid:17) u s = J s (cid:80) r(cid:96) =1 (cid:101) A i(cid:96) (cid:16)(cid:80) rj =1 (cid:101) A ( m ) (cid:96)j (cid:17) u s = J s (cid:16)(cid:80) rj =1 (cid:101) A ( m +1) ij (cid:17) u s . Consequently, from (23) we get: (cid:16)(cid:80) rj =1 A ( m +1) ij (cid:17) u s = J s (cid:16)(cid:80) rj =1 A ( m +1) ij (cid:17) u s = (cid:16)(cid:80) rj =1 (cid:101) A ( m +1) ij (cid:17) u s . This ends the proof of the lemma. (cid:3)
With the help of Lemma 4.3, as in the orientable case we obtain:
Corollary 4.4. h top (Φ P − n ) = log max { ρ ( C n ) , } . Computation of the topological entropy — secondreduction: Super compacting the matrix C n The super compacted matrix of rank n is the n × n matrix SC n = ( s ij )defined as follows:(25) s ij = i ≤ n − j = i + 1 or i = n ,2 if i = n − j = n ,2 n − i = n − j < n ,2 n − i = n − j = n , and0 otherwise.In matrix form we have: SC n = · · · · · · · · · ... ... ...... ... ... · · · . . . ... ... ...0 0 0 0 · · · · · · n − n − n − n − · · · n − n − n −
41 1 1 1 · · ·
In this section we prove
Proposition 5.1.
For every n ≥ , max { ρ ( C n ) , } = max { ρ ( SC n ) , } . To prove the above result we need another intermediate matrix which weobtain from C n . We introduce the divided compacted matrix of rank n of size n × n, denoted by DC n = ( d ij ) , which we define as follows:(26) d ij = c ij if i < n and j ≤ n , c i,j − if i < n and j ≥ n + 1, c i − ,j if i > n + 1 and j ≤ n , c i − ,j − if i > n + 1 and j ≥ n + 1,1 if i = n and j ≤ n or i = n + 1 and j ≥ n + 1,0 if i = n and j ≥ n + 1 or i = n and j ≤ n ,where C n = ( c ij ) . In matrix form, DC n is (compare with the definition ofthe matrix C n in Page 18): · · · · · · · · · · · · ... ... ... ... · · · ... ... ... ... ... ... ... ... ...... ... ... · · · ... ... ... ... ... ... ... ... ... ... · · · · · · n − n − n − · · · n − n − n − n − n − n − · · · n − n − n −
11 1 1 · · · · · · · · · · · · n − n − n − · · · n − n − n − n − n − n − · · · n − n − n −
20 0 0 · · · · · · ... ... ... ... ... ... ... ... ... ... · · · ... ... ...... ... ... ... ... ... ... ... ... ... · · · ... ... ... · · · · · · · · · · · ·
Notice that the matrix DC n is indeed the Markov matrix of a topologicalmodel obtained by subdividing the central interval of the topological modelfrom Remark 3.8 at a fixed point (that exists because the central intervalcovers itself). Proof of Proposition 5.1.
First we will prove thatSpec( DC n ) = Spec( C n ) ∪ { } , where Spec( · ) denotes the set of eigenvalues of a matrix. To do this observethat 1 is an eigenvalue of DC n with eigenvector (0 , , . . . , , , − , . . . , , , where the two non-zero elements of this vector are at the n and n + 1 entries.Also, if µ is an eigenvalue of C n with eigenvector ( v , v , . . . , v n − ) , then,it follows directly from the definition of the matrix DC n that µ is also aneigenvalue of DC n with eigenvector (cid:0) v , v , . . . , v n − , v n , v n , v n +1 , . . . , v n − (cid:1) . Conversely, if µ is an eigenvalue of DC n with eigenvector ( v , v , . . . , v n ) , then, again from the definition of the matrix DC n , it follows that µ is alsoan eigenvalue of C n with eigenvector( v , v , . . . , v n − , v n + v n +1 , v n +2 , . . . , v n ) . This proves the statement.The second step of the proof will be to show that ρ ( DC n ) = ρ ( SC n ) . Todo this it is convenient to write the matrix DC n in block form, with blocksof size n × n : DC n = (cid:18) D D D D (cid:19) . OLUME ENTROPY FOR SURFACE GROUPS 27
Observe that DC n is symmetric with respect to the central point. So, D = J n D J n and D = J n D J n , which amounts to: DC n = (cid:18) D D J n D J n J n D J n (cid:19) . Now let us consider the block matrix of size 2 n × n defined by Z := (cid:18) I n n n J n (cid:19) , where I n denotes the identity matrix of size n × n. Clearly Z is non-singularand Z − = Z . Hence, Z DC n Z − = (cid:18) D D J n D J n D (cid:19) because J n J n = I n . Observe that Z DC n Z − is a non-negative block circulantmatrix. Thus, ρ ( DC n ) = ρ (cid:0) Z DC n Z − (cid:1) = ρ ( D + D J n )by Lemma 3.4. Moreover, by direct inspection it follows that D + D J n = SC n . Summarizing, we have provedmax { ρ ( C n ) , } = max { ρ ( DC n ) , } = max { ρ ( SC n ) , } . (cid:3) Remark 5.2.
