Separating subgroups of mapping class groups in homological representations
SSEPARATING SUBGROUPS OF MAPPING CLASS GROUPSIN HOMOLOGICAL REPRESENTATIONS
ASAF HADARI
Abstract.
Let Γ be either the mapping class group of a closed surface ofgenus ≥
2, or the automorphism group of a free group of rank ≥
3. Givenany homological representation ρ of Γ corresponding to a finite cover, andany term I k of the Johnson filtration, we show that ρ ( I k ) has finite indexin ρ ( I ), the Torelli subgroup of Γ. Since [ I : I k ] = ∞ for k >
1, this impliesfor instance that no such representation is faithful. Introduction
Let X = X ng,b be an oriented surface of genus g with b boundary componentsand n punctures. The mapping class group of X , or Mod( X ) = Mod ng,b is thegroup of orientation preserving diffeomorphisms X → X that fix the boundaryand punctures pointwise, up to isotopies that fix the boundary point wise.Mapping class groups have a large collection of finite dimensional repre-sentations called homological representations . For every finite characteristiccover f : Y → X we can associate a representation ρ = ρ f : Mod n +1 g,b → GL( H ( Y, Z )) in the following way.Pick a point (cid:63) ∈ X . Suppose f : Y → X corresponds to characteristicsubgroup K ≤ π ( X, (cid:63) ). Let Γ ∼ = Mod n +1 g,b be the mapping class group of thesurface X (cid:48) which is obtained from X by adding a puncture at (cid:63) . We can thinkof Γ as the group of orientation preserving diffeomorphisms X → X that fixthe boundary, punctures, and (cid:63) pointwise up to isotopies that fix the boundaryand (cid:63) point wise. There is a natural map Γ → Aut( π ( X, (cid:63) )). Since, K is acharacteristic subgroup, restriction gives a map Γ → Aut(K). This induces amap Γ → Aut( H ( K, Z )) ∼ = GL( H ( Y, Z )).Our goal in this paper is to address two basic questions about homologicalrepresentations. These questions fit into a larger meta-question: Question 1.1.
Which properties of the mapping class group can we discernin its homological representations?The main question question we address in this paper is the following:
Date : September 5, 2019. a r X i v : . [ m a t h . G T ] S e p ASAF HADARI
Question 1.2.
Let H ≤ G ≤ Γ be subgroups of Γ such that [ G : H ] = ∞ .Is there a homological representation ρ f such that [ ρ f ( G ) : ρ f ( H )] = ∞ ?In other words, can the homological representation theory of mapping classgroups discern whether or not one subgroup has infinite index in another?When H is the trivial group, the answer to question 1.2 is yes. For example in[4] we showed that if the surface X has boundary components then any elementof Γ with positive topological entropy has infinite order in some homologicalrepresentation (and in fact, has eigenvalues off of the unit circle). Yi Liuproved a similar theorem in [9] that holds for closed surfaces as well. In [5],we proved an even stronger result - if H is the trivial group and G is a non-amenable subgroup of Γ, then there is some homological representation ρ f such that ρ f ( G ) is non-amenable (and in particular infinite).This paper is concerned with the opposite direction - what happens when H is a large subgroup of the mapping class group? I turns out that in thiscase a very different phenomenon occurs in which the answer to Question 1.2can be negative. Namely, we show the following: Theorem 1.3.
Suppose that X is a closed surface of genus ≥ . Then thereexist subgroups H ≤ G ≤ Γ such that [ G : H ] = ∞ and [ ρ ( G ) : ρ ( H )] < ∞ forevery homological representation ρ . The subgroups we use in our proof of Theorem 1.3 are well known subgroups.Let I = I be the Torelli subgroup of Γ, and I k be the k -th term in the Johnsonfiltration (these groups are defined below in 2). We prove the following: Theorem 1.4.
Let X be a closed surface of genus ≥ .For every k > , andevery homological representation ρ of Γ , [ ρ ( I ) : ρ ( I k )] < ∞ . This is enough to prove theorem 1.3 since [ I : I k ] = ∞ for every i ≥ F n ) with n ≥
3. Givenany characteristic subgroup
K < F n , we get a representation ρ : Aut( F n ) → GL( H ( K, Z )). The group Aut( F n ) also has subgroups I k (which we define in2). We prove the following theorem: Theorem 1.5.
