Arithmetic quotients of the mapping class group
Fritz Grunewald, Michael Larsen, Alexander Lubotzky, Justin Malestein
AArithmetic quotients of the mapping class group
Fritz Grunewald ∗ , Michael Larsen †2 , Alexander Lubotzky ‡3 , and Justin Malestein §41 Mathematisches Institut der Heinrich-Heine-Universität Düsseldorf, Universitätsstr. 1,40225 Düsseldorf, Germany Department of Mathematics, Indiana University, Bloomington, IN, 47405, USA Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem,91904, Israel Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn,Germany
Abstract
To every (cid:81) -irreducible representation r of a finite group H , there corresponds asimple factor A of (cid:81) [ H ] with an involution τ . To this pair ( A , τ ) , we associate anarithmetic group Ω consisting of all ( g − ) × ( g − ) matrices over a natural order O op of A op which preserve a natural skew-Hermitian sesquilinear form on A g − . Weshow that if H is generated by less than g elements, then Ω is a virtual quotient ofthe mapping class group Mod (Σ g ) , i.e. a finite index subgroup of Ω is a quotientof a finite index subgroup of Mod (Σ g ) . This shows that Mod (Σ g ) has a rich familyof arithmetic quotients (and “Torelli subgroups”) for which the classical quotientSp ( g , (cid:90) ) is just a first case in a list, the case corresponding to the trivial group H andthe trivial representation. Other pairs of H and r give rise to many new arithmeticquotients of Mod (Σ g ) which are defined over various (subfields of) cyclotomic fieldsand are of type Sp ( m ) , SO ( m , 2 m ) , and SU ( m , m ) for arbitrarily large m .
1. Introduction
Let
Σ = Σ g be a closed surface of genus g ≥ (Σ) the mapping class group of Σ , which isthe group of orientation-preserving homeomorphisms of Σ modulo those isotopic to the identity.It is well-known that there is an epimorphism from Mod (Σ) onto the arithmetic group Sp ( g , (cid:90) ) .The purpose of this paper is to show that many more arithmetic groups are quotients of Mod (Σ) .Specifically, for every pair ( H , r ) , where H is a finite group minimally generated by d ( H ) < g elements and r is a nontrivial irreducible (cid:81) -representation of H , we associate a finite index sub-group Γ H of Mod (Σ) , an arithmetic group Ω H , r (whose exact structure will be described below) ∗ Paper written posthumously † [email protected] ‡ [email protected] § [email protected] a r X i v : . [ m a t h . G T ] A p r nd a homomorphism ρ H , r : Γ H → Ω H , r whose image is of finite index in Ω H , r . We call such ahomomorphism a virtual epimorphism of Mod (Σ) onto Ω H , r . (Of course, via the induced repre-sentation, every virtual homomorphism also gives a representation of the full group Mod (Σ) , butit is more natural to study the original representation.) The homomorphism Mod (Σ) → Sp ( g , (cid:90) ) is the special case where H is the trivial group. The homomorphism from a finite index subgroupof Mod (Σ g ) to Sp ( ( g − ) , (cid:90) ) studied in [ ] and [ ] is the one corresponding to the case H = (cid:90) / (cid:90) and r the nontrivial one-dimensional representation. The first to show that Mod (Σ) has a large collection of virtual arithmetic quotients beside Sp ( g , (cid:90) ) was Looijenga [ ] who an-alyzed the case where H is abelian. In [ ] , [ ] , [ ] , all representations have degree boundedby 2 g and the target arithmetic groups are limited, while we obtain irreducible representationsof arbitrarily large degree and a wider range of target arithmetic groups.The homomorphism ρ is obtained from the action of Mod (Σ) (or more precisely, of a finiteindex subgroup of Mod (Σ) ) on the first integral homology group of a finite Galois cover of Σ corresponding to H ; equivalently, on R def = R / [ R , R ] where R is a finite index normal subgroupof T g = π (Σ g ) with T g / R ∼ = H . This action on the homology of finite covers was also studiedby Koberda, but with the goal of showing that one can determine whether a mapping class isperiodic, reducible or pseudo-Anosov by considering enough finite covers [ ] .Our work is also related to that of Grunewald–Lubotzky [ ] whose results we will use inour proofs. Recall that by the Dehn-Nielsen-Baer Theorem, Mod (Σ g ) is naturally isomorphicto an index 2 subgroup of Out ( T g ) , the outer automorphism group of T g = π (Σ g ) . (Allow-ing orientation-reversing homeomorphisms would give all of Out ( T g ) .) In [ ] , Grunewald andLubotzky studied the analogous problem where Out ( T g ) is replaced by Aut ( F n ) the automor-phism group of F n , the free group of rank n . While there are a few similarities between thepapers, the theory developed in the current paper requires several advances (which we will dis-cuss), and the main techniques of proof in [ ] , as far as we can determine, are unusable inthis case. The difference is nicely illustrated already in the easiest case; while the fact that π : Aut ( F n ) → GL ( n , (cid:90) ) is surjective is easy to establish, the fact that Mod (Σ g ) → Sp ( g , (cid:90) ) is aclassical but nontrivial result due to Burkhardt and Clebsch-Gordan [ ] We begin by describing the general procedure for obtaining representations to arithmeticgroups from Mod (Σ g , ∗ ) , the mapping class group fixing the point ∗ ∈ Σ g up to isotopy fixing ∗ .We will denote these representations also by ρ . A short argument (Section 8.2) shows that therepresentations descend (virtually) to Mod (Σ g ) . Via the natural map Mod (Σ g , ∗ ) → Aut ( T g ) , wecan identify Mod (Σ , ∗ ) with its image, which is of index 2 and which we denote by Aut ( T g ) + . Let p : T g → H be a surjective homomorphism with kernel R onto a finite group H , and let Γ H , p = { γ ∈ Aut ( T g ) + | p ◦ γ = p } . Then, Γ H , p is a finite index subgroup of Aut ( T g ) + which preserves R and acts on R as a (cid:90) [ H ] -module. Thus, Γ H , p acts by (cid:81) [ H ] -automorphisms on ˆ R = R ⊗ (cid:90) (cid:81) andwe obtain a representation ρ H , p : Γ H , p → Aut (cid:81) [ H ] (ˆ R ) . (See Section 6 for a discussion of theseand some other properties of ρ H , p .)By an analogue of a classical result of Gaschütz, the module ˆ R can be identified precisely. Notethat here, and elsewhere unless stated otherwise, (cid:81) is considered to be the trivial (cid:81) [ H ] -module. Proposition 1.1.
Let T = T g be the fundamental group of a surface Σ of genus g ≥ . Let R bea finite index normal subgroup of T and H = T / R. Then ˆ R = (cid:81) ⊗ (cid:90) ( R / [ R , R ]) is isomorphic as a (cid:81) [ H ] -module to (cid:81) ⊕ (cid:81) [ H ] g − . [ ] . We give an alternate proof in Section 2. The group ring (cid:81) [ H ] decomposes as a direct sum of simple algebras (cid:81) [ H ] = (cid:81) ⊕ (cid:76) (cid:96) i = A i where each A i is the ring of m i × m i matrices over a division algebra D i with a number field L i / (cid:81) as its center. Consequently, ˆ R ∼ = (cid:81) g ⊕ (cid:76) (cid:96) i = A g − i , and so we can project to representations ρ H , p , i : Γ H , p → Aut A i ( A g − i ) .The action on (cid:81) g is the standard symplectic representation.Up to now, the above procedure follows that of [ ] with the important exception thatGrunewald and Lubotzky use the actual theorem of Gaschütz in place of Proposition 1.1. Via suchrepresentations, they show that Aut ( F n ) virtually surjects onto a rich class of arithmetic groups,including SL (cid:96) ( n − ) ( (cid:90) ) and SL (cid:96) ( n − ) ( O m ) where (cid:96) ranges over all positive integers, m ranges overall integers ≥
3, and O m is the ring of integers in the number field generated by (cid:81) and a primitive m th root of unity.What makes the surface case much more difficult (and interesting) is the fact that in ourcase ˆ R is equipped with a (cid:81) [ H ] -valued skew-Hermitian sesquilinear form 〈− , −〉 on ˆ R with re-spect to the standard involution τ on (cid:81) [ H ] which is defined by τ ( h ) = h − . This form can bedefined in terms of the group action H and the natural symplectic structure on ˆ R coming fromits identification with the rational first homology of the covering surface. (See Section 3 for adetailed discussion of this form.) As we will see in Section 6, ρ H , p (Γ H , p ) preserves this form. Thesesquilinear form descends to a nondegenerate A i -valued form on each factor M i = A g − i , andthe image of ρ H , p , i lies in Aut A i ( A g − i , 〈− , −〉 ) .On account of this, we obtain an even richer class of arithmetic quotients. We illustrate ithere by some simple to state examples obtained by appropriate choices of H and p . See Section9 for more. We denote by ζ n the primitive n th root of unity, and by (cid:81) ( ζ n ) + we denote the index2 subfield of (cid:81) ( ζ n ) which is fixed by the order 2 Galois automorphism of (cid:81) ( ζ n ) mapping ζ n to ζ − n . By SU ( m , m , O ) , we denote the subgroup of SL m ( O ) preserving the Hermitian form ofsignature ( m , m ) , namely 〈 ( a , . . . , a m , b , . . . , b m ) , ( a (cid:48) , . . . , a (cid:48) m , b (cid:48) , . . . , b (cid:48) m ) 〉 = m (cid:88) i = ( a i a (cid:48) i − b i b (cid:48) i ) where O is the ring of integers in (cid:81) ( ζ n ) and is the order 2 automorphism of (cid:81) ( ζ n ) just de-scribed. Theorem 1.2.
For a fixed g ≥ , there are virtual epimorphisms of Mod (Σ g ) onto the followingarithmetic groups:(a) Sp ( m ( g − ) , (cid:90) ) for all m ∈ (cid:78) ,(b) for all m ∈ (cid:78) and n ≥ , the group Sp ( m ( g − ) , O ) where O is the ring of integers in (cid:81) ( ζ n ) + ,(c) for all m ∈ (cid:78) and n ≥ , the group SU ( m ( g − ) , m ( g − ) , O ) where O is the ring of integersin (cid:81) ( ζ n ) ,(d) for all m ∈ (cid:78) and n ≥ , an arithmetic group of type SO ( m ( g − ) , 2 m ( g − )) (whoseprecise description will be given in Section 9). Theorem 1.3.
Let L be an arbitrary subfield of a finite cyclotomic extension of (cid:81) and O the ring ofintegers of L. Then there is some s L and N L such that for every g ≥ N L and every m ∈ (cid:78) , there is avirtual epimorphism of Mod (Σ g ) onto Sp ( ms L ( g − ) , O ) if L is a totally real field • SU ( ms L ( g − ) , ms L ( g − ) , O ) if L is a totally imaginary field. Note that any subfield of a cyclotomic field is necessarily either totally real or totally imagi-nary. (See Lemma 4.2.)Taking g = m = [ ] and by McCarthy via similartechniques [ ] . Corollary 1.4.
There is a virtual epimorphism from
Mod (Σ ) onto Sp ( (cid:90) ) = SL ( (cid:90) ) . In partic-ular, there is a virtual epimorphism of Mod (Σ ) onto a free group, and hence Mod (Σ ) is large. We also deduce the main result of [ ] in Section 9.7. Corollary 1.5.
