Torsions of 3-Dimensional Small Covers

Jiming MA Fangting ZHENG

1 Introduction

1.1 Small covers

Small covers,or Coxeter orbifolds,were studied by Davis and Januszkiewicz[7].They are a class of manifolds which admit locally standard Zn2-actions,such that the orbit spaces are n-dimensional simple polyhedra.The algebraic and topological properties of a small cover are closely related to the combinatorics of the orbit polyhedron and the coloring on its boundary.In this paper,we focus on the 3-dimensional case.

definition 1.1LetPbe a3-dimensional polytope,Γbe a trivalent graph in∂Pwhich gives a cell decomposition of∂P.A-coloring is a mapλ :∂P −Γ →such thatλ(fi1),λ(fi2)andλ(fi3)generatewhenandare sharing a common vertex,wherefijis a connected component in∂P−Γandis the corresponding closure.

From a-coloring λ and the trivial principal-bundle over P,we can get a 3-manifold which depends only on the coloring λ.Preparing eight copies of P,namely P ×,then a quotient space M(P,λ)can be constructed under the following equivalent relation:

Here Gfis the subgroup generated by λ(fi1),···,λ(fik),whereis the only i-face,0≤ i≤ 2,that contains x as an interior point.It is easy to see that M(P,λ)is a closed 3-manifold and we call it a small cover over P.

For example,if we consider a coloring on a tetrahedron that the four faces are colored by e1,e2,e3and e1+e2+e3respectively,then following the construction above,we can get a closed orientable 3-manifold RP3.It should be noticed that a tetrahedron admits a unique right-angled spherical structure.And those spherical structures on the four copies of the tetrahedron are glued together to form the unique spherical structure on RP3.

Choi-park[6]once discussed the torsions of real topological toric manifolds.For any positive odd number q,they constructed a real topological toric manifold N whose integral cohomology has a q-torsion.For a large q,the manifold being constructed will be of large dimension.What’s more,they gave a formula for the cohomology groups of real topological toric manifolds with coefficient Zp.Even though not being stated explicitly,from[6,Theorem 4.6]we can see that there are no odd torsions in a 3-dimensional small cover.Letting P be a 3-dimensional polytope and M=M(P,λ)be any small cover over P with at most 2-torsion in cohomology,Trevisan[22]gave out all the possible integral homology and cohomology groups.And it is still unknown about the existence of any 2k-torsion for k≥2 in a 3-dimensional small cover.In this paper,we show the following result.

Theorem 1.1LetM=M(P,λ)be a3-dimensional small cover.Then there are onlyZ2-torsions inH1(M;Z).

1.2 Asymptotic behaviors of mod 2 Betti numbers of finite covers

Let G be an in finite group,Gi＜G be a sequence of finite-index subgroups of G.If Gi+1＜ Gi,then we sayis a tower of G.thenis co- final.If Gi◁ G,thenis a regular sequence of G.

The asymptotic behavior of algebraic invariants in finite covers Miof a 3-manifold M depends on the sequenceof M.For any co- final regular towerof a hyperbolic 3-manifoldquals to the L2-Betti number of H3.And it is zero as shown in[17],namely the normalized first Betti number converges to zero for co- final regular towers.But this is not true for all the co- final sequences of M,see Gir˜ao[11,Theorem 3.1]as well as[9,13]for related topics.And Gir˜ao[10–11]also studied the rank gradients of some hyperbolic 3-manifolds.

There are many works on the asymptotic behavior of homology torsions in finite covers of a 3-manifold,see[4,14,16,18,21].In particular,it is conjectured that torsion growth of a co- final normal subgroup sequence of a hyperbolic 3-manifold M is related to the volume of M(see[4,18]).It is also conjectured that exponential torsion growth for any sequence(might not be normal,even not co- final)of a fi bered 3-manifold Nφis related to the virtual homology entropy of φ(see[14]).We show the following theorem.

Theorem 1.2LetMbe a small cover over a right-angled hyperbolic polytope.ThenMhas a co- final finite-cover sequenceMisuch thatH1(Mi;Z2)has exponential growth.

Theorem 1.2 can be compared with Theorem 1.2 of[15],where Lackenby proved that any finitely generated,discrete,non-elementary subgroup of PSL(2,C)with torsions has a tower of subgroups with linear-increasing mod p homologies for some prime p.See also[19,Theorem 2.2]as well as an example in[5]where the closed hyperbolic 3-manifold M has a normal cofinal sequence Misuch that H1(Mi;Z3)is alwaysIn[8,p.64],it is stated that“At the same time,very deep recent work of Wise on quasi-convex hierarchies combined with a theorem of Lackenby implies that for every hyperbolic 3-manifold group G and every prime p,the pgradient of G is zero”.The p-gradient of G is defined aswhere H runs over all subnormal subgroups of finite p-power index in G and dp(H)is the rank ofSo Theorem 1.2 shows that there are differences between subnormal sequence and general sequence in considering their p-gradients.