Let n ≥ . For every k and every homological representation ρ of Aut( F n ) , [ ρ ( I ) : ρ ( I k )] < ∞ . As part of our proof of Theorem 1.4, we answer an even more basic questionwhose answer was surprisingly not in the literature - namely: is any homolog-ical representation faithful?
Theorem 1.6. If X is a closed surface of genus ≥ then no homologicalrepresentation is faithful. Theorem 1.6 follows from Theorem 1.4 since [ I : I ] = ∞ . It’s also proveddirectly in Lemma 5.1. EPARATING SUBGROUPS 3
Our final observation is that while our theorems show that the image of onegroup has finite index in the other, this index need not be 1.
Theorem 1.7.
Let I , I , I , . . . be the Johnson filtration of either Mod( X ) for X a closed surface of genus ≥ or Aut( F n ) for n ≥ . Then there exists ahomological representation ρ with the following property: for any k > thereexists a number N ≥ k such that if j > N then [ ρ ( I k ) : ρ ( I j )] > . the image of homological representations. Theorem 1.4 fits a con-jectural description of the image of homological representations. Suppose f : Y → X is a finite characteristic cover with deck group D . Given amapping class φ ∈ Γ, any lift (cid:101) φ of φ to Y normalizes the deck group D . Inparticular, ρ f ( φ ) normalizes D ∗ , the image of D in Sp( H ( Y, Z )). It is naturalto ask the following question: Question 1.8. Is ρ f (Γ) a finite index subgroup of the normalizer of D ∗ inSp( H ( Y, Z ))? Does a similar phenomenon hold for homological representa-tions of Aut( F n )?McMullen addressed this question in [10]. He showed that when the genusof X is zero, the answer to question 1.8 can be negative. However, in everysingle known case in genus ≥
2, the answer to this question is positive. Forexample, Gr¨unewald, Larsen, Lubotzky, and Malestein showed this for theclass of redundant covers of closed surfaces in [3] and Looijenga showed itfor the class of abelian covers of closed surfaces in [8]. The correspondingquestion for Aut( F n ) representations also has a positive answer in every singleknown case (for example, Gr¨unewald and Lubotzky proved this for the classof redundant covers in in [2]).Our Theorems 1.4 and 1.5 can be viewed as evidence of a positive answerto Question 1.8 in genus ≥ F n with n ≥
3. The normalizers thatappear in the question are lattices in high rank semi-simple Lie groups. SinceMod( X ) / I ∼ = Sp(2 g, Z ), we get that a positive answer to the mapping classgroup portion of Question 1.8 implies that the image of I is also a lattice in ahigh rank semi-simple Lie group. By the Margulis normal subgroup theorem,every normal subgroup of such a lattice is either finite or has finite index. Inparticular, all of its quotients are either finite or semi-simple. Since I k (cid:1) I forevery k and I / I k is a nilpotent group, we must have that the image of I k hasfinite index in the image of I . Thus, a positive answer to Question 1.8 wouldimply our Theorem 1.4. Acknowledgments
We would like to thank Dan Margalit for helpful com-ments about the paper.
ASAF HADARI
Sketch of Proof of Theorem 1.4.