For every genus g ≥ and every finite group G, there is a finite index subgroup of Mod (Σ g ) which surjects onto G. We now explain the main technical result of the paper. The standard involution τ of (cid:81) [ H ] ,defined above, descends to an involution on each simple factor A i of (cid:81) [ H ] (see Lemma 3.2). Let K i = L τ i be the subfield of the center L i fixed by τ , and let O i be the image of (cid:90) [ H ] in A i whichis an order in A i . Let G H , i be the K i -defined algebraic group Aut A i ( A g − i , 〈− , −〉 ) , and let G H , i bethose elements of reduced norm 1 over L i (see Section 3.4 for the definition of reduced norm).Let G H , i ( O i ) be the arithmetic subgroup G H , i ∩ Aut O i ( O g − i ) . Our main result says that, under asuitable condition, the image of Γ H , p contains a finite index subgroup of G H , i ( O i ) . Namely, thesuitable condition is that p be φ -redundant , which means that p factors through a surjective map φ : T g → F g where F g is the rank g free group and the induced map p (cid:48) : F g → H is redundant ,i.e. p (cid:48) contains a free generator in its kernel. Theorem 1.6.
Suppose g ≥ and p : T g → H is φ -redundant. Then, for ρ H , p , i , Γ H , p , G H , i ( O i ) asdefined above, ρ H , p , i (Γ H , p ) is commensurable with G H , i ( O i ) . It is the structure of ( A g − i , 〈− , −〉 ) which ultimately determines G H , i ( O i ) , and this structureis intimately related to the representation theory of the finite group H . The structure is deter-mined by the pair ( A i , τ | A i ) , and the simple factors A i of (cid:81) [ H ] are in natural one-to-one corre-spondence with the nontrivial irreducible (cid:81) -representations r i of H (and the factor (cid:81) of (cid:81) [ H ] corresponds to the trivial representation). We can also extract information about the structure(Theorem 1.7 below) by extending scalars to get (cid:82) -algebras and (cid:67) -representations as follows.Let n i = dim L i ( A i ) . As we will see later (Proposition 4.3), K i is a totally real field. As A i is simple,so are A i ⊗ L i (cid:67) and A i ⊗ K i (cid:82) ; consequently, A i ⊗ L i (cid:67) ∼ = Mat n i ( (cid:67) ) , and A i ⊗ K i (cid:82) is isomorphic to oneof Mat n i ( (cid:82) ) , Mat n i ( (cid:67) ) , or Mat n i / ( (cid:72) ) where (cid:72) denotes the Hamiltonian quaternions. Note thatthe projection (cid:81) [ H ] → A i induces a representation H → ( A i ⊗ L i (cid:67) ) × which by the isomorphism A i ⊗ L i (cid:67) ∼ = Mat n i ( (cid:67) ) gives an irreducible (cid:67) -representation r i , (cid:67) : H → GL n i ( (cid:67) ) .After extending scalars to (cid:67) , the group G H , i “becomes” one of the classical algebraic groupsover (cid:67) . The following theorem describes the structure after extending scalars and summarizes,essentially, how we obtain the different types of arithmetic groups (Sp, O, U ) in Theorem 1.2.(For definitions of first / second kind and symplectic / orthogonal type, see Section 3.3.)4 heorem 1.7. The group G H , i is the group of K i -points of a K i -defined complex algebraic group Gwith the additional property that • G ∼ = Sp ( g − ) n i ( (cid:67) ) ⇔ r i , (cid:67) ( H ) preserves a nondegenerate symmetric bilinear form ⇔ A i ⊗ K i (cid:82) ∼ = Mat n i ( (cid:82) ) ⇔ τ | A i is of first kind and orthogonal type. • G ∼ = O ( g − ) n i ( (cid:67) ) ⇔ r i , (cid:67) ( H ) preserves a nondegenerate alternating bilinear form ⇔ A i ⊗ K i (cid:82) ∼ = Mat n i / ( (cid:72) ) ⇔ τ | A i is of first kind and symplectic type. • G ∼ = GL ( g − ) n i ( (cid:67) ) ⇔ r i , (cid:67) ( H ) preserves no nonzero bilinear form ⇔ A i ⊗ K i (cid:82) ∼ = Mat n i ( (cid:67) ) ⇔ τ | A i is of second kind. Theorem 1.6 and its proof show that we have the following “procedure” to obtain a rich col-lection of arithmetic quotients of Mod (Σ g ) . We present this procedure in the form of a theorem.Note that the homomorphism ρ H , A below is explicit and constructive. Theorem 1.8.
Let H be a finite group with d ( H ) < g generators and let A be a nontrivial sim-ple component of (cid:81) [ H ] . Let τ be the involution on A induced by the standard one of (cid:81) [ H ] ,let L be the center of A, and let K = L τ be the τ -fixed subfield. Set M = A g − with free ba-sis x , . . . , x g − , y , . . . , y g − , and endow M with the skew-Hermitian sesquilinear form satisfying 〈 x i , y j 〉 = δ i j , 〈 y i , x j 〉 = − δ i j , 〈 x i , x j 〉 = and 〈 y i , y j 〉 = . Set G to be the K-algebraic groupof the A-automorphisms of ( M , 〈− , −〉 ) and G to be its elements of reduced norm over L. Set Ω H , A = G ( O ) = G ∩ Aut O ( O g − ) where O is the order in A which is the image of (cid:90) [ H ] . Then,there is a finite index subgroup Γ H < Mod (Σ g ) and a homomorphism ρ H , A : Γ H → Ω H , A whoseimage is of finite index. Note that since there is a bijection between nontrivial simple components A of (cid:81) [ H ] andnontrivial irreducible (cid:81) -representations r of H , we could, with appropriate rewording, replace A with r in the above theorem. Thus, if A corresponds to r , then ρ H , r = ρ H , A and Ω H , r = Ω H , A are respectively the representation and arithmetic group promised at the beginning. Note, inaddition, that in Theorem 1.8, we did not reference the φ -redundant homomorphism p : T g → H .The homomorphism ρ does depend on the choice of p , but under certain conditions (such as g (cid:29) d ( H ) ) we know that different choices of p lead to representations ρ which are equivalentin a natural way. In Section 8.4, we discuss the notion of equivalence of representations ρ andwhen they are known to be equivalent.Theorem 1.2 is deduced from Theorem 1.8 by making special choices of finite groups H andsimple components A of (cid:81) [ H ] and then analyzing the resulting arithmetic groups, in some casesusing Theorem 1.7. (See Section 9 for the corresponding irreducible representations r .) For (a),we use H = Sym ( m + ) and A = Mat m ( (cid:81) ) . For (b) and m ≥
3, we use H = Alt ( m + ) × Dih ( n ) where Dih ( n ) is the dihedral group of order 2 n , and A = Mat n ( L ) where L is the field (cid:81) ( ζ n ) + .For (c) and m ≥
2, we use H = Sym ( m + ) × (cid:90) / n (cid:90) and A = Mat n ( (cid:81) ( ζ n )) . For (d) and m ≥ H = Alt ( m + ) × Dic ( n ) where Dic ( n ) is the dicyclic group of order 4 n (definition inSection 9) and A = Mat n ( D ) where D is the quaternion algebra D = (cid:168)(cid:130) α β − β α (cid:140) ∈ Mat ( (cid:81) ( ζ n )) | α , β ∈ (cid:81) ( ζ n ) (cid:171) .For smaller m in cases (b),(c),(d), see Section 9 for the corresponding H . Theorem 1.3 is alsodeduced from Theorem 1.8, and the fact that, for such L , the algebra Mat s L ( L ) appears as a5imple factor of (cid:81) [ H ] for some finite group H and some s L . The reader may notice that thegroup G obtained is always of type A n , C n , or D n but never B n , G , F , E , E or E .The above mentioned results open a lot of questions. They show that the classical Torelligroup of Mod (Σ) is just a first in a list of countably many “generalized Torelli subgroups” –ker ( ρ H , r ) as above. Are these subgroups (or any of them) finitely generated? Note that these arevery different from the “higher Torelli groups” (or equivalently, groups in the Johnson filtration)obtained from nilpotent quotients of T g ; in fact, for any group I in the Johnson filtration and any T H , r = ker ( ρ H , r ) for H nontrivial, the product I T H , r is of finite index in Mod (Σ) . (See Section8.6.)Our theorem can be useful also toward solving the long-standing problem whether Mod (Σ g ) ,for g ≥
3, can virtually surject onto (cid:90) . In [ ] , Putman and Wieland showed, roughly speaking,that if ρ H , p (Γ H , p ) has no finite orbits for all p and H (a collection for which ker ( p ) is cofinalwould suffice), then the mapping class group (for a surface of genus 1 greater) does not virtuallysurject to (cid:90) . Theorem 1.6 easily implies this condition for φ -redundant p . This is just a stepin this direction since the subgroups ker ( p ) for all such p do not form a cofinal family. (To getthe conclusion that Mod (Σ) does not virtually surject onto (cid:90) , one must prove a similar result forcovers of a surface with one boundary component. See [ ] for the precise formulation.)The key idea for proving Theorem 1.6 is that the handlebody subgroup of the mapping classgroup behaves like a maximal parabolic subgroup of a semisimple Lie group. Let us elaborate.Along the way, we will indicate where each step in the argument is proved. As explained above,we are getting a representation ρ H , p , i of Mod (Σ) , or to be more precise a finite index subgroupof it, into G H , i .Let Σ ∼ = ∂ H be an identification inducing φ : T g → π ( H ) ∼ = F g where H is a genus g handlebody. From [
20, Theorem 5.2 ] , it follows that all surjective homomorphisms φ : T g → F g arise this way. We analyze first the image of Map ( H ) , the handlebody subgroup of Mod (Σ) fora handlebody H with boundary Σ . (More accurately, we will study Map ( H , ∗ ) , the handlebodygroup with a fixed point.) This is the subgroup of Mod (Σ) consisting of all isotopy classes con-taining homeomorphisms which extend to H . This important subgroup of Mod (Σ) has beenactively studied in recent years – see for example [