The topological model in Remark 3.8 has a fixed point andcommutes with the symmetry of degree -1 with respect to this fixed point.The quotient space obtained by identifying each orbit of the symmetry toa point is a closed interval, and the induced map on this quotient space isalso a Markov map. The matrix SC n is in fact the Markov matrix of thisquotient map.6. The spectral radius of SC n and proof of Theorem 1.1 To compute the spectral radius of SC n we will use the rome method pro-posed in [2]. To this end we have to introduce some notation.Let M = ( m ij ) be a k × k matrix. Given a sequence p = ( p j ) (cid:96)j =0 ofelements of { , , . . . , k } we define the width of p , denoted by w ( p ) , as thenumber (cid:81) (cid:96)j =1 m p j − p j . If w ( p ) (cid:54) = 0 then p is called a path of length (cid:96) . Thelength of a path p will be denoted by (cid:96) ( p ). A loop is a path such that p (cid:96) = p i.e. that begins and ends at the same index.A subset R of { , , . . . , k } is called a rome if there is no loop outside R ,i.e. there is no path ( p j ) (cid:96)j =0 such that p (cid:96) = p and { p j : 0 ≤ j ≤ (cid:96) } is disjointfrom R . For a rome R a path ( p j ) (cid:96)j =0 is called simple if p i ∈ R for i = 0 , (cid:96) and p i / ∈ R for i = 1 , , . . . , (cid:96) − R = { r , r , . . . , r (cid:96) } is a rome of a matrix M then we define an (cid:96) × (cid:96) matrix-valued real function M R ( x ) by setting M R ( x ) = ( a ij ( x )), where a ij ( x ) = (cid:80) p w ( p ) · x − (cid:96) ( p ) , where the summation is over all simple pathsoriginating at r i and terminating at r j . · · · n − nn −
11 1 1 1 12 1 1 2 n − n − n − Figure 10.
The combinatorial graph associated to SC n . Thearrows ending at braces indicate multiple arrows with thesame weight, each one directed towards a node under thebrace.
Theorem 6.1 (Theorem 1.7 of [2]) . If R is a rome of cardinality (cid:96) of a k × k matrix M then the characteristic polynomial of M is equal to ( − k − (cid:96) x k det( M R ( x ) − I (cid:96) ) . To use Theorem 6.1 it is helpful to represent the matrix M in form of acombinatorial graph which amounts to draw all paths of length 1 associatedto M. To do this we introduce the following notation. A path ( i, j ) of length1 will be written as i w −→ j, where w denotes the width of the path. Forthe matrix M the width w of the path i w −→ j, is just the entry ( M ) i,j (cid:54) = 0.Observe that, with this notation, a path p = ( p j ) kj =0 is written as p w −→ p w −→ · · · p k − w k − −−−→ p k and w ( p ) = (cid:81) k − i =0 w i .We will compute the spectral radius of SC n in Proposition 6.3 by usingTheorem 6.1. In Figure 10 we show the combinatorial graph associated to SC n . Remark 6.2.
The combinatorial graph associated to SC n shown in Fig-ure 10 is in fact the generalized Markov graph of the topological modelobtained in Remark 5.2. Proposition 6.3.
The spectral radius of SC n is the largest root of the poly-nomial Q n ( x ) . Proof.
By direct inspection of the graph of Figure 10 it follows that SC n isan irreducible non-negative integer matrix. Hence, by the Perron-FrobeniusTheorem (see [11]), we get that the spectral radius of SC n is the largesteigenvalue of the characteristic polynomial of SC n . It is larger than 1 andsimple. So, to prove the theorem, we have to show that the characteristicpolynomial of SC n is Q n ( x ) . OLUME ENTROPY FOR SURFACE GROUPS 29
Clearly R = { n − , n } is a rome of SC n (see Figure 10). Hence, M R ( x ) = (cid:18) β (cid:0) x − + z ( x ) (cid:1) ( β − x − + 2 βz ( x ) x − + z ( x ) x − + 2 z ( x ) (cid:19) where β = 2 n − , z ( x ) := (cid:80) n − (cid:96) =2 x − (cid:96) . By Theorem 6.1, the characteristic polynomial of SC n is( − n − x n (cid:12)(cid:12)(cid:12)(cid:12) β (cid:0) x − + z ( x ) (cid:1) − β (cid:0) x − + z ( x ) (cid:1) − ( β + 1) x − x − + z ( x ) 2 (cid:0) x − + z ( x ) (cid:1) − x − − (cid:12)(cid:12)(cid:12)(cid:12) = x n (cid:16)(cid:0) x − − β − (cid:1)(cid:0) x − + z ( x ) (cid:1) + x − + 1 (cid:17) = x n (cid:0) x − − (2 n − (cid:1) (cid:32) n − (cid:88) (cid:96) =1 x − (cid:96) (cid:33) + x n − + x n = x n − (2 n − n − (cid:88) j =1 x j + 1 . This ends the proof of the proposition. (cid:3)
Next we prove a technical lemma that studies the polynomial (1) andgives the bounds for λ n . Lemma 6.4.