The group Γ acts on I / I by con-jugation. Since I acts trivially on this quotient, I / I becomes a Sp(2 g, Z )-module. Denote H = H ( X, Z ). By work of Dennis Johnson, it is known that I / I ∼ = Λ H as a Sp(2 g, Z )-representation (see [6]).This representation decomposes as a sum of irreducibles as Λ H ∼ = H ⊕ Λ H/H . The first factor is the image of the point pushing subgroup and thesecond factor is the image of those elements of I that are not in the kernel ofthe map Mod( X (cid:48) ) → Mod( X ) given by forgetting the puncture (cid:63) .In Section 4 we discuss point pushing maps and a related notion called curvepushing maps. We also give an explicit description of the image of certain pushmaps under homological representations. In Section 5, we use this descriptionto construct for each homological representation ρ elements φ, ψ ∈ Γ such that:(a) φ, ψ ∈ I \ I .(b) φ, ψ ∈ ker( ρ ) (which immediately proves Theorem 1.6)(c) φ is a point pushing map that projects to the first irreducible factor of I / I .(d) ψ is a curve pushing map that projects to the second irreducible factorof I / I .Since φ, ψ ∈ ker( ρ ), their conjugacy classes are as well. This means thatthe Sp(2 g, Z ) orbits in I / I are in the kernel of the map I → ρ ( I ) /ρ ( I ).Irreducibility now gives the result.For k ≥
2, we have that I / I k is a nilpotent group. In Lemma 3.2, we showthat if N is a finitely generated nilpotent group and K (cid:1) N projects to a finiteindex subgroup of N/ [ N, N ] then K has finite index in N . Applying this toker( ρ ) now gives the result.In Section 3 we carry out a similar proof for the Aut( F n ) case. Thiscase is much simpler. The representation Λ H is an irreducible SL( n, Z )-representation, so we only need to construct one map φ ∈ I \ I ∩ ker( ρ ).Furthermore, this map ends up being easier to construct - it’s a Nielsen trans-formation that is simple to describe.2. The Torelli group and the Johnson filtration
Let Γ be either Aut( F n ), or Mod( X (cid:48) ) where X is a closed surface of genus ≥ X (cid:48) is obtained from X by adding the puncture (cid:63) . Let π be either F n or π ( X, (cid:63) ).There is a natural map Γ → Aut( π ). Whenever L (cid:1) π is a characteristicsubgroup, we get a map Γ → Aut( π/L ). The sequence of groups L = [ π, π ], L i +1 = [ π, L i ] is called the lower central series of π . All the groups L i arecharacteristic subgroups of π . EPARATING SUBGROUPS 5
The kernel of the map Γ → Aut( π/L k ) is called the k-th term of the Johnsonfiltration . We denote this kernel I k . When k = 1 the group is called the Torellisubgroup of
Γ and we denote it I .We will require three standard facts about the Johnson filtration:(a) I / I k is a finitely generated nilpotent group.(b) The group I / I is abelian and the map I / [ I , I ] → I / I has finitekernel. ([7]).(c) The group Γ acts on I / I by conjugation. As a Γ / I -module, I /CI ∼ =Λ H , where H = π/ [ π, π ].3. The
Aut( F n ) case Let n ≥ K (cid:1) F n be a characteristic subgroup, and ρ K : Aut( F n ) → GL( H ( K, Z )) be the corresponding homological representation. Proposition 3.1.
In the notation above [ ρ K ( I ) : ρ K ( I )] < ∞ .Proof. Let m = [ F n : K ], and F n = (cid:104) a , a , . . . , a n (cid:105) . For every fixed 1 ≤ i (cid:54) = j ≤ n the endomorphism F n → F n that sends a i → a i a j and a k → a k for k (cid:54) = i is called a Nielsen transformation , and is well known to be an automorphism.Let φ : F n → F n be given by φ ( a ) = a [ a m , a m ], and φ ( a k ) = a k for k > φ can be written as a product of Nielsen transformations,and is thus an automorphism.Claim 1: φ ∈ Ker( ρ K ). To see this claim, let X = n (cid:87) S be a join of n circlesat a point which we call p . Then π ( X, p ) ∼ = F n , and the automorphism φ isinduced by a homotopy equivalence ϕ : X → X that fixes p . Let X → X bethe finite sheeted cover corresponding to the subgroup K < F n . Since K ischaracteristic, we can lift ϕ to a map ϕ : X → X fixing some lift p of p .The spaces X and X are graphs, and are thus also simplicial complexes. Let C ∗ be the chain complex of simplicial chains in X . Denote by ϕ : C → C the map induced by ϕ . Note that a m , a m ∈ K . Thus, [ a m , a m ] ∈ [ K, K ].Given an edge r in X that is a lift of an edge corresponding to a i for i >
1, wehave that ϕ ( r ) = r . Given an edge r that is a lift of a , we have that ϕ ( r )is an edge path consisting of r followed by a cycle whose class in H ( X ) istrivial. Thus ϕ is the identity map.Since H ( K ) can be identified with the subspace of 1-cycles in C , and ρ K ( φ )is given by restricting ϕ to this subspace, we get that ρ k ( φ ) is the identitymap. ASAF HADARI
Claim 2: φ ∈ I \ I . Note that , a − ( φ ( a )) = [ a m , a m ] and a − i φ ( a i ) for i ≥ a m , a m ] ∈ [ F n , F n ] we get that φ ∈ I . Since [ a m , a m ] / ∈ [ F n , [ F n , F n ]] weget that φ / ∈ I .We are now ready to complete the proof. The quotient I / I , as a Aut( F n )module is Λ H . Since I acts trivially on this module, I / I is a Aut( F n ) / I ∼ = SL ( n, Z )-module.Let [ φ ] be the image of the automorphism φ constructed above in I / I . Byclaim 1, [ φ ] is in the kernel of the map I / I → ρ K ( I ) /ρ K ( I ). Thus, the entireSL n ( Z ) orbit of [ φ ] is in this kernel. By claim 2, [ φ ] (cid:54) = 0.Since Λ H is an irreducible SL( n, Z ) representation, we have that SL( n, Z )[ φ ]generates a finite index subgroup of Λ H . Thus ρ K ( I ) /ρ K ( I ) is finite, asdesired. (cid:3) Proposition 3.1 is a subcase of Theorem 1.5, namely - the case where k = 2.Since I / I k is nilpotent for every k , Theorem 1.5 follow directly from Proposi-tion 3.1 and from the following lemma: Lemma 3.2.
Let N be a finitely generated nilpotent group. Let K (cid:1) N be asubgroup such that the image of K in N/ [ N, N ] has finite index in N/ [ N, N ] .Then [ N : K ] < ∞ .Proof. Denote by N i the i -th term in the lower central series of N . Let a , . . . , a m ∈ N be a set of elements that project to a free generating setof N/ [ N, N ]. For elements g , . . . , g k ∈ N , denote by [ g , . . . , g k ] the repeatedcommutator [ g , [ g , . . . , g k ] . . . ]. The following two facts are standard:(a) The group N /N i +1 is an abelian group generated by the images of allelements of the form [ x , . . . , x i ] where x , . . . , x i ∈ { a , . . . , a n } .(b) The map N i → N i /N i +1 given by ( x , . . . , x i ) → [ x , . . . , x i ] is a homo-morphism in each coordinate.We now proceed by induction on the length of the lower central series of N .The lemma is obviously true for N abelian. Suppose it is true for N with alower central series of length i . The multi linearity of the repeated commutator,together with our inductive hypothesis give that [ N i : N i ∩ K ] < ∞ .Given any x ∈ N , the inductive hypothesis gives that there exists a number l such that the image of x l in N/N i − is contained in the image of K in thisgroup. This means that there is a y such that y l ∈ K and xy − ∈ N i . Write x = yh . Since [ N i : N i ∩ K ] < ∞ , there is a m such that h m ∈ K . Since N i is central in N , we have that ( yh ) lm = ( y l ) m ( h m ) l ∈ K . A finitely generatednilpotent group of bounded exponent is finite, which concludes the proof. (cid:3) EPARATING SUBGROUPS 7 Pushing maps
Point pushing and curve pushing.