17, 18 ] . Via its action on the fundamentalgroup of H , it is mapped onto Out ( F g ) , the outer automorphism group of the free group on g generators. Following carefully the definitions, one sees that when p : T g → H is φ -redundant, ρ H , p , i ( Map ( H )) acts on a submodule of M i in precisely the same way as ρ ( Out ( F g )) in [ ] (Sec-tion 6). We can therefore appeal to the results of [ ] to deduce that ρ H , p , i ( Map ( H )) containsan arithmetic subgroup of the Levi factor L of a suitable maximal parabolic subgroup P = LN + of G H , i (Section 6).Moreover, we use again the φ -redundant assumption to show that the image of Γ H , p < Mod (Σ g , ∗ ) contains nontrivial unipotent elements in the unipotent radical N + of P as well as of N − , its opposite subgroup (upper triangular versus lower triangular). At this point, we use bothresults as well as the fact that L acts irreducibly on N + and on N − , to deduce that the image ofMod (Σ) contains finite index subgroups of U + ( O ) = U + ∩ G ( O ) and U − ( O ) = U − ∩ G ( O ) where U + and U − are opposite maximal unipotent subgroups of G H , i (Section 7). Now, we can appeal tothe result of Raghunathan [ ] (see also Venkataramana [ ] ) that two such subgroups generatea finite index subgroup of G H , i ( O ) and Theorem 1.6 is deduced (Section 8).The paper is organized as follows. In Section 2, we prove an analogue of Gaschütz’ Theoremfor surface groups, and in Section 3, we develop what we call “Symplectic Gaschütz theory”,6.e. identifying the structure of ˆ R = R ⊗ (cid:90) (cid:81) , not merely as a (cid:81) [ H ] -module but also as a modulewith a skew-Hermitian form over (cid:81) [ H ] – an algebra with involution. We relate its structure tothe representation theory of the finite group H . In Section 4, we elaborate on this connection.Most of the material in Section 4 is likely known to experts on simple algebras, but we choseto include a quick presentation which seems not to be available in this form. This material isuseful when one comes to producing examples of arithmetic quotients out of finite groups H andtheir representations. But the reader can skip this section on a first reading, going right away toSection 5.In Section 5, we describe a submodule of ˆ R isomorphic to (cid:81) [ H ] which leads (in Section 7)to an identification of two unipotent elements in ρ H , p , i (Γ H , p ), and we prove that the sesquilinearform 〈− , −〉 on M i = A g − i has an isotropic submodule M (cid:48) i isomorphic to A g − i . The parabolicsubgroup P mentioned above consists precisely of those elements of G H , i preserving M (cid:48) i . Some ofthe content of Sections 6, 7, and 8 have been indicated above; additionally, Section 6 establishesthe basic properties of ρ H , p , and in Section 8, we finish the proof of Theorem 1.8. We end inSection 9 with proofs of the remaining results claimed in the introduction and some discussionof the arithmetic groups Ω H , A as ( H , A ) ranges over all pairs of finite groups H and simple com-ponents A of (cid:81) [ H ] . For the convenience of the reader, we collect here some of the more important notation that isconsistent throughout the paper.Mod (Σ) : the mapping class group of the closed surface Σ Mod (Σ , ∗ ) : the mapping class group of the closed surface Σ fixing the point ∗ g : genus of the surface H : a finite group r , r i : an irreducible (cid:81) -representation of HA , A i : a simple (cid:81) -algebra (almost always denoting a component of (cid:81) [ H ] ) M , M i : modules over A , A i (almost always denoting A g − , A g − i ) L , L i : the center of A , A i τ : the canonical involution on (cid:81) [ H ] and each A , A i (See Lemma 3.2.) K , K i : the τ -fixed subfield of L , L i 〈− , −〉 : the sesquilinear pairing on A g − i (or ˆ R ; see Section 3.1.) G H , i : the algebraic group Aut A i ( A g − i , 〈− , −〉 ) (defined over K i ) G H , i : the elements in G H , i of reduced norm 1 O , O i : an order in A , A i which is the image of (cid:90) [ H ]Ω , Ω H , r , Ω H , A : the arithmetic group consisting of O , O i points of G H , i . (where A = A i and r is therepresentation (cid:81) [ H ] → A ) T g : = π (Σ g ) p : T g → H : some surjective homomorphism R : = ker ( p ) R = R / [ R , R ]ˆ R = R ⊗ (cid:81) H , p = { γ ∈ Aut ( T g ) + = Mod (Σ , ∗ ) | p ◦ γ = p } ρ H , p : Γ H , p → Aut (cid:81) [ H ] (ˆ R ) : the representation for the induced action of Γ H , p on ˆ R ρ H , p , i : Γ H , p → G H , i : the map ρ H , p followed by the projection onto the action of the i th isotypiccomponent of ˆ R Γ H : a finite index subgroup of Mod (Σ) (of some relation to Γ H , p ; see Section 8.2.) ρ H , r , ρ H , A : alternative names for the representation Γ H → Ω H , A = Ω H , r where r : (cid:81) [ H ] → A isan irreducible representation. H , H g : the genus g handlebodyMap ( H ) , Map ( H , ∗ ) : the mapping class group of the handlebody (also called the handlebodygroup) respectively without and with a fixed point. F g : the free group of rank g φ : T g → F g : a surjective homomorphism p (cid:48) : F g → H : a surjective homomorphism satisfying p (cid:48) ◦ φ = p (when p is φ -redundant) S : = ker ( p (cid:48) ) S = S / [ S , S ]ˆ S = S ⊗ (cid:81) P : a maximal parabolic in G H , i L : a Levi factor of G H , i N + : the unipotent radical of PN − : the opposite subgroup of N + U + , U − : opposite maximal unipotent radicals of G H , i The authors acknowledge useful discussions with Ursula Hamenstädt and Sebastian Hensel onthe handlebody groups and with V. Venkataramana on generators of arithmetic groups. We arealso grateful to Andrei Rapinchuk, Eli Eljadeff, and Uriah First for various discussions on algebrasand arithmetic groups. The work of the 3rd and 4th authors is supported by the ERC, and thework of the 2nd and 3rd authors by the NSF and BSF.
2. A theorem of Gaschütz and a generalization to surface groups
Suppose we have an exact sequence of groups1 → R → T p → H → T is the fundamental group of a closed orientable surface of genus g and H is a finitegroup. The action of T on R = R / [ R , R ] by conjugation descends to an action of H . Thus, R hasthe structure of a (cid:90) [ H ] module. In this section, we prove Proposition 1.1 from the introduction.We recall the proposition here. Proposition 1.1.
Let T = T g be the fundamental group of a surface Σ of genus g ≥ . Let R bea finite index normal subgroup of T and H = T / R. Then ˆ R = (cid:81) ⊗ (cid:90) ( R / [ R , R ]) is isomorphic as a (cid:81) [ H ] -module to (cid:81) ⊕ (cid:81) [ H ] g − . As mentioned in the introduction, this is a known result due to Chevalley–Weil. We adopt atopological viewpoint to give an alternative proof of Proposition 1.1, and we begin by translating8lgebraic objects into topological ones. For any normal finite index subgroup R < T , there is acorresponding finite index regular cover ˜ Σ → Σ such that the image π ( ˜ Σ) → π (Σ) is precisely R . Using this, one can identify H ( ˜ Σ , (cid:90) ) with R and H ( ˜ Σ , (cid:81) ) ∼ = ˆ R . The group H = T / R acts on thecover by deck transformations and thereby induces an action of H on H ( ˜ Σ , (cid:81) ) . The isomorphismH ( ˜ Σ , (cid:81) ) ∼ = ˆ R is an isomorphism of (cid:81) [ H ] -modules. In the case where T is replaced by a free group F , the description of ˆ R is a classical result ofGaschütz. Since we require its use in later sections, we present Gaschütz’s theorem. We alsoprovide a new topological proof of the theorem which we then adapt for the analagous theoremfor surfaces. Theorem 2.1. (Gaschütz) Suppose → R → F n p → H → is a short exact sequence where F n is thefree group on n generators and H is a finite group. Let R = R / [ R , R ] and ˆ R = R ⊗ (cid:90) (cid:81) . Then, there isa (cid:81) [ H ] -module isomorphism: ˆ R ∼ = (cid:81) [ H ] n − ⊕ (cid:81) Proof.
To prove this, let us first identify F n with the fundamental group of an n -petalled rose Y with oriented edges where each edge is one of the free generators x i of F n . Let (cid:101) Y → Y be thecover corresponding to R . Lift the orientation of Y to an orientation of the edges of (cid:101) Y .We compute H ( (cid:101) Y , (cid:81) ) via cellular homology. Let C i ( (cid:101) Y , (cid:81) ) denote formal sums with (cid:81) coef-ficients of i -cells of (cid:101) Y . Since there are no 2-cells, H ( (cid:101) Y , (cid:81) ) is the kernel of the boundary map ∂ . Pick some vertex ˜ ∗ of the graph (cid:101) Y , and let e , . . . , e n be the (oriented) edges going out from˜ ∗ where e i covers the edge corresponding to the generator x i of F n . The group H acts freely onthe orbit of e i , the H -orbit of e i and e j are disjoint if i (cid:54) = j , and the H -orbits of all the e i cover (cid:101) Y .Thus, we have an internal (cid:81) [ H ] -module direct sum decomposition of the space of 1-chains C ( (cid:101) Y , (cid:81) ) = n (cid:77) i = (cid:81) [ H ] · e i ∼ = (cid:81) [ H ] n Furthermore, C ( (cid:101) Y , (cid:81) ) = (cid:81) [ H ] · ˜ ∗ ∼ = (cid:81) [ H ] . The boundary map is a (cid:81) [ H ] -homomorphism, andfurthermore, ∂ ( e i ) = ( h i − ) · ˜ ∗ where h i = p ( x i ) .The above argument shows that the image of ∂ lies in h · ˜ ∗ where h is the augmentation ideal;i.e. h is the kernel of the augmentation map ε : (cid:81) [ H ] → (cid:81) defined by (cid:80) h ∈ H α h h (cid:55)→ (cid:80) h ∈ H α h .Since we know that dim (cid:81) ( H ( (cid:101) Y , (cid:81) )) =
1, the image of ∂ must be the entire augmentation ideal.Semi-simplicity of (cid:81) [ H ] -modules implies (cid:81) [ H ] n ∼ = C ( (cid:101) Y , (cid:81) ) ∼ = H ( (cid:101) Y , (cid:81) ) ⊕ h .Note that ε is a (cid:81) [ H ] -module homomorphism from (cid:81) [ H ] to the trivial module so (cid:81) [ H ] ∼ = (cid:81) ⊕ h and H ( (cid:101) Y , (cid:81) ) ∼ = (cid:81) [ H ] n − ⊕ (cid:81) . We now prove Proposition 1.1. Let Y be the 2 g -petalled rose with oriented loops, and label theloops by a i , b i for i =
1, . . . g . Let Y be the 2-dimensional CW-complex obtained by gluing a 2-cellalong its boundary to the path [ a , b ][ a , b ] . . . [ a g , b g ] in Y where [ x , y ] = x y x − y − . It is9ell known that Y is a closed, genus g surface. Let (cid:101) Y be the cover corresponding to the subgroup R < T = π ( Y ) .As in the above proof, ∂ ( C ( (cid:101) Y , (cid:81) )) = h still holds. However, in this case, ∂ ( C ( (cid:101) Y , (cid:81) )) isnontrivial, so we must determine it to compute H ( (cid:101) Y , (cid:81) ) . The group H acts freely and transitivelyon the 2-cells of (cid:101) Y , so C ( (cid:101) Y , (cid:81) ) ∼ = (cid:81) [ H ] · c ∼ = (cid:81) [ H ] where c is some oriented 2-cell of (cid:101) Y . By semi-simplicity then, finding ∂ ( C ( (cid:101) Y , (cid:81) )) is equivalent to determining ker ( ∂ ) which, since there areno 3-cells, is H ( (cid:101) Y , (cid:81) ) . Since (cid:101) Y is a surface, H ( (cid:101) Y , (cid:81) ) ∼ = (cid:81) as a (cid:81) -vector space. However, we mustverify that the (cid:81) [ H ] -module structure is trivial. The action of h ∈ H on H ( (cid:101) Y , (cid:81) ) is multiplicationby the degree of the map, and since H acts by orientation-preserving homeomorphisms, thatdegree is necessarily 1.Thus, the image of ∂ is isomorphic to h , and by semi-simplicity of (cid:81) [ H ] -modules, (cid:81) [ H ] g ∼ = C ( (cid:101) Y , (cid:81) ) ∼ = H ( (cid:101) Y , (cid:81) ) ⊕ h ,and the desired result follows.