2 Preliminaries

2.1 Small cover

For an n-dimensional simple polytope P in Lobachevski n-space Hn,Davis and Januszkiewicz showed that if there is a 2n-index torsion-free subgroup Γ of the Coxeter group over P,then the manifold corresponding to this subgroup,namely the Clifford-Klein space form Hn/Γ,is a small cover over P.It is a G-manifold with group action.

Moreover,there is another equivalent but more practical way in describing small cover by using the language of coloring:Let F(P)={F1,F2,···,Fm}be the set of all co-dimensional one faces of P.Such face is named as facet.Then we define a-coloring characteristic function

where λ(F1),λ(F2),···,λ(Fn)generatewhen the facets F1,F2,···,Fnshare a common vertex.This condition is called the non-singular condition.And the corresponding characteristic matrix is defined to be the matrix obtained by placing the image of facets F1,···,Fmunder λ column by column.By the way,such function λ is not destined to exist and its existence is concerned with the Buchstaber invariant.

If characteristic function λ is defined successfully,then we can construct manifold M(P,λ):=by the following equivalent relation:

where f=Fi1∩···∩Fin−k,0≤k≤ n−1,is the unique co-dimensional(n−k)-face that contains x as an interior point,and Gfis the subgroup generated by λ(Fi1),λ(Fi2),···,λ(Fin−k).M(P,λ)is called a small cover over P.For example,defining a Z22-coloring characteristic function λ on the square as show in Figure 1,where(1,0)=e1,(0,1)=e2are the standard basis of Z22.

Figure 1 A coloring on the square.

Then by gluing the four pieces together along the facets according to the equivalent relation,we can finally get the Klein bottle.

3 Torsions in Small Covers

In this section,we show that there are only 2-torsions in H1(M;Z)for a 3-dimensional small cover M,which can be viewed as a refinement of Theorem 3.1 of[7].We start from a construction in[7].

Lemma 3.1There is a presentation matrixHforH1(M;Z),whose non-trivial entries are either2or−2.Moreover,there are at most two non-trivial entries in each row.If a row has exactly two non-trivial entries,then they must be2and−2.

Proof Let P be a 3-dimensional polytope and we embed P in R3.Choosing a vectorµin R3which is generic to P.Then adopting a function φ :R3→ R by φ(x)= 〈x,µ〉,where 〈x,µ〉is the inner product.Now using φ(x)as a height function,we can get a directed graph on the 1-skeleton of P.

There is a unique vertex,such that all the three adjacent edges point away from it.We denote this vertex by I and this is the unique“bottom”vertex.There is also a unique vertex,such that all the three adjacent edges point towards it.We denote this vertex by T and this is the unique“top” vertex.For other vertices,say O,there are two possibilities:

Case 1 We have two of the three edges adjacent to O point away from it while one points towards it.By h-vector and some simple combinatorial analysis,it is easy to see there are totally m−3 such type of vertices,where m=|F(P)|.We denote these vertices by V1,V2,V3,···,Vm−3.

Case 2 We have one of the three edges adjacent to O points away from it while the other two point towards it.It is easy to see that there are still m−3 vertices of this type.We denote them by W1,W2,W3,···,Wm−3.

For each vertex Vi,we take Eito be the unique closed edge that runs towards Vi.defining G to be the union of all of these Ei.Then G is a connected graph in the 1-skeleton of P which contains I and does not contain T.We use a cube D to explain all these notions in Figure 2.

Figure 2 Directed graph of a cube D and its union of Ei.

Considering π−1(G)in M,where π :M → P is the projection map.Now in the small cover M=M(P,λ),π−1(G)is a graph which is a double of G along the vertex set(see[7,Lemma 1.3].Namely,as shown in Figure 3,there are two copies of G,which are denoted by G′and G′′respectively,in the corresponding small cover M.Their vertices are marked byAnd their edges are labeled byand

Figure 3 G′,G′′and π−1(G)of cube D.

For each edge Eiwith respect to Vi,we label the colorings of faces adjacent to Viby α,β and γ.Among them, α and β are the two colorings of faces that are adjacent to Ei.Then Ei×{1},Ei× {α},Ei×{β}and Ei×{α+β}are glued together to form an edge in π−1(G).This edge is what we denote byin π−1(G).Another edge in π−1(G)with the representative Ei×{γ}is exactly what we mean byin π−1(G).