We begin by recalling a standardtheorem from differential topology:
Theorem 4.1. ( The isotopy extension theorem ) Let M be a compact manifold(possibly with boundary) and N a boundary less sub manifold. Let H : N × [0 , → M be a smooth homotopy such that H t ( x ) = H ( x, t ) : N → M is anembedding for each t and H is the inclusion of N into M . Then H can beextended to a smooth isotopy (cid:101) H : M × [0 , → M where (cid:101) H t ( x ) = (cid:101) H ( x, t ) : M → M is a diffeomorphism for each t , and (cid:101) H is the identity map. Extensions of homotopies H that have the added property that H is theidentity map on N allow us to construct several interesting families of mappingclasses, which we will use to mimic the proof of Theorem 1.5 for mapping classgroups.The first such family is very well known. Suppose M is a surface, and p ∈ M is a point. Let N = { p } . A homotopy H satisfying the condition H = H = Id is just a closed curve γ that is based at p . Let (cid:101) H be the extendedisotopy. Let M = M \ { p } . The diffeomorphism (cid:101) H can be restricted to M .This restriction is known as the point pushing map about the curve γ . Byconstruction (cid:101) H is isotopic to (cid:101) H = Id as maps from M → M . However, if γ isnot null-homotopic then their restrictions are not isotopic as maps M → M .Now, suppose N ⊂ M consists of a finite set of points N = { p , . . . , p r } andonce again assume that H = H . This means that each point p i traces a closedcurve γ i . We call the restriction of the diffeomorphism (cid:101) H to M = M \ M a multi-point pushing map about the curves γ , . . . , γ r .Let X be a surface and δ a non-peripheral, non-separating simple closedcurve. Let X be the surface obtained from X by cutting along δ . The surface X has two more boundary components (which we call δ , δ ) than X , and itsgenus is one less than the genus of X . Let X be the surface obtained from X by gluing a disk to δ , and adding a marked point p in this disk.Choose a closed curve γ in X that is based at p . The curve γ gives ahomotopy of embeddings of the point p into X . Extend this homotopy to aisotopy (cid:101) H : X × [0 , → X . If B is a sufficiently small ball centered at p , wecan modify H so that H is the identity map when restricted to B .Let f = (cid:98) H . The surface X can be obtained from X by removing the interiorof B , and gluing ∂B to δ . Since f fixes ∂B and δ point wise, it defines amap f : X → X , which we call a curve pushing map . We say that f pushesthe curve δ along the curve γ .Note that γ is not a closed curve in X , but we can think of it as a closedcurve in ( X, δ ). Throughout our discussion, when we refer to γ as a closedcurve we mean it in this sense. ASAF HADARI
We can make a similar definition when the curve δ is separating. Supposeit separates X into two components, X , X . We choose one of them, say Z and glue in a disk to X along δ equipped with a marked point p . We pick aclosed curve γ in X that is based at p and proceed as before.By replacing the curve δ with a multi-curve (a finite collection of mutuallydisjoint simple closed curves) we can define a mutli-curve pushing map .4.2. The image of push maps under homological representations.
The action of point and multi-point pushing maps on homology.
Let X be a surface with at least one puncture. Let φ ∈ Mod( X ) be a point pushingmap about the closed curve γ which is based at the puncture p and whosehomology class (relative to the puncture p ) we denote by c . Let d be thehomology class of a small positively oriented loop about the puncture p . If thesurface X has only one puncture, then every point pushing map acts triviallyon H ( X, Z ). This no longer holds when X can have multiple punctures.Suppose first that γ is a simple closed curve. Let α be a closed curve in X with homology class a . The curve φ ( α ) is obtained from α in the followingway. For every intersection of γ and α , two strands that run alongside γ inopposite orientations are attached to α using the surgery depicted in the figure1. If we use [ · ] to denote the homology class of a curve, and (cid:98) i ( · , · ) to denotethe oriented intersection pairing we get that: φ ( a ) = a + (cid:98) i ( a, c ) d Figure 1.
EPARATING SUBGROUPS 9
Suppose now that the curve γ has self intersections. Perform the surgerydescribed above by attaching two copies that run along γ in opposite orien-tations to each intersection point. Label the intersection points cyclically by x , . . . , x s . At the intersection point x i add 2 i strands ( i in each orientation)parallel to the portion of γ that exits the intersection the second time it pasesthrough it. To find the image φ ( c ), at each self intersection of γ perform thesurgery described in Figure 2. At each such intersection there are an equalnumber of positively oriented and negatively oriented strands parallel to γ ineach direction. Thus, even if γ has self intersection, the same formula holds: φ ( a ) = a + (cid:98) i ( a, c ) d Figure 2.
Now suppose that φ is a multi-point push map, in which the punctures p , . . . , p r are pushed about the curves γ , . . . , γ r (whose homology classes wedenote by c , . . . , c r ). Let d , . . . , d r be small, positively oriented loops aboutthe punctures p , . . . , p r . The same argument that we used for the point push-ing map gives a similar description (this is illustrated in Figure 3). Lemma 4.2.
In the notation above, for every curve α , we have that: φ ( a ) = a + r (cid:88) i =1 (cid:98) i ( a, c i ) d i The action of curve and multi-curve pushing maps on homology.