3. Symplectic Gaschütz Theory
Let us fix R < T , a finite index normal subgroup, and set H = T / R and ˆ R = ( R / [ R , R ]) ⊗ (cid:90) (cid:81) asabove. For surface groups T , the (cid:81) [ H ] -module ˆ R has a richer structure than in the analogoussituation for free groups. Specifically, ˆ R admits a natural (cid:81) [ H ] -valued sesquilinear form 〈− , −〉 .In this section, we describe the form, and we exhibit a decomposition of the pair (ˆ R , 〈− , −〉 ) intofactors. Later in Section 6, we will see that this structure is preserved by the action of ρ H , p (Γ H , p ) (Lemma 6.2). ˆ R We first define a few necessary terms. An anti-homomorphism τ : A → A of a (cid:81) -algebra is a (cid:81) -linear map of A such that τ ( a a ) = τ ( a ) τ ( a ) for all a , a ∈ A ; furthermore τ is an involution if τ = Id. Suppose M is an A -module. A form 〈− , −〉 : M × M → A is sesquilinear (relative to theinvolution τ ) if it is (cid:81) -bilinear and for any r , s ∈ A , m , m (cid:48) ∈ M 〈 r m , sm (cid:48) 〉 = r 〈 m , m (cid:48) 〉 τ ( s ) The form furthermore is skew-Hermitian if 〈 m , m (cid:48) 〉 = − τ ( 〈 m (cid:48) , m 〉 ) and nondegenerate if for allnonzero m ∈ M , there is an m (cid:48) ∈ M such that 〈 m , m (cid:48) 〉 (cid:54) = (cid:81) [ H ] admits a canonical involution τ defined by setting τ ( h ) = h − for h ∈ H and extending linearly. Recall that ˆ R is naturally identified with H ( ˜ Σ , (cid:81) ) which has analternating intersection form, and we denote this form by 〈− , −〉 Sp . In a similar way to [ ] , [ ] , we define the following (cid:81) [ H ] -valued form on ˆ R : 〈 x , y 〉 = (cid:88) h ∈ H 〈 x , h y 〉 Sp h . (1) Lemma 3.1.
The form 〈− , −〉 is nondegenerate, sesquilinear with respect to τ , and skew-Hermitian.Proof. Nondegeneracy follows from the nondegeneracy of the symplectic form. It can be readilychecked that sesquilinearity follows from the fact that H preserves the symplectic form. The10ction of H preserves the symplectic form as it is equivalent to the action of deck transformationson H ( ˜ Σ , (cid:81) ) and deck transformations act by orientation preserving homeomorphisms. The form 〈− , −〉 is skew-Hermitian since 〈− , −〉 Sp is alternating. (cid:81) [ H ] We now recall some basic facts about the group ring (cid:81) [ H ] and its modules. The ring (cid:81) [ H ] is asemisimple (cid:81) -algebra and thus is isomorphic to a finite product of simple (cid:81) -algebras (cid:81) [ H ] = (cid:81) × (cid:96) (cid:89) i = A i .Moreover, for each i , we have A i ∼ = Mat m i ( D i ) for some finite-dimensional division algebra D i with center L i which is a finite-dimensional field extension of (cid:81) . For all i , the algebra A i actson the (cid:81) -vector space V i = D m i i by left multiplication; via the projection to A i , each V i is a (cid:81) [ H ] -module which is furthermore irreducible. We say V i is the irreducible module (or repre-sentation) corresponding to A i . Moreover, every irreducible (cid:81) [ H ] -module (or representation of H ) is isomorphic to one of the V i , and note that as a (cid:81) [ H ] -module, A i ∼ = V m i i .We describe how τ acts on the decomposition in the following lemma. Lemma 3.2.
Each A i in the decomposition of (cid:81) [ H ] is τ -invariant.Proof. Note that each A i is a minimal two-sided ideal of (cid:81) [ H ] . For the purpose of the proof, set A to be the trivial factor (cid:81) in the decomposition of (cid:81) [ H ] . Since τ is an anti-homomorphism, τ sends minimal two-sided ideals to minimal two-sided ideals. Since τ is order 2, there are twopossibilities; for any i either τ ( A i ) = A i , or there is a j (cid:54) = i such that τ ( A i ) = A j and τ ( A j ) = A i .We must rule out the second possibility. (Our proof depends on the base field being the totallyreal field (cid:81) . The lemma is false, for instance, over (cid:67) .)Let e i be the unit in A i . It suffices to show that τ ( e i ) A i (cid:54) =
0. Consider the representation Ψ : (cid:81) [ H ] → End (cid:81) ( A i ) ∼ = Mat t ( (cid:81) ) given by left multiplication where t = dim (cid:81) ( A i ) . Viewing the matrices with entries in (cid:67) yieldsa representation of H to GL t ( (cid:67) ) , and so χ ( h ) = χ ( h − ) where χ is the character for Ψ andindicates complex conjugation. Since the traces are necessarily rational, in fact χ ( h ) = χ ( h − ) ,and linearity of χ implies χ ( r ) = χ ( τ ( r )) for any r ∈ (cid:81) [ H ] . Thus, 0 (cid:54) = χ ( e i ) = χ ( τ ( e i )) and leftmultiplication on A i by τ ( e i ) is non-zero.Recall from Theorem 2.1 that ˆ R ∼ = (cid:81) [ H ] g − ⊕ (cid:81) . Thus, as a (cid:81) [ H ] -module ˆ R ∼ = (cid:81) g ⊕ ( m (cid:77) i = A g − i ) Let M i ⊆ ˆ R be the submodule such that M i ∼ = A g − i . Each A i is isomorphic to several copies ofan irreducible (cid:81) [ H ] -module V i and V i (cid:29) V j for i (cid:54) = j . Hence, the submodule M i is unique, andfurthermore any (cid:81) [ H ] -automorphism ϕ of ˆ R restricts to an automorphism of each M i . Since all A j except A i annihilate M i , we obtain a representationAut (cid:81) [ H ] (ˆ R , 〈− , −〉 ) → Aut A i ( M i , 〈− , −〉 ) .11ur next task then is to understand this automorphism group. (Note that we omitted the modulecorresponding to the trivial representation. This is because the automorphism group turns out tobe Sp ( g , (cid:81) ) and the representation Γ H , p → Sp ( g , (cid:81) ) is the standard symplectic representationof the mapping class group.) M i We first need to know the properties of 〈− , −〉 restricted to each M i . By abuse of notation, wewill view τ as an involution on A i . Lemma 3.3.
For any m , m (cid:48) ∈ M i , the pairing 〈 m , m (cid:48) 〉 lies in A i . Furthermore, 〈− , −〉 : M i × M i → A i is a nondegenerate, sesquilinear, skew-Hermitian form when viewed as a form with values in A i .Proof. Let e i be the unit in A i . When restricted to M i , the form takes values in A i since sesquilin-earity implies that 〈 m , m (cid:48) 〉 = 〈 e i m , e i m (cid:48) 〉 = e i 〈 m , m (cid:48) 〉 τ ( e i ) ∈ A i When viewed as a form with values in A i , it is clear that the form remains sesquilinear and skew-Hermitian. Since 〈− , −〉 is nondegenerate on all of ˆ R , to show that it is nondegenerate on M i ,we only need to show that the M i are mutually perpendicular. This is verified by essentially thesame computation. If m ∈ M i and m (cid:48) ∈ M j and i (cid:54) = j , then 〈 m , m (cid:48) 〉 = 〈 e i m , e j m (cid:48) 〉 = e i 〈 m , m (cid:48) 〉 τ ( e j ) Lemma 3.2 implies that τ ( e j ) = e j , so the product is 0.The group Aut A i ( M i , 〈− , −〉 ) is the set of the K i -points of an algebraic group defined over K i where K i = L τ i , the fixed field of τ acting on L i . Our next goal is to determine the structure ofthis algebraic group; to this end, we will use some of the basic theory of involutions and followthe exposition in [
35, Section 2.3.3 ] . Involutions
In this subsection, we utilize the dictionary between nondegenerate bilinear formsand involutions to describe the structure of G H , i = Aut A i ( M i , 〈− , −〉 ) . For convenience, we willfix A = A i , M = M i , and G = G H , i as we will only be considering each algebra individually, so M ∼ = A ( g − ) with the sesquilinear form as described above. We set L to be the center of A and K = L τ the subfield of L fixed by τ . The dictionary is that the sesquilinear form goes over tothe unique involution σ : End A ( M ) → End A ( M ) satisfying the following for all m , m (cid:48) ∈ M and C ∈ End A ( M ) : 〈 C m , m (cid:48) 〉 = 〈 m , σ ( C ) m (cid:48) 〉 .This is the adjoint involution associated to the sesquilinear form. Now,Aut A ( M , 〈− , −〉 ) = { C ∈ End A ( M ) | 〈 C m , C m (cid:48) 〉 = 〈 m , m (cid:48) 〉 ∀ m , m (cid:48) ∈ M } = { C ∈ End A ( M ) | 〈 m , σ ( C ) C m (cid:48) 〉 = 〈 m , m (cid:48) 〉 ∀ m , m (cid:48) ∈ M } = { C ∈ End A ( M ) | σ ( C ) C = Id } = G In other words, those automorphisms preserving the form can be entirely determined by theinvolution alone. 12nvolutions of finite-dimensional simple algebras over a field k form a trichotomy based onkind / type. The first distinction is that of kind. An involution τ on a finite-dimensional simplealgebra A over a field k is of the first kind if it fixes the center of A and is of the second kind if it doesnot. Furthermore, if A is of the first kind and n = dim L ( A ) , then either dim L ( A τ ) = n ( n + ) / L ( A τ ) = n ( n − ) / L is the center of A and A τ is the subspace of A fixed by τ . Ifthe former is true, we say τ is of orthogonal type and if the latter is true, τ is of symplectic type .It is a standard fact that these are the only two possible types of involutions of the first kind andthis fact is, in particular, implied by Lemma 3.6. We note that in [ ] , orthogonal and symplectictype are called first and second type respectively.We now set B = End A ( M ) and seek to understand the kind and type of the involution σ interms of the kind and type of τ . First, however, we prove that B is itself a simple algebra. Since M is a free A -module of rank 2 g − B is isomorphic to Mat g − ( A op ) and we henceforth fix somesuch identification. Lemma 3.4.
B is simple.Proof.
Since A is a simple finite dimensional (cid:81) -algebra, so is A op , and so A op is isomorphic to amatrix algebra Mat m ( D ) over a division ring D . Thus, B ∼ = Mat g − ( A op ) ∼ = Mat ( g − ) m ( D ) whichis simple.We now convert our terminology into that of matrices to establish the relation between thetype and kind of τ and σ . Note that τ is also an involution on A op of the same kind and type.For a matrix C = ( c i j ) ∈ Mat g − ( A op ) = B , we set C ∗ = ( τ ( c ji )) . Letting e , . . . , e g − be thestandard free basis of A g − , we define a matrix F = ( f i j ) where f i j : = 〈 e j , e i 〉 . It can be checkedthat 〈 C v , w 〉 = 〈 v , Dw 〉 for all v , w ∈ M if and only if F C = D ∗ F . Moreover, F ∗ = − F as 〈− , −〉 isskew-Hermitian. Thus, since ( F C ) ∗ = C ∗ F ∗ = − C ∗ F and ( D ∗ F ) ∗ = − F D , we have σ ( C ) = F − C ∗ F . Lemma 3.5.