Now for each vertex Wj,there is a 2-cell corresponding to it.Assume that the two edges running towards Wjare l′and l′′.The face containing l′and l′′is Fj.And the colorings of the two faces which are adjacent to l′and l′′respectively are µ and ν.In 3-manifold M,four copies of Fj− (∂Fj− l′− l′′)are glued together to build an open embedded disk along pre-images of l′and l′′under π,then we denoted this disk by Dj.Illustrations about all these descriptions are shown in Figures 4–5.

Figure 4 Denotation illustrations.

Furthermorean open 3-ball,which is the union of eight copies of P−Cl(∂P−U1−U2−U3),where U1,U2,U3are the three faces adjacent to the vertex T.

Now H1(M;Z)can be obtained by quotienting Djout from H1(π−1(G)),whereis a basis of H1(π−1(G);Z)and each Djgives a relation.For a vertex Viin Fj,where Fjis the face corresponding to the vertex Wj,we haveEi×{µ+ν}− Ei×{ν},following the locating relations as shown in Figure 5.

Figure 5 Building up a disk.

Now the matrixis a presentation matrix of H1(M;Z).We will furtherlyfigure out that the non-trivial entries of

We picture the relative locations of Eiand Fjas well as some related colorings in Figure 6,here Eiis on the boundary of Fj,namely Fjwould contribute to the relation for quotient.We firstly adjust the colorings of the three facets adjacent to Vito be e1,e2and e3.This can be realized by simply performing a suitable coordinate transformation.And the other two faces adjacent to Wjare denoted as d and e.

Figure 6 Relative locations with fixed coloring basis.

We list out all the possible colorings on face d and face e in Table 1 based on the non-singular condition.There are totally 24 cases.The colorings of face e placed on the right of a certain row are the only four choices when face d is colored by the coloring placed in the left column of that row.For example,when d is colored by e3,then e can only be colored by e1,e1+e2,e1+e3and e1+e2+e3.

Table 1 All possible colorings for(d,e)and corresponding

Table 1 All possible colorings for(d,e)and corresponding

Colorings on face d Colorings on face c and corresponding??∂Wj∂Vi??e3 e1 e1+e2 e1+e3 e1+e2+e3 0 0 2(E′− E′′) 2(E′− E′′)e1 e3 e1+e3 e2+e3 e1+e2+e3 0 0 0 0 e1+e2 e3 e1+e3 e2+e3 e1+e2+e3 0 0 0 0 e1+e3 e1 e3 e2+e3 e1+e2 0 2(E′− E′′) 2(E′− E′′) 0 e2+e3 e1 e1+e2 e1+e3 e1+e2+e3 0 0 2(E′− E′′) 2(E′− E′′)e1+e2+e3 e1 e3 e2+e3 e1+e2 0 2(E′− E′′) 2(E′− E′′) 0

And then we can discuss all the possibleThe results are placed just below the coloring cases respectively as shown in Table 1.

Therefore,using the notations claimed before and marking them in Figure 7,we can make the conclusion as follows.

Figure 7 Relative locations for general case.

(1)Ifµ ∈ {α,β,α +β},then

(2)If ν∈ {α,β,α + β},thenas well.

(3)If{µ,ν} ∩ {α,β,α + β}= Ø, µ and ν both lie in{γ,α + γ,β + γ,α + β + γ},soµ + ν ∈ {α,β,α + β}.And then in π−1(G),Ei× {1}=Ei× {µ + ν},Ei× {µ}=Ei× {ν}.Thus Ei× {1}−Ei×{µ}+Ei×{µ+ν}− Ei× {ν}=2(Ei×{1}−Ei×{µ}).

We always see P from the outside,which means that we always orient the boundary of Fj,the face corresponds to Wj,anti-o’clockly.For the edge Ei,which corresponds to a vertex Viin ∂Fj,its orientation may or may not be the same with the orientation derived form ∂Fj.So we should add either plus or minus sign to the absolute value of.Namely we have

Therefore,in the presentation matrixthe non-trivial entries are either 2 or−2.There are only two faces,denoted by Fj1and Fj2,that are adjacent to Ei.If the orientation of Eiagrees with the orientation of∂Fj1,then it will de finitely disagree with the orientation of∂Fj2.Thus there are at most two non-trivial entries in each row.Furthermore,if a row possesses two non-trivial entries,then they must be 2 and−2.