Sup-pose that φ is the map given by pushing the curve δ (whose homology classwe denote by d ) along the curve γ . For any curve α that does not intersect Figure 3. δ , the same picture as the point pushing case holds here (this is illustrated inFigure 4). We get that once again, φ ( a ) = a + (cid:98) i ( a, c ) d . Figure 4.
In the curve pushing case we have an added complication: the curve pushingmap is defined by cutting X along δ to form the surface X , pushing along thecurve γ in X , and regluing two copies of δ to re-form the surface X . As wesee in figure 4, calculating φ ( c ) is very similar to the the point pushing case,as long as α is a closed curve in X . We need to separately determine φ ( α ) forarcs in X whose endpoints lie on the two different copies of δ . Let δ , δ bethose copies. We need to calculate the action of φ on H ( X, δ ∪ δ ). EPARATING SUBGROUPS 11
Suppose first that γ is a simple closed curve and α is an arc, one of whoseendpoints lies on δ . To obtain φ ( α ) we add to α one strand parallel to γ to α using the surgery depicted in figure 5. Figure 5.
Now suppose that γ has self intersections: x , . . . , x s arranged from thebeginning of γ till its end. To find φ ( α ) add strand parallel copy to γ asabove. Then, following the order x , . . . , x s , at x i add two parallel in oppositedirection along the section of γ exiting the intersection for the second time.Then perform the surgery described in Figure 6. Figure 6.
As opposed to the point pushing case, it is no longer the case that at eachintersection there are an equal number of positively oriented and negativelyoriented strands parallel to γ in each direction. For each x j , let I j = (cid:98) i ( e j , f j )where e j is the portion of γ that hits the intersection first, and f j is the portion of γ that hits the intersection second. Let I γ = (cid:80) I j . This number determineshow many copies of d are added at the j -th intersection.Now let α be a closed curve in X . Putting the descriptions above together,we get that φ ( a ) = a + (cid:98) i ( a, c ) d + (cid:98) i ( a, d ) (cid:0) c + I γ d (cid:1) Now let δ , . . . , δ r be pairwise disjoint simple closed curves with homologyclasses d , . . . , d r and take φ to be the multi-curve pushing map that pushes δ j about γ j (whose homology class is denoted c j ). We get the following. Lemma 4.3.
In the notation above, for any curve α in X : φ ( a ) = a + (cid:88) j (cid:98) i ( a, c j ) d j + (cid:98) i ( a, d j ) (cid:0) c j + I γ j d j (cid:1) Lifts of push maps to covers.
Point, multi-point, curve, and multi-curve pushing maps can have very complicated actions on the homology ofcovers of a surface. For instant, the point pushing subgroup of X containsmany pseudo-Anosov elements (a point pushing map is pseudo-Anosov when-ever the pushing curve fills the surface). By [4], for any pseudo-Anosov elementof the point pushing subgroup there is a finite cover Y → X to which f lifts,such that the action of the lift of f on H ( Y, Z ) has eigenvalues off the unitcircle. As a further example, by [5] there is a finite cover Y → X where the im-age of the entire point pushing subgroup under the homological representationis non-solvable.This complexity is not immediately apparent from the descriptions given inLemmas 4.2 and 4.3. Part of the issue is that the lift of a point (resp. curve)pushing map to a cover need not be a point (resp. curve) or even a multi-point(resp. multi-curve) pushing map. In the point pushing case this is caused bythe fact that if Y → X is a finite cover and p is a puncture of X then thecovering map may be branched over p . If p is pushed about the curve γ , and (cid:101) p is a lift of the puncture p in Y , there may not be only one lift of the curve γ originating at (cid:101) p Nevertheless, sometimes lifts of push maps to finite covers are themselvespush maps, and their images under homological representations can thus bedescribed by Lemmas 4.2 and 4.3.We begin by describing a sufficient criterion for this to happen in the curvepushing case. Let δ be a simple closed curve in X . Identify δ with R / Z . Let γ be a curve in X originating at a point in δ corresponding to p ∈ R / Z andterminating at the point q = p + .Let Y → X be a regular finite cover. Let σ be the element of the deck groupof Y → X corresponding to δ , and let s be its