The involutions σ and τ are of the same kind. The type of σ is opposite that of τ .Proof. It is clear that L σ = L τ so they are of the same kind. A straightforward argument countingdimension shows that the involution ∗ on B has the same type as τ if they are both first kind.Because 〈− , −〉 is skew-Hermitian, it follows that F ∗ = − F . Then, a computation shows that if C ∗ = − C , then σ ( F − C ) = F − C and if C ∗ = C , then σ ( F − C ) = − F − C . Hence, via F − , the + ∗ is isomorphic to the − σ and vice versa. This implies σ hasthe opposite type to ∗ . Extending scalars
Consider now the following three examples of involutions ν for matrix al-gebras over (cid:67) .(1) ν : Mat n ( (cid:67) ) → Mat n ( (cid:67) ) defined by ν ( N ) = N t .(2) ν : Mat n ( (cid:67) ) → Mat n ( (cid:67) ) defined by ν ( N ) = J N t J − where n is even and J = (cid:130) I − I (cid:140) .(3) ν : Mat n ( (cid:67) ) × Mat n ( (cid:67) ) → Mat n ( (cid:67) ) × Mat n ( (cid:67) ) defined by ν ( A , B ) = ( B t , A t ) .13ote that if we consider the elements satisfying ν ( M ) M = I , then from (1), we obtain O n ( (cid:67) ) ,from (2), Sp n ( (cid:67) ) and from (3), GL n ( (cid:67) ) . Moreover, it can be easily checked that the involution in(1) is of orthogonal type and the involution in (2) is of symplectic type. We quote the followingresult [ ] which essentially tells us these are the only involutions after extending scalars. Lemma 3.6.
Let B be a finite-dimensional simple (cid:81) -algebra with center L and involution σ . LetK = L σ and let ˜ σ be the unique (cid:67) -linear extension of σ to B ⊗ K (cid:67) . Set n = dim L ( B ) . • If σ is of the first kind, then there is a (cid:67) -algebra isomorphism ϕ : B ⊗ K (cid:67) ∼ = Mat n ( (cid:67) ) such that ν = ϕ ˜ σϕ − is one of the involutions (1) or (2). • If σ is of the second kind, then there is a (cid:67) -algebra isomorphism ϕ : B ⊗ K (cid:67) ∼ = Mat n ( (cid:67) ) × Mat n ( (cid:67) ) such that ν = ϕ ˜ σϕ − is the involution (3). At this point, we can already prove part of Theorem 1.7. Specifically, we can show that thetype of G is determined by the type and kind of τ . We will be able to prove the full theorem afterestablishing further results in Section 4. G ( O ) We conclude this section with a few facts on reduced norms and apply them to G ( O ) = Aut O ( O g − , 〈− , −〉 ) ⊆ End A ( A g − ) where, as defined in the introduction, O is the image of (cid:90) [ H ] in A . Our goal is to prove Propo-sition 3.9, which gives an “upper bound” on the image of the representation ρ which we study(while Theorem 1.6 gives the “lower bound”). Recall from the introduction that G ( O ) = G H , i ( O ) is defined to be the subgroup of G ( O ) consisting of elements of reduced norm 1.The reduced norm (over L ) of a finite-dimensional central simple L -algebra B is defined asfollows. Let E be any field extension of L such that B ⊗ L E splits , i.e. there is some isomorphism ϕ : B ⊗ L E → Mat n ( E ) for some n . Then the reduced norm (over L ) for b ∈ B isnrd B / L ( b ) = det ( ϕ ( b ⊗ )) .The reduced norm is independent of the isomorphism ϕ , and nrd B / L ( B ) ⊆ L . (See e.g. [ ] .) In later sections, we also use the reduced trace which, for b ∈ B , istrd B / L ( b ) = Tr ( ϕ ( b ⊗ )) where Tr is the trace of the E -linear map ϕ ( b ) . Just as for reduced norm, the reduced tracealways lies in L and is independent of ϕ . From the definitions, it is clear that trd B / L is L -linear. Lemma 3.7.
Let B = Mat g − ( A op ) . Let b ∈ G = Aut A ( A g − , 〈− , −〉 ) ⊆ B and λ = nrd B / L ( b ) .Then, τ ( λ ) λ = . In particular, if τ is of the first kind, then λ = ± . roof. We know that b ∈ G satisfies 1 = σ ( b ) b , and so1 = nrd B / L ( σ ( b )) nrd B / L ( b ) = nrd B / L ( σ ( b )) λ .Thus, it suffices to show that nrd B / L ( σ ( b )) = τ ( λ ) .Let E be a finite Galois field extension of L such that there is an isomorphism ϕ : B ⊗ L E ∼ = Mat n ( E ) for some n . Let φ be the composition B → B ⊗ L E ∼ = Mat n ( E ) . Let ν = τ | L , and let ˜ ν bean extension of ν to a Galois automorphism of E . For C = ( c i j ) ∈ Mat n ( E ) , define C ∗ = ( ˜ ν ( c ji )) .Now, let ψ : B → Mat n ( E ) be the map defined by b (cid:55)→ ( φ ( σ ( b ))) ∗ . Note that since σ and ∗ are both anti-homomorphisms, ψ is a homomorphism, and moreover, one can check that ψ isan L -algebra homomorphism. By the Skolem-Noether Theorem, ψ and φ are the same up toconjugation by an element of Mat n ( E ) . Thus,nrd B / L ( σ ( b )) = det ( φ ( σ ( b ))) = ν − ( det ( ψ ( b ))) = ν − ( det ( φ ( b ))) = ν − ( nrd B / L ( b )) .Since nrd B / L ( b ) ∈ L , we can replace ν − with τ − or equivalently τ .We next show that elements in the ring of integers in a cyclotomic field E of “absolute value” ± Lemma 3.8.
Let E = (cid:81) ( ζ ) be a cyclotomic field, let O E its ring of integers, and let ι : E → (cid:67) besome embedding into (cid:67) . If λ ∈ O E and | ι ( λ ) | = , then λ is a root of unity.Proof. According to a theorem of Kronecker, if all the Galois conjugates of an algebraic integer λ have absolute value ≤
1, then λ is a root of unity. Let ν be the order 2 Galois automorphism thatis the restriction of complex conjugation (via ι ). As | ι ( e ) | = ι ( ν ( e ) e ) for any e ∈ E , it will sufficeto show that ν ( σ ( λ )) σ ( λ ) = σ of E . Since E / (cid:81) is cyclotomic, itsGalois group is abelian, and thus ν ( σ ( λ )) σ ( λ ) = σ ( ν ( λ )) σ ( λ ) = σ ( ν ( λ ) λ ) = σ ( ) = Remark.
The above lemma holds for any CM-field. The essential point is that complex conjugationalways restricts to the same automorphism of E regardless of the embedding of E into (cid:67) . Proposition 3.9.
The group G ( O ) is of finite index in G ( O ) .Proof. It suffices to show that the image of G ( O ) under nrd B / L consists only of roots of unity.If τ is of the first kind, this is obvious by Lemma 3.7. Suppose τ is of the second kind. Any λ ∈ nrd B / L ( G ( O )) lies in O L where O L is the ring of integers in L , and by Lemma 3.7, λ satisfies τ ( λ ) λ =
1. Fix some embedding ι : L → (cid:67) . By Proposition 4.3 below, τ | L extends to complexconjugation, and so 1 = ι ( τ ( λ ) λ ) = ι ( λ ) ι ( λ ) = | ι ( λ ) | . Since L is a subfield of a cyclotomic field E (to which ι extends), by Lemma 3.8, λ is a root of unity in L .15 . Determining type and kind of τ The kind of algebraic group into which ρ H , p , i maps depends on τ | A i . The kind and type of τ depends on the representation theory of the finite group H . We therefore dedicate this sectionto study this dependency. This will be important later when we come in Section 9 to producespecific arithmetic groups as (virtual) quotients of Mod (Σ) . However, this section may be skippedon a first reading if the reader wants to see first why the image of ρ H , p , i is an arithmetic group.Before beginning, we note additionally that the material presented here is known to experts, butwe include it for the convenience of the non-expert and for lack of a convenient reference. As inthe previous section, we will focus on one nontrivial component of (cid:81) [ H ] and drop the subscript i . Since we fix some A , by abuse of notation, we will refer to τ | A as τ .Recall that there is a trichotomy: τ can be of the first kind and orthogonal type, of the firstkind and symplectic type, or of the second kind. This trichotomy corresponds in a nice wayto other trichotomies in the representation theory of the finite group H . The first trichotomyarises from the various kinds of invariant bilinear forms for complex representations. Fix someembedding L (cid:44) → (cid:67) and some isomorphism A ⊗ L (cid:67) ∼ = Mat n ( (cid:67) ) . Then the composition H → A → A ⊗ L (cid:67) → Mat n ( (cid:67) ) yields an action of H on V = (cid:67) n which in turn is an irreducible (cid:67) [ H ] -module.There are three possibilities: V has an H -invariant non-degenerate symmetric bilinear form, an H -invariant non-degenerate alternating bilinear form, or no nonzero H -invariant bilinear form.This corresponds to the type and kind of τ as follows. Proposition 4.1.
Keeping notation as above, we have that, restricted to A, the involution τ is • of the first kind and orthogonal type if and only if V has an H-invariant nondegeneratesymmetric bilinear form, • of the first kind and symplectic type if and only if V has an H-invariant nondegenerate alter-nating bilinear form, • of the second kind if and only if V has no invariant non-zero H-invariant nondegeneratebilinear form. Before proving this proposition, we first describe two other trichotomies closely related to thetrichotomy of kind and type. One of these is a trichotomy involving the real representations of H . This trichotomy also relates to a dichotomy involving the center L of A . In order to presentthe dichotomy, we first recall some elementary facts about L . A short proof of the lemma is givenbelow. Lemma 4.2.
Let A be a simple component of (cid:81) [ H ] with center L. Then, L is a subfield of acyclotomic field extension of (cid:81) . Moreover, L is either totally imaginary (there are no embeddingsL (cid:44) → (cid:82) ) or totally real (all embeddings L (cid:44) → (cid:67) have image in (cid:82) ). The next proposition presents the relation to kind and type. The isomorphisms in the propo-sition below are as (cid:82) -algebras.
Proposition 4.3.
Let notation be as above, let n = dim L ( A ) and let e : L (cid:44) → (cid:67) be an arbitraryembedding. Then, the embedding satisfies K = e − ( (cid:82) ) , i.e. τ , restricted to A, is of the first kind ifand only if L is totally real. If τ is of the second kind, then τ | L extends to complex conjugation viathe embedding e. Moreover, restricted to A, the involution τ is of the first kind and orthogonal type if and only if A ⊗ K , e (cid:82) ∼ = Mat n ( (cid:82) ) , • of the first kind and symplectic type if and only if A ⊗ K , e (cid:82) ∼ = Mat n / ( (cid:72) ) , • of the second kind if and only if A ⊗ K , e (cid:82) ∼ = Mat n ( (cid:67) ) . Another trichotomy relates to the Frobenius-Schur indicator ιχ for any character χ of H ,defined to be ιχ = | H | (cid:88) h ∈ H χ ( h ) .We will be quoting some results from [ ] where ιχ is defined differently and only for irreducible (cid:67) -characters (over (cid:67) ). However, the definitions coincide for irreducible characters (see the proofof Theorem 23.14 in [ ] ), and this is all we require. Proposition 4.4.