For every n ≥ , Q n ( x ) has a unique real root λ n larger thanone. Moreover, for n ≥ , n − − n − n − < λ n < n − . Proof.
Observe that Q n (0) = 1 , Q n (1) = − n ( n − < Q (cid:48) n ( x ) = n x n − − n − n n − (cid:88) j =1 jx j − ≤ n (cid:18) x n − − n − n (cid:19) . Since x n − − n − n is negative for every for n ≥ x ∈ [0 , , it followsthat Q n has a unique root in (0 , Q n ( x ) = x n Q n ( x − ) (that is, Q n is a reciprocal polynomial ) and, hence, z is a root of Q n if and only if z − isa root of Q n . Consequently, Q n ( x ) has a unique real root larger than one.Also, Q n (2 n −
1) = (2 n − n − n −
1) (2 n − n − (2 n − n −
1) + 1 = 2 n. So, λ n < n − . To end the proof of the lemma it is enough to show that Q n (cid:16) n − − n − n − (cid:17) < n ≥ . We have Q n ( z ) = z n − n − s z n − z n − s − − z n − n ( n − s + 2 n − n − s − , where z := 2 n − − s and s := (2 n − n − . Since 2( n − s − > n − > , z n − n ( n − s > Q n ( z ) < . An exercise shows that z n > (2 n − n − n (2 n − . Hence, z n − n ( n − s > (2 n − n − (cid:18) − n (2 n − n − (cid:19) and z n − n ( n − s > n ≥ . So, Q n ( z ) < (cid:3) Proof of Theorem 1.1.
It follows from Theorem 2.2, Corollaries 3.7 and 4.4,Propositions 5.1 and 6.3, and Lemma 6.4. (cid:3)
References [1] Llu´ıs Alsed`a, Jaume Llibre, and Micha(cid:32)l Misiurewicz.
Combinatorial dynamics andentropy in dimension one , volume 5 of
Advanced Series in Nonlinear Dynamics . WorldScientific Publishing Co. Inc., River Edge, NJ, second edition, 2000.[2] Louis Block, John Guckenheimer, Micha(cid:32)l Misiurewicz, and Lai Sang Young. Periodicpoints and topological entropy of one-dimensional maps. In
Global theory of dynamicalsystems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979) , volume819 of
Lecture Notes in Math. , pages 18–34. Springer, Berlin, 1980.[3] Rufus Bowen and Caroline Series. Markov maps associated with Fuchsian groups.
Inst. Hautes ´Etudes Sci. Publ. Math. , (50):153–170, 1979.[4] D. Calegari. The ergodic theory of hyperbolic groups.
Contemp. Math. , 597:15–52,2013.[5] J. W. Cannon. The growth of the closed surface groups and the compact hyperboliccoxeter groups. 1980.[6] J. W. Cannon and Ph. Wagreich. Growth functions of surface groups.
Math. Ann. ,293(2):239–257, 1992.[7] James W. Cannon. The combinatorial structure of cocompact discrete hyperbolicgroups.
Geom. Dedicata , 16(2):123–148, 1984.[8] Pierre de la Harpe.
Topics in geometric group theory . Chicago Lectures in Mathemat-ics. University of Chicago Press, Chicago, IL, 2000.[9] David B. A. Epstein, James W. Cannon, Derek F. Holt, Silvio V. F. Levy, Michael S.Paterson, and William P. Thurston.
Word processing in groups . Jones and BartlettPublishers, Boston, MA, 1992.[10] William J. Floyd and Steven P. Plotnick. Growth functions on Fuchsian groups andthe Euler characteristic.
Invent. Math. , 88(1):1–29, 1987.[11] F. R. Gantmacher.
The theory of matrices. Vols. 1, 2 . Translated by K. A. Hirsch.Chelsea Publishing Co., New York, 1959.[12] R. I. Grigorchuk. On the Milnor problem of group growth.
Dokl. Akad. Nauk SSSR ,271(1):30–33, 1983.[13] M. Gromov. Hyperbolic groups. In
Essays in group theory , volume 8 of
Math. Sci.Res. Inst. Publ. , pages 75–263. Springer, New York, 1987.[14] Mikhael Gromov. Groups of polynomial growth and expanding maps.
Inst. Hautes´Etudes Sci. Publ. Math. , (53):53–73, 1981.[15] J´erˆome Los. Volume entropy for surface groups via Bowen-Series-like maps.
J. Topol. ,7(1):120–154, 2014.[16] John Milnor and William Thurston. On iterated maps of the interval. In
Dynamicalsystems (College Park, MD, 1986–87) , volume 1342 of
Lecture Notes in Math. , pages465–563. Springer, Berlin, 1988.[17] Michael Shub.
Stabilit´e globale des syst`emes dynamiques , volume 56 of
Ast´erisque .Soci´et´e Math´ematique de France, Paris, 1978. With an English preface and summary.
OLUME ENTROPY FOR SURFACE GROUPS 31 [18] John Stillwell.
Classical topology and combinatorial group theory , volume 72 of
Grad-uate Texts in Mathematics . Springer-Verlag, New York, second edition, 1993.
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