order. Suppose that δ s lifts to Y for some s . Fix a lift (cid:101) δ of δ s and let (cid:101) p , . . . , (cid:103) p s − , (cid:101) q , . . . , (cid:103) q s − ∈ R /s Z be the EPARATING SUBGROUPS 13 lifts of p , q in this (cid:101) δ arranged cyclically. Lift the curve γ to (cid:101) γ originating at p . Suppose this curve terminates at (cid:101) q j .We say that γ satisfies the cyclic generation criterion if the integer j gen-erates the group Z /s Z . If this is the case, then: s − (cid:83) i =0 σ i (cid:101) γ is a (cid:104) σ (cid:105) invariant setconsisting of a single curve which we call (cid:101) γ s .This curve passes through all thelifts (cid:101) p , . . . , (cid:103) p s − , (cid:101) q , . . . , (cid:103) q s − . The curve (cid:101) γ s is simply a lift of γ s to Y , and byconstruction there is a unique such lift incident at the curve (cid:101) δ . Since Y isregular, this is the case at every other lift of δ s .Let φ ∈ Mod( X ) be the push map given by pushing δ about γ s . By con-struction, the lift of φ to Y is simply the multi-curve pushing map that pusheseach lift of δ about each lift of (cid:101) γ s . Figure 7 shows that if γ does not satisfythe cyclic generation criterion there can be multiple lifts of the curve γ s at theloop (cid:101) δ and hence the lift of φ is not a multi-curve pushing map. Figure 7. proof of Theorem 1.4 We are now ready to construct push maps that mimic the properties of themap constructed in the proof of Theorem 1.5. Let X be a closed surface ofgenus ≥
2, and X (cid:48) = X \ { (cid:63) } for some basepoint (cid:63) . Let Γ = Mod( X (cid:48) ). Let Y → X be a characteristic cover. Let ρ be the corresponding homologicalrepresentation. Lemma 5.1.
There exist maps φ, ψ ∈ ( I \ I ) ∩ ker ρ such that φ is a pointpushing map and ψ is a curve pushing map whose image in Mod( X ) is non-trivial.Proof. Pick a element of π ( X, (cid:63) ) whose projection to H = H ( X, Z ) is notzero. Let φ be the point pushing map about this curve. Since φ is a pointpushing map, we have that φ ∈ I . As stated in the introduction, we have that I / I ∼ = H ⊕ Λ /H , where point pushing maps are sent to the first factor usingthe abelianization map π ( X, (cid:63) ) → H . By our choice of a curve in X , we havethat φ / ∈ I .Finally, note that since the cover Y → X is not branched over the point (cid:63) ,the lift of φ to Mod( Y ) is a multi-point pushing map. Since Y does not havepunctures, Lemma 4.2 gives that φ ∈ ker ρ .To construct the map ψ , start with a separating simple closed curve δ in X (we can choose one because the genus of X is at least 2). Let X be thesurface described in section 4, and let Y → X be the corresponding (possiblybranched) cover. Choose a curve γ in X with the following properties:(a) There exists a lift of γ to Y that is a closed curve that passes throughevery lift of δ , and satisfies the cyclic generation criterion at each suchlift.(b) The homology class c of γ is not equal to 0.Let ψ ∈ Mod( X ) be the push map that pushes the curve δ along the curve γ . Let T δ be the Dehn twist about δ . By our assumption on γ , ψ s lifts to amulti-curve pushing map for some s .Let (cid:101) δ , . . . , (cid:101) δ r be the lifts of δ , and γ , . . . , γ r be the curves about which theyare being pushed. The curves γ , . . . , γ r all have the same homology class - alift of γ s which we denote by (cid:101) γ . Since r (cid:83) j =1 (cid:101) δ j separates Y , and c j ’s are all equalto each other, the term (cid:80) j (cid:98) i ( a, d j ) (cid:0) c j + I γ j d j (cid:1) that appears in Lemma 4.3 is0. Similarly, since γ passes through all the lifts of δ and these separate Y , weget that the term (cid:80) j (cid:98) i ( a, c j ) d j is also 0. Lemma 4.3 now gives that ψ ∈ ker ρ .This implies that ψ ∈ I .It now remains to be seen that we can take ψ / ∈ I . Recall that δ sepa-rates X into two subsurfaces - X , X , where X is pushed about a curve in EPARATING SUBGROUPS 15 X . Let a , b , . . . , a g , b g be a standard generating set for π ( X, (cid:63) ) such that a , b , . . . , a j , b j is a standard generating set for π ( X , (cid:63) ). It is a standardcalculation that the image of ψ in I / I ∼ = Λ H is j (cid:80) i =1 [ a ] ∧ [ b i ] ∧ c (cf. Section6.6.2 in [1]). Since this is not 0, we have that ψ / ∈ I . (cid:3) We are now ready to prove Theorem 1.4.