Continuing our notation as above, let χ be the character corresponding to theirreducible (cid:81) -representation V of H. Then, restricted to A, the involution τ is • of the first kind and orthogonal type if and only if ιχ > , • of the first kind and symplectic type if and only if ιχ < , • of the second kind if and only if ιχ = τ is the unique involution on A with a certain property. Lemma 4.5.
Suppose k is some field and ψ : k [ H ] → A is a surjective homomorphism of k-algebras.Then, there is at most one k-algebra involution ν on A such that ν ( ψ ( h )) ψ ( h ) = Id .Proof. The elements ψ ( h ) generate A as a k -vector space. Proof of Proposition 4.1.
Suppose τ is of the first kind. Then, τ is L -linear and the involutionextends to a ( (cid:67) -algebra) involution ˜ τ of A ⊗ L (cid:67) . By Lemma 3.6, there is an isomorphism ϕ : A ⊗ L (cid:67) ∼ = Mat n ( (cid:67) ) such that ν = ϕ ˜ τϕ − is the standard symplectic or orthogonal involution. If ψ is the composition (cid:81) [ H ] → A → A ⊗ L (cid:67) , then it is the case that ˜ τ ( ψ ( h )) ψ ( h ) = Id and so ν ( ϕ ( ψ ( h ))) ϕ ( ψ ( h )) = Id. This implies ϕ ( ψ ( h )) preserves a nondegenerate symmetric bilinearform (resp. alternating form) if τ is of orthogonal (resp. symplectic) type.Suppose that V has a non-zero H -invariant bilinear form. Then, since V is irreducible, theform must be nondegenerate because a nontrivial null-space would necessarily be H -invariant.Now, let ψ be the projection (cid:81) [ H ] → A . For all h ∈ H , the adjoint involution ν associated tothe form satisfies ν ( ψ ( h ) ⊗ ) = ( ψ ( h ) ⊗ ) − . Thus, ν preserves A = A ⊗
1, and by Lemma 4.5 ν | A = τ . Since ν is (cid:67) -linear, L ⊗ = ⊗ L , and ν | L ⊗ = τ , the involution τ is L -linear andthus of the first kind. Moreover, ν is the extension of τ , so they are of the same type and ν isof orthogonal type (resp. symplectic type) when the invariant bilinear form is symmetric (resp.alternating).Before moving on to the next characterization, we first prove Lemma 4.2 and a lemma aboutinvolutions when projecting (cid:82) [ H ] to a simple (cid:82) -algebra.17 roof of Lemma 4.2. First we show L is a subfield of a cyclotomic field. The reduced trace (over L ) is an L -linear map trd A / L : A → L , and it is clear that it is surjective. (Indeed, trd A / L ( L ) = L .)Choose some embedding L → (cid:67) and an isomorphism ϕ : A ⊗ L (cid:67) ∼ = Mat n ( (cid:67) ) , and let r : (cid:81) [ H ] → A be the projection. Some power of the image ϕ ( r ( h ) ⊗ ) is the identity, and so its trace (whichis trd A / L ( r ( h )) by definition) is a sum of roots of unity. The composition trd A / L ◦ r is (cid:81) -linear and H spans the image L as a (cid:81) -vector space. It is clear from this that L is a subfield of a cyclotomicfield. The second claim follows from the fact that subfields of cyclotomic fields are all Galoisextensions of (cid:81) . Lemma 4.6.
Suppose there is a surjection of (cid:82) -algebras ψ : (cid:82) [ H ] → B where B is simple. There isan (cid:82) -linear involution ν on B such that ν ( ψ ( h )) ψ ( h ) = Id for every h ∈ H and • ν is of the first kind and orthogonal type if and only if B ∼ = Mat m ( (cid:82) ) for some m ∈ (cid:78) , • ν is of the first kind and symplectic type if and only if B ∼ = Mat m ( (cid:72) ) for some m ∈ (cid:78) , • ν is of second kind if and only if B ∼ = Mat m ( (cid:67) ) for some m ∈ (cid:78) , in which case the center of Bis (cid:67) and (cid:67) ν = (cid:82) .Proof. First, note that since the only finite-dimensional division algebras over (cid:82) are (cid:82) , (cid:67) , and (cid:72) , it follows that B is isomorphic to one of Mat m ( (cid:82) ) , Mat m ( (cid:67) ) , Mat m ( (cid:72) ) for some m ∈ (cid:78) . Letrespectively D be one of (cid:82) , (cid:67) , or (cid:72) . We can without loss of generality identify B with Mat m ( D ) ,and thus obtain a corresponding action on D m .In the cases of D = (cid:82) , (cid:67) , we can average the standard inner or Hermitian product on D m under H to obtain an H -invariant inner or Hermitian product. In the case of (cid:72) , there is also astandard Hermitian product, namely v · w = (cid:80) mi = v i w i where w is the involution on (cid:72) given by a + bi + c j + d k = a − bi − c j − d k . In this case also, averaging the standard Hermitian productyields an H -invariant Hermitian product. In each case v · v > v and the averagehas the same property. By a D -linear change of basis via Gram-Schmidt, we may assume the H -invariant inner or Hermitian products are the standard ones. We can take ν ( b ) = b t in thecase of D = (cid:82) and ν ( b ) = b t in the cases of D = (cid:67) , (cid:72) . This involution is easily checked to satisfythe claimed properties. Uniqueness of ν by Lemma 4.5 allows us to conclude the “only if” in eachof the three statements. Proof of Proposition 4.3.
Let K (cid:48) = e − ( (cid:82) ) . Implicitly, when we tensor over K (cid:48) or L , we will do itvia the map e . If K (cid:48) (cid:54) = L , then we have an isomorphism e (cid:48) : L ⊗ K (cid:48) (cid:82) → (cid:67) satisfying e (cid:48) ( (cid:96) ⊗ λ ) = e ( (cid:96) ) λ , and so by associativity of tensor products, there are isomorphisms A ⊗ K (cid:48) (cid:82) ∼ = A ⊗ L ( L ⊗ K (cid:48) (cid:82) ) ∼ = A ⊗ L (cid:67) .In either case ( L = K (cid:48) or L (cid:54) = K (cid:48) ), A ⊗ K (cid:48) (cid:82) is isomorphic to A tensored over its center L witha simple (cid:82) -algebra ( (cid:82) or (cid:67) ) and hence is simple. We have A ⊗ K (cid:48) (cid:82) ∼ = Mat n ( (cid:82) ) , Mat n ( (cid:67) ) , orMat n / ( (cid:72) ) .Let ϕ : (cid:81) [ H ] → A be the projection. There is an induced homomorphism (cid:82) [ H ] = (cid:81) [ H ] ⊗ (cid:81) (cid:82) → A ⊗ K (cid:48) (cid:82) , and since the image of H generates A over (cid:81) (and K (cid:48) ), this map is surjective.Let ν be the involution on A ⊗ K (cid:48) (cid:82) as in Lemma 4.6, which is unique by Lemma 4.5. Since ν ( ϕ ( h )) = ϕ ( h ) − and ϕ ( h ) ∈ A ⊗
1, the involution ν restricts to an involution on A = A ⊗ K (cid:48) -linear (hence (cid:81) -linear). By uniqueness (Lemma 4.5), ν | A = τ . By Lemma 4.6, the ν -fixed subfield of the center in all cases is (cid:82) whose intersection with e ( L ) is e ( K (cid:48) ) , and agreementof ν and τ on A implies K (cid:48) = K .The above argument implies that if L = K , then all embeddings of L in (cid:67) are real-valued;consequently, A ⊗ K (cid:82) has an involution of the first kind and orthogonal or symplectic type and so A ⊗ K (cid:82) is isomorphic to either Mat n ( (cid:82) ) or Mat n / ( (cid:72) ) respectively. Conversely, if A ⊗ K (cid:82) is eitherMat n ( (cid:82) ) or Mat n / ( (cid:72) ) , then ν fixes the center and so fixes L which implies τ is of the first kind.The remaining cases are L (cid:54) = K and A ⊗ K (cid:82) ∼ = Mat n ( (cid:67) ) and so they coincide.To prove Proposition 4.4, we first establish some facts for ιχ when χ is an irreducible (cid:67) -character. We recall some theorems relating ιχ to the existence of certain invariant bilinearforms on irreducible representations and translate them into facts about involutions. We willthen bootstrap the results to (cid:81) to prove the proposition. We recall Theorem 23.16 from [ ] with some mild modification. Note that for irreducible characters (over (cid:67) ), ιχ takes only valuesin {−
1, 0, 1 } . Theorem 4.7.
Suppose V is an irreducible (cid:67) [ H ] -module with character χ . Then ιχ (cid:54) = if andonly if V admits a non-zero H-invariant bilinear form, and in this case, the form is nondegenerate.Furthermore, the form is symmetric if and only if ιχ = and skew-symmetric if and only if ιχ = − .Proof. This is precisely Theorem 23.16 from [ ] except for our claim that the forms are nonde-generate. (In [ ] , it only says “non-zero”.) The form must be non-degenerate by Proposition4.1.By Proposition 4.1, proving Proposition 4.4 requires only that we translate information aboutcomplex traces to rational ones. Recall that A is a simple factor in the decomposition of (cid:81) [ H ] .Let W be the corresponding irreducible (cid:81) -representation. Let A = A ⊗ L (cid:67) . We have a surjection (cid:67) [ H ] ∼ = (cid:81) [ H ] ⊗ (cid:81) (cid:67) → A ∼ = Mat n ( (cid:67) ) and a corresponding irreducible (cid:67) -representation V . Un-fortunately, it is not always the case that V is W tensored with (cid:67) , so we must expend some effortto relate the characters.First, we recall the definitions of various traces for algebras over fields and some well-knownfacts relating them. Suppose A is a finite-dimensional (cid:81) -algebra with center L . Then, A acts onitself by left multiplication and this gives a representation η (cid:96) : A → End L ( A ) . Define the trace of a ∈ A over L to be Tr A / L ( a ) = tr ( η (cid:96) ( a )) .Furthermore, for any field k contained in L , left multiplication by a is k -linear and we canconsider the trace as a k -linear map on the k -vector space A . Denote this new trace Tr A / k ( a ) .If k is a subfield of L , then as for A , we can view elements in L as k -linear maps on L by leftmultiplication. As before we have Tr L / k ( α ) for α ∈ L . We recall the following elementary factabout traces [
40, Section 9 ] . In particular, for the algebra A , we have Tr A / k ( a ) = Tr L / k ( Tr A / L ( a )) . Lemma 4.8.
Suppose V is an L-vector space and T : V → V is L-linear. Let T k be T viewed as ak-linear map on the k-vector space V . Then, tr ( T k ) = Tr L / k ( tr ( T ) . ) We relate the trace Tr A / L and the corresponding character for irreducible representations.19 emma 4.9. Suppose V is k-vector space and H → GL ( V ) is an irreducible k-representation andsuppose A is the corresponding simple k-algebra in the decomposition of k [ H ] . Let χ be the characterof V . Then, for a ∈ A, m χ ( a ) = Tr A / k ( a ) where A ∼ = V m as k [ H ] -modules.Proof. By definition of character, χ ( a ) is the trace of a via the action of k [ H ] , and hence A , on V . The action of A on itself block-diagonalizes as m copies of its action on V and the result easilyfollows.Recall the reduced trace trd A / L for a central simple L -algebra A from Section 3.4. In additionto the properties discussed in that section, we have that Tr A / L = n trd A / L where n = dim L ( A ) .(See [
40, Section 9 ] .) Now, we have the tools to prove Proposition 4.4. Proof of Proposition 4.4.