Proof of Theorem 1.1 By transposing H and rearranging the rows,we get a new matrix(A(m−3)×m1|B(m−3)×m2|C(m−3)×m3),where m1+m2+m3=m−3.Here A is a zero matrix,B has only one non-trivial entry,2 or−2 in each column,and C is a matrix with exactly two non-trivial entries,2 and−2,in every column.

We first multiply(−1)if necessary to make all the non-trivial entries in B to be 2.Furthermore we suitably replace some column i by column i+(±1)×column j,where m1+1≤i≤m−3 m1+1≤j≤m1+m2,and reorder the columns to obtain a new matrix,also called by H,in the following form

where each column of C1has exactly two non-trivial entries(see[20,Chapter 8]).

Let C1=(ci,j),and assuming c1,1=2 and c1,2= −2.We add C1’s first row to its second row and get a matrix C2.Moreover by adding the first column of C2(might times with−1)to some other columns of C2,we obtain a matrix,denoted also by C2,of the following form

Now we can easily see that,in each column of C3,there are still at most two non-trivial entries,2 and−2,in each column.By re performing the processes that were applied for B and C1,we can finally get a presentation matrix of the form

Thus there are only 2-torsions in H1(M;Z).

Remark 3.1 Our proof above only holds in 3-dimensional small covers,and it is not true for higher dimensions.

4 Mod 2 Betti Numbers of Finite Covers of Small Covers.

The following simple lemma is well-known,for example,see[12].

Lemma 4.1LetPbe a3-dimensional right-angled hyperbolic polytope.ThenPhas at least one pentagon face.Moreover,there is no faceFofP,such that every pentagon inPis adjacent toF.

Proof Since P is a right-angled hyperbolic polytope,by Andreev’s theorem(see[2]),there is no triangle or quadrilateral in F(P).Denoted by fkthe number of k-gons among the faces of P,k ≥ 5,then a simple calculation by means of Euler’s formula implies that f5is non-zero.

Moreover,if P has a face F which is adjacent to every pentagon in P,then by doubling P along F,we can get a right-angled hyperbolic polytope Q such that every face of it has at least six edges,contradicting the previous fact.

Theorem 4.1LetPbe a3-dimensional right-angled hyperbolic polytope,G(P)be the Coxeter group associated toP.Then there are hyperbolic polytopesPi,where eachPiis a doubling ofPi−1along a face ofPi−1,such that

wheremis the number of faces ofP,andP0is defined to beP.Proof The Coxeter group of P is given by

So H1(G(P))=Now let#{P(2)}be the number of faces of P,and Pi+1be a doubling of Pialong a pentagon.We haveThen

Remark 4.1 Comparing to Giro’s approaches(see[10–11])on rank gradients of small covers,Atkinson’s result(see[3])on the relationship between volume and the number of vertices of a hyperbolic polytope is not necessary in our proof.

We now make a refinement of the proof and result of Theorem 4.1.

Theorem 4.2LetPbe a3-dimensional right-angled hyperbolic polytope,G(P)be the Coxeter group associated toP.Then there are hyperbolic polytopesPi,where eachPiis a doubling ofPi−1along a face ofPi−1,such thatG(Pi)is a co- final sequences inG(P)with

wheremis the number of faces ofPandP0is defined to beP.That is,the homology torsion of finite covers grows exponentially.

Proof We embed P in H3and fix a point x in the interior of P.Denoted by F1,F2,···,Fmthe faces of P and it is satisfied that d(F1,x)≤ d(F2,x)···≤ d(Fm,x).We assume that F1has a edges.If F1is a pentagon,then we double P along F1and denote the resulting polyhedron by Q1.Otherwise,from Lemma 4.1,there will be a minimum i such that Fiis a pentagon and not adjacent to F1.

We now double P along Fiwhile the initial F1remains in Q1.Then there is another face of Q1which is a pentagon and is not adjacent to F1.We double Q1along that face and get a polyhedron Q2.Denote the polyhedron,that results from doubling P for k times,by Qk.For a n arbitrary∈,we can make k large enough to satisfyIt can be calculated that Qkhas 2km−(2k−1)7 faces.We now double Qkalong F1and denote the resulting polyhedron by P1.Then P1has 2(2km−(2k−1)7)−a−2 faces.We have

Now x is also in the interior of P1,and the interior of F1lies in the interior of P1.We take the minimum i when Filies in the boundary of P1and double P1along pentagons many times as above to obtain a polytope Q.Furthermore we double Q along the face that contains F1and obtain a polytope P2,such that

As d(x,∂P) ≥ d(x,∂P1) ≥ d(x,∂P2),we can get a polytope R by repeating the above process for at most m times,such that

Now we have d(x,∂P)≥ d(x,∂R).Then by taking R as the initial P and applying previous operations,we can have a polytope S,such that

In fact,the distance between x and the boundary of above polytopies diverges to in finite by repeating the above process,then as in[11,Section 5](which is contributed by Agol[1]),the Coxeter groups related to the polytopies we construct form a co- fi nial sequence.Now the sequence above have torsion growth(m − 7)log2 by the arbitrariness of∈.