Proof.
As we did before, decompose the Sp(2 g, Z )-representation Λ H intoirreducible representations as Λ H ∼ = H (cid:76) Λ H/H where the first factor isthe image of the point-pushing maps and the second factor is the image ofmaps that are not in the kernel of the forgetful map Mod( X (cid:48) ) → Mod( X ),where X (cid:48) is the surface obtained from X by puncturing at (cid:63) . Lemma 5.1 givesa point pushing map φ ∈ I that has non-zero image in the first factor anda curve pushing map ψ ∈ I that has non-zero image in the second factor.Denote by [ φ ] , [ ψ ] the images of φ and ψ in I / I . Neither of these images is 0.Thus, the Sp(2 g, Z )-orbits of [ φ ] , [ ψ ] generate a finite index subgroup of I / I .This entire subgroup is in the kernel of the map I / I → ρ ( I ) /ρ ( I ). Thisshows that [ ρ ( I ) : ρ ( I )] < ∞ .As in the proof of Theorem 1.5, the k ≥ (cid:3) Proof of Theorem 1.7
Proof.
Let G be either the fundamental group of the closed surface X withgenus ≥ F n with n ≥
2. Pick a prime p . Let K be the kernel of the map G → H ( G, Z /p Z ). For any integer m , let L m (cid:1) K be the kernel of the map K → H ( K, Z /p m Z ). Let ρ be the homological representation correspondingto the cover given by the subgroup K .Since K is a characteristic subgroup of G and L m is a characteristic subgroupof K , we get that L m (cid:1) K . Furthermore, since G/K and
K/L m are p -group,we have that G/L m is a p -group, and is thus nilpotent.Denote by d ( m ) the nilpotence degree of G/L m . By definition, the group I d ( m ) acts trivially on G/L m . Thus, it implies that I d ( m ) acts trivially on H ( K, Z /p m Z ). This means that the elements of ρ ( I d ( m ) ) are in the p m -congruence subgroup of GL( H ( K, Z )).Fix k >
0. There exists g ∈ I k and i > ρ ( g ) is not in the p i -congruence subgroup of GL( H ( K, Z )). This implies that ρ ( I k ) (cid:54) = ρ ( I j ) forevery j such that j > d ( i ), ρ ( I j ) (cid:54) = ρ ( I k ), as required. (cid:3) References [1] B. Farb and D. Margalit.
A primer on mapping class groups, volume 49 of Princetonmathematical series.
Princeton University Press, Princeton, NJ, 2012. ISBN 978-0-691-14794-9.[2] F. Gr¨unewald and A. Lubtozky.
Linear representations of the automorphism group ofa free group.
Geom. Func. Anal. 18(5):1564–1608, 2009.[3] F. Gr¨unewald, M. Larsen, A. Lubotzky, and J. Malestein.
Arithmetic quotients of themapping class group.
To appear in Geom. Func. Anal.[4] A. Hadari.
Homological eigenvalues of lifts of pseudo-Anosov mapping classes to finitecovers (preprint) arxiv: 1712.01416[5] A. Hadari
Non virtually solvable subgroups of mapping class groups have non virtuallysolvable representations , to appear in Groups, Geom., and Dynamics.[6] D. Johnson.
An abelian quotient of the mapping class group I g . Math. Ann. 249(3):225–242, 1980.[7] D. Johnson. Conjugacy relations in subgroups of the mapping class group and a group-theoretic description of the Rochlin invariant.
Math. Ann. 249(3): 243–263, 1980.[8] E. Looijenga.
Prym representations of mapping class groups.
Geom. Dedicata 64(1):69–83, 1997.[9] Yi. Liu.
Virtual homological spectral radii for automorphisms of surfaces , arxiv:https://arxiv.org/abs/1710.05039.[10] C. T McMullen.