As above, set A = A ⊗ L (cid:67) which has an isomorphism ϕ : A → Mat n ( (cid:67) ) ,and let V be the corresponding irreducible (cid:67) -representation of H . Let ψ be the correspondingcharacter. Proposition 4.1 and Theorem 4.7 imply that it is sufficient to show that ιψ and ιχ arepositive multiples of each other.Now suppose η A : H → A × is the representation from H to the invertible elements of A and η A : H → A × is the composition of η A with the canonical A → A ⊗ L (cid:67) = A . Note that for all h ∈ H , by definition, ψ ( h ) is the trace of η A ( h ) which is the reduced trace (over L ) of η A ( h ) .Thus, | H | ιψ = (cid:88) h ∈ H ψ ( h ) = (cid:88) h ∈ H trd A / L ( η A ( h )) .By the properties of reduced trace, (cid:88) h ∈ H trd A / L ( η A ( h )) = n (cid:88) h ∈ H Tr A / L ( η A ( h )) .Now, suppose m is such that A ∼ = V m where V is the irreducible representation corresponding to A . Since ιψ ∈ {−
1, 0, 1 } , we further have that if d = dim (cid:81) ( L ) , | H | ιψ = | H | d Tr L / (cid:81) ( ιψ ) = nd Tr L / (cid:81) ( (cid:88) h ∈ H Tr A / L ( η A ( h )))= nd (cid:88) h ∈ H Tr A / (cid:81) ( η A ( h )) = mnd (cid:88) h ∈ H χ ( h ) = | H | mnd ιχ .where the fourth equality follows from Lemma 4.9.We now prove Theorem 1.7. As we have been doing, we drop the subscript i in our proof. Proof of Theorem 1.7.
Recall that G = Aut A ( A g − , 〈− , −〉 ) , and that σ is the adjoint involutiondefined on B = Mat g − ( A op ) and associated to 〈− , −〉 . Extend σ (uniquely) to a (cid:67) -linear invo-lution ˜ σ on B ⊗ K (cid:67) and let G = { C ∈ B ⊗ K (cid:67) | ˜ σ ( C ) C = I } .Then, G is a K -defined algebraic group whose K -points are G . Let n = dim L ( A ) .20irst, suppose that τ is of the first kind. By Lemma 3.6, there is an isomorphism ϕ : B ⊗ K (cid:67) → Mat ( g − ) n ( (cid:67) ) such that ν = ϕ ˜ σϕ − is either involution (1) or (2). The involution ν has thesame type as ˜ σ and σ , which, by Lemma 3.5, have the opposite type of τ . Thus ϕ maps G isomorphically onto Sp ( g − ) n ( (cid:67) ) if τ is of orthogonal type and onto O ( g − ) n ( (cid:67) ) otherwise.If τ is of the second kind, then there is an isomorphism ϕ : B ⊗ K (cid:67) → Mat ( g − ) n ( (cid:67) ) × Mat ( g − ) n ( (cid:67) ) such that ν = ϕ ˜ σϕ − is the involution (3). Arguments similar to the above show G ∼ = GL ( g − ) n ( (cid:67) ) . The remaining equivalences follow from Propositions 4.1 and 4.3.
5. Submodules of M i In this section we analyze further the structure of ˆ R , when p is φ -redundant. In this case, weshow that ˆ R decomposes into two isomorphic totally isotropic submodules. Later, in Section 6,we will see that the image of the handlebody group under ρ preserves one of the submodulesand thus has a block uppertriangular form. Moreover, we will show that when p is φ -redundant,there is an explicit rank two (cid:81) [ H ] -submodule of ˆ R which allows us (in Section 6) to produce twotightly controlled unipotent elements in the image of ρ H , p .Recall that if φ : T g → F g is a surjective homomorphism, we say that an epimorphism p : T g → H is φ -redundant if p factors through an epimorphism p (cid:48) : F g → H which is redundant(i.e. the kernel contains a free generator). As mentioned in the introduction, every epimorphism φ : T g → F g arises as follows [
20, Theorem 5.2 ] . Let H = H g be a genus g handlebody and picksome identification of ∂ H with Σ g . The inclusion Σ g (cid:44) → H induces a map on the fundamentalgroups which is surjective. Since the fundamental group of H is a free group, this induces asurjective map φ : T g → F g = π ( H ) where F g is the free group of rank g . Henceforth, we willsimply refer to π ( H ) as F g .Now fix some φ : T g → F g arising from an identification Σ ∼ = ∂ H , fix some φ -redundant p ,and set S = ker p (cid:48) for the corresponding p (cid:48) : F g → H . Let ˜ Σ → Σ be the cover corresponding to p . Aut ( F g ) In this section, we recall a basic fact about handlebody groups. Let ∗ be some point on theboundary of H . The handlebody group with fixed point, Map ( H , ∗ ) , is the subgroup of thosemapping classes in Mod (Σ , ∗ ) which contain a representative homeomorphism extending to ahomeomorphism of H . Recall that the handlebody group (without fixed point) Map ( H ) is thesimilarly defined subgroup of Mod (Σ) . Theorem 5.1. [
15, 33, 45 ] The natural homomorphisms
Map ( H , ∗ ) → Aut ( F g ) and Map ( H ) → Out ( F g ) are surjective. We start by finding a special torus with one boundary component embedded in Σ . Let ∗ ∈ Σ = ∂ H be a basepoint for the fundamental groups of both H and Σ . Lemma 5.2.
Let p : T g → H be φ -redundant and p (cid:48) : F g → H the induced map. Then, Σ containsa subsurface Σ (cid:48) homeomorphic to a torus with one boundary component such that the image of π (Σ (cid:48) , ∗ ) in π (Σ , ∗ ) = T g lies in the kernel of p. Moreover, there are simple closed curves a , b in Σ (cid:48) such that a and b intersect once, and 〈 a , b 〉 Sp = • the homology classes of a , b generate H (Σ (cid:48) , (cid:81) ) • b bounds a disc in H • a , b avoid ∗ .Proof. Let α be a free basis element of F g lying in the kernel of p (cid:48) . This exists by the assumptionthat p is φ -redundant. We first show that the homotopy class α contains some simple closedcurve a supported on Σ . Moreover, we show that there is an oriented simple closed curve b on Σ passing through ∗ such that b intersects a transversely at one point and the based homotopyclass of b lies in the kernel of φ . (Note that since we define Mod (Σ) as Homeo + (Σ) / Homeo (Σ) ,we are implicitly working in the category of topological spaces and so “transverse intersection”strictly speaking has no meaning. However, Mod (Σ) is also Diff + (Σ) / Diff (Σ) , and we can justas well work in the smooth category.) To see all this, note that by assumption there is some freebasis α = α , . . . , α g of F g . Recall that φ : T g → F g is the natural map on fundamental groups.By explicit construction, one can find a (potentially different) free basis α (cid:48) , . . . , α (cid:48) g of F g suchthat each homotopy class α (cid:48) i contains a simple closed curve a (cid:48) i on Σ ; moreover, we can ensurethere is some oriented simple closed curve b (cid:48) passing through ∗ such that a (cid:48) and b intersect oncetransversely and b (cid:48) bounds a disc in H (so b (cid:48) lies in the kernel of φ ). See Figure 1.Figure 1: The curves a (cid:48) , . . . , a (cid:48) g and b (cid:48) in the proof of Lemma 5.2 for g = ( H , ∗ ) → Aut ( F g ) is surjective (Theorem 5.1), there is some homeomorphism ϕ ofthe handlebody fixing ∗ and mapping α (cid:48) to α . Let a = ϕ ( a (cid:48) ) and b = ϕ ( b (cid:48) ) . Note that b boundsa disc in H as well.Since a and b intersect once transversely, a regular neighborhood Σ (cid:48) of a and b in Σ is a toruswith one boundary component and with fundamental group generated by the homotopy classesof a and b . Since (the homotopy classes of) a and b both lie in the kernel of p , it follows that(the image of) π (Σ (cid:48) , ∗ ) lies in the kernel. 22s it stands, the curves a , b satisfy the first three properties required by the lemma but a and b do not avoid ∗ . This is easily fixed by an isotopy of the curves.Let Σ (cid:48) , a , b be as in the lemma. Then, the preimage of Σ (cid:48) under the cover ˜ Σ → Σ consists of | H | disjoint homeomorphic copies of Σ (cid:48) . Label these surfaces as ˜ Σ (cid:48) h as h ranges over H so that d h ( ˜ Σ (cid:48) H ) = ˜ Σ (cid:48) h where d h is the deck transformation induced by h ∈ H .Similar to Σ (cid:48) , the preimages of a and b each consist of | H | simple closed curves which projecthomeomorphically to a and b via the covering map. Call the preimages ˜ a h and ˜ b h as h rangesover H so that a h , b h lie in ˜ Σ (cid:48) h . Then, a h = d h ( a H ) and b h = d h ( b H ) . For a simple closed curve c ,we denote its homology class by [ c ] . Lemma 5.3.
Let notation be as above. Then [ ˜ a H ] (similarly [ ˜ b H ] ) generates a submodule of H ( ˜ Σ , (cid:81) ) isomorphic to (cid:81) [ H ] . Together, [ ˜ a H ] and [ ˜ b H ] generate a submodule isomorphic to (cid:81) [ H ] . Moreover, 〈 [ ˜ a H ] , [ ˜ b H ] 〉 = .Proof. Consider the disjoint union ˜ Σ (cid:48) = ∪ h ∈ H Σ (cid:48) h . From the Mayer-Vietoris sequence, one candeduce that H ( ˜ Σ (cid:48) , (cid:81) ) is a direct summand as a (cid:81) -vector space of H ( ˜ Σ , (cid:81) ) . Since the actionof H preserves ˜ Σ (cid:48) and its complement, H ( ˜ Σ (cid:48) , (cid:81) ) is a summand as a (cid:81) [ H ] -module. As ˜ Σ (cid:48) isthe disjoint union of the ˜ Σ (cid:48) h , it follows that H ( ˜ Σ (cid:48) , (cid:81) ) ∼ = (cid:76) h ∈ H H ( ˜ Σ (cid:48) h , (cid:81) ) . Each H ( ˜ Σ (cid:48) h , (cid:81) ) is2-dimensional so dim (cid:81) H ( ˜ Σ (cid:48) , (cid:81) ) = | H | .Now, we know that for each h ∈ H , the classes [ ˜ a h ] and [ ˜ b h ] generate H ( ˜ Σ (cid:48) h , (cid:81) ) , andso [ ˜ a H ] , [ ˜ b H ] generate H ( ˜ Σ (cid:48) , (cid:81) ) as a (cid:81) [ H ] -module. Because of dimension, we must have [ ˜ a H ] , [ ˜ b H ] freely generate, and H ( ˜ Σ (cid:48) , (cid:81) ) ∼ = (cid:81) [ H ] . It follows that each of [ ˜ a H ] and [ ˜ b H ] generate a module isomorphic to (cid:81) [ H ] . For the last claim, recall that by definition, 〈 [ ˜ a H ] , [ ˜ b H ] 〉 = (cid:88) h ∈ H 〈 [ ˜ a H ] , h · [ ˜ b H ] 〉 Sp h Since h · [ ˜ b H ] = [ ˜ b h ] is disjoint from ˜ a H , the only term which survives is that for h = H . Since 〈 [ ˜ a H ] , [ ˜ b H ] 〉 Sp = 〈 [ a ] , [ b ] 〉 Sp =
1, the claim follows.