Proof of Theorem 1.2 We proved in Theorem 4.1 that for a right-angled hyperbolic polytope P and the Coxeter group G(P)associated to P,there are hyperbolic polytopes Pi,where each Piis a doubling of Pi−1along a face of Pi−1and P0=P,such that G(Pi)is a co- final sequence in G(P)with the numbers of facets of Pigrowing exponentially.Thus for any small cover Miover Pi,by[7,Theorem 3.1],H1(Mi;Z2)has an exponential torsion growth with π1(Mi)a co- final sequence.

Acknowledgement The authors would like to thank Zhi L¨u for introducing them to the topic on small covers.

[1]Agol,I.,Criteria for virtual fibering,J.Topol.,1(2),2008,269–284.

[2]Andreev,E.,On convex polyhedra of finite volume in Lobaˇcevski˘ı space,Math.USSR Sbornik,12(3),1971,225–259.

[3]Atkinson,C.,Volume estimates for equiangular hyperbolic Coxeter polyhedra,Algebraic&Geometric Topology,9,2009,1225–1254.

[4]Bergeron,N.and Venkatesh,A.,The asymptotic growth of torsion homology for arithmetic groups,J.Inst.Math.Jussieu,12(2),2013,391–447.

[5]Boston,N.and Ellenberg,J.,Pro-p groups and towers of rational homology spheres,Geomery&Topology,10,2006,331–334.

[6]Choi,S.and Park,H.,On the cohomology and their torsion of real toric objects,Forum.Math.,29(3),2017,543–553.

[7]Davis,M.and Januszkiewicz,T.,Convex polytopes,Coxeter orbifolds and torus actions,Duke Math.J.,62,1991,417–451.

[8]Ershov,M.,Golod-Shafarevich groups:A survey,International Journal of Algebra and Computation,22(5),2012,68.

[9]Friedl,S.,A note on the growth of Betti numbers and ranks of 3-manifold groups,Bulletin des Sciences Mathmatiques,138,2014,63–70.

[10]Gir˜ao,D.,Rank gradient of small covers,Pacific Journal of Mathematics,266(1),2013,23–29.

[11]Gir˜ao,D.,Rank gradient in co- final towers of certain Kleinian groups,Groups,Geometry,and Dynamics,8(1),2014,143–155.

[12]Inoue,T.,Organizing volumes of right-angled hyperbolic polyhedra,Algebraic&Geometric Topology,8,2008,1523–1565.

[13]Kionke,S.and Schwermer,J.,On the growth of the first betti number of arithmetric hyperbolic 3-manifolds,Groups,Geometry,and Dynamics,9(2),2015,531–565.

[14]Koberda,T.,Homological eigenvalues of mapping classes and torsion homology growth for fi bered 3-manifolds,2012,arXiv:1205.0215.

[15]Lackenby,M.,Covering spaces of 3-orbifolds,Duke Math.J.,136,2007,181–203.

[16]Liu,Y.,Immersing quasi-fuchsian surfaces of odd euler characteristic in closed hyperbolic 3-manifolds,J.Differential Geom.,to appear.

[17]Lott,J.and L¨uck,W.,L2-topological invariants of 3-manifolds,Invent.Math.,120(1),1995,15–60.

[18]Lk,W.,Approximating L2-invariants and homology growth,Geom.Funct.Anal.,23,2013,622–663.

[19]Reid,A.,The geometry and topology of arithmetic hyperbolic 3-manifolds,Proc.Symposium Topology,Complex Analysis and Arithmetic of Hyperbolic Spaces,Kyoto 2006,RIMS Kokyuroku Series,1571,2007,31–58.

[20]Rolfsen,D.,Knots and links,corrected reprint of the 1976 original,Mathematics Lecture Series,7,Publish or Perish,Inc.,Houston,TX,1990.

[21]Sun,H.,Virtual homology torsion of closed hyperbolic 3-manifolds,J.Diffenential Geome.,100(3),2015,547–583.

[22]Trevisan,A.,Generalized Davis-Januszkiewicz spaces and their applications to algebra and topology,Ph.D.thesis,Vrije Universiteit Amsterdam,Amsterdam,2012.