The inclusion Σ g (cid:44) → H induces a surjective map H (Σ g , (cid:81) ) → H ( H , (cid:81) ) . It follows that the actionof a homeomorphism of the handlebody preserves the kernel. It is easy to see that the kernel ofthis map is totally isotropic relative to the alternating intersection form 〈− , −〉 Sp . I.e. for any twoelements β , β (cid:48) in the kernel, 〈 β , β (cid:48) 〉 Sp =
0. Moreover, this subspace has a complementary totallyisotropic subspace which projects onto H ( H , (cid:81) ) . We want to show similarly that ˆ R = H ( ˜ Σ , (cid:81) ) has an isotropic submodule and decomposition.Now, let ˆ S = ( S / [ S , S ]) ⊗ (cid:90) (cid:81) which is naturally identified with H ( ˜ H , (cid:81) ) , the rational firsthomology of the cover ˜ H → H corresponding to the inclusion S → F g . Let ˆ P be the kernel ofthe map ˆ φ R : ˆ R → ˆ S induced by φ . Note that this kernel is equivalent to the kernel of the mapH ( ˜ Σ , (cid:81) ) → H ( ˜ H , (cid:81) ) . Lemma 5.4.
The subspace ˆ P is a (cid:81) [ H ] -submodule which is totally isotropic relative to the sesquilin-ear form 〈− , −〉 . Moreover, as (cid:81) [ H ] -modules, ˆ R ∼ = ˆ P ⊕ ˆ S and both ˆ P and ˆ S are isomorphic to (cid:81) [ H ] g − ⊕ (cid:81) .
23e remark that below we show that the isomorphism ˆ P ⊕ ˆ S ∼ = ˆ R can be chosen such that theimage of ˆ S is totally isotropic relative to 〈− , −〉 . Proof.
As mentioned above, the map H (Σ , (cid:81) ) → H ( H , (cid:81) ) has totally isotropic kernel relative tothe alternating intersection form. Applying the same fact to the map on the covers H ( ˜ Σ , (cid:81) ) → H ( ˜ H , (cid:81) ) implies that ˆ P is totally isotropic relative to 〈− , −〉 Sp . Since the sesquilinear form isdefined as a sum with coefficients in terms of 〈− , −〉 Sp , it follows that ˆ P is also totally isotropicrelative to the sesquilinear form.It follows from the definitions that ˆ φ R is a (cid:81) [ H ] -homomorphism, so its kernel, ˆ P , is a (cid:81) [ H ] -submodule. By Theorem 2.1, ˆ S ∼ = (cid:81) [ H ] g − ⊕ (cid:81) . This fact combined with Proposition 1.1 andsurjectivity of ˆ φ R implies the rest of the Lemma since (cid:81) [ H ] is semisimple.Now consider the above situation projected to the i th isotypic component M i of ˆ R usingthe notation of the previous sections. The above decomposition projects to a decomposition of M i , and we can conclude a stronger statement relative to the sesquilinear form. Let M (cid:48) i be theprojection of ˆ P to M i . Lemma 5.5.
Let ˜ a H , ˜ b H be as in Lemma 5.3 and ˜ α i , ˜ β i the projection of their homology classes toM i . The module M (cid:48) i is totally isotropic relative to 〈− , −〉 , and there is a subspace M (cid:48)(cid:48) i of M i such that • M (cid:48)(cid:48) i is totally isotropic relative to 〈− , −〉 , • M i = M (cid:48) i ⊕ M (cid:48)(cid:48) i • there are free A i -bases ˜ β i = m (cid:48) i ,1 , m (cid:48) i ,2 , . . . , m (cid:48) i , g − (resp. ˜ α i = m (cid:48)(cid:48) i ,1 , m (cid:48)(cid:48) i ,2 , . . . , m (cid:48)(cid:48) i , g − ) of M (cid:48) i (resp. M (cid:48)(cid:48) i ) such that 〈 m (cid:48)(cid:48) i , j , m (cid:48) i , k 〉 = δ j , k Proof.
The module M (cid:48) i is totally isotropic since ˆ P is. Notice that by the choice of ˜ b H , its homologyclass lies in ˆ P . The homology class of ˜ b H generates a copy of (cid:81) [ H ] in ˆ P , and so [ ˜ b H ] necessarilygenerates a copy of A i in M i . Thus, ˜ β i can be extended to some basis ˜ β i = m (cid:48) i ,1 , m (cid:48) i ,2 , . . . , m (cid:48) i , g − of M (cid:48) i . Furthermore, we can ensure 〈 ˜ α i , m (cid:48) i , j 〉 = δ j . (If it is not already true, we can alter m (cid:48) i , j for j > β i .)Lemma 5.4 implies that M (cid:48) i is isomorphic to A g − i , and so we immediately have some comple-mentary subspace N i which is isomorphic to A g − i (but not necessarily isotropic). Moreover, since˜ α i does not lie in M (cid:48) i , we can further arrange that N i contains ˜ α i . Consider the homomorphismto the dual space of N i M (cid:48) i → Hom A i ( N i , A i ) m (cid:48) i (cid:55)→ 〈− , m (cid:48) i 〉 (2)Note that this is an A i -module homomorphism where the (left) A i -module structure on Hom A i ( N i , A i ) is given by ( a · f )( n ) = ( f ( n )) τ ( a ) for all n ∈ N i . Since M (cid:48) i is totally isotropic and 〈− , −〉 is nonde-generate, each m (cid:48) ∈ M (cid:48) i must pair nontrivially with some element in N i . Consequently, the mapin (2) has trivial kernel, and the fact that Hom A i ( N i , A i ) ∼ = A g − i implies (2) is an isomorphism.There is therefore a unique basis n i ,1 , . . . , n i , g − of N i such that 〈 n i , j , m (cid:48) i , k 〉 = δ j , k . Moreover,since ˜ α i lies in N i and 〈 α i , m (cid:48) i , j 〉 = δ j , it follows that n i ,1 = ˜ α i . From N i , we can construct asubmodule M (cid:48)(cid:48) i which has the same properties as N i but is also isotropic.24et c j , k = 〈 n i , j , n i , k 〉 for j (cid:54) = k , and let c j , j = 〈 n i , j , n i , j 〉 . Note that since 〈− , −〉 is skew-Hermitian, τ ( c j , k ) = − c k , j . Let m (cid:48)(cid:48) i , j = n i , j + (cid:80) (cid:96) ≤ j c j , (cid:96) m (cid:48) i , (cid:96) and M (cid:48)(cid:48) i be the submodule generated by m (cid:48)(cid:48) i ,1 , . . . , m (cid:48)(cid:48) i , g − . Note that m (cid:48)(cid:48) i ,1 = n i ,1 = ˜ α i . It follows from the properties of the n i , j and the factthat M (cid:48) i is isotropic that 〈 m (cid:48)(cid:48) i , j , m (cid:48) i , k 〉 = δ jk . Moreover, one computes that for all pairs j > k : 〈 m (cid:48)(cid:48) i , j , m (cid:48)(cid:48) i , k 〉 = 〈 n i , j , n i , k 〉 + 〈 (cid:80) (cid:96) ≤ j c j , (cid:96) m (cid:48) i , (cid:96) , n i , k 〉 + 〈 n i , j , (cid:80) (cid:96) ≤ k c k , (cid:96) m (cid:48) i , (cid:96) 〉 = c j , k + 〈 c j , k m (cid:48) i , k , n i , k 〉 = c j , k − c j , k = 〈 m (cid:48)(cid:48) i , j , m (cid:48)(cid:48) i , j 〉 = 〈 n i , j , n i , j 〉 + 〈 c j , j m (cid:48) i , j , n i , j 〉 + 〈 n i , j , c j , j m (cid:48) i , j 〉 = c j , j − c j , j + τ ( c j , j ) = M (cid:48)(cid:48) i has all the desired properties. Corollary 5.6.
The submodule ˆ P has a complementary totally isotropic isomorphic submodule in ˆ R.Proof.
By Lemma 5.5, there is a direct sum decomposition of ˆ R as a (cid:81) [ H ] -module: ˆ R = (cid:81) g ⊕ (cid:96) (cid:77) i = M i = (cid:81) g ⊕ (cid:96) (cid:77) i = ( M (cid:48) i ⊕ M (cid:48)(cid:48) i ) where M (cid:48) i and M (cid:48)(cid:48) i are isotropic and M (cid:48) i = ˆ P ∩ M i . Similarly, (cid:81) g ∩ ˆ P ∼ = (cid:81) g and there is acomplimentary isotropic subspace B isomorphic to (cid:81) g . The submodule complementary to ˆ P is B ⊕ ( (cid:76) (cid:96) i = M (cid:48)(cid:48) i ) .
6. The representation ρ and the image of the handlebody group We now provide more details about the representations ρ H , p . As mentioned in the introduction,we will work with Aut ( T g ) + = Mod (Σ g , ∗ ) instead of Mod (Σ g ) , and at the end (Section 8.2) wewill use (virtual) arithmetic quotients of the former to obtain (virtual) arithmetic quotients ofthe latter. As in the previous section, let p : T g → H be a φ -redundant surjective homomorphism,and let S be the kernel of the induced map p (cid:48) : F g → H . We have the following commutativediagram and short exact sequences.1 −−−→ R −−−→ T g p −−−→ H −−−→ (cid:121) (cid:121) φ (cid:13)(cid:13)(cid:13) −−−→ S −−−→ F g p (cid:48) −−−→ H −−−→ Γ H , p = { f ∈ Aut ( T g ) + | p ◦ f = p } .Note that if f ∈ Γ H , p , then it follows automatically that f ( R ) = R . For such f , we can restrict theautomorphism to R and then project to an automorphism of the abelianization R . This, in turn,induces an automorphism of ˆ R . Denote this map as ρ H , p : Γ H , p → Aut (ˆ R ) . For the remainder ofthis section, we will suppress the subscripts H and p , so henceforth we fix some p and H , and set ρ = ρ H , p and Γ = Γ H , p .As we have seen, ˆ R has a (cid:81) [ H ] -module structure, and this is preserved by such f .25 emma 6.1. The image ρ (Γ) lies in Aut (cid:81) [ H ] (ˆ R ) .Proof. Let f ∈ Γ , and let f be the induced action of f on R . It suffices to check that f commuteswith the action of H . Suppose r ∈ R and r ∈ R is its image in the quotient. For any h ∈ H ,the action on f ( r ) = f ( r ) is h · f ( r ) = h f ( r ) h − . Since f ∈ Γ , it follows that f is a trivialautomorphism mod R , and so r f ( h ) = h for some r ∈ R and h f ( r ) h − = r f ( hrh − ) r − = f ( hrh − ) = f ( h · r ) .To show that ρ ( f ) additionally preserves the form 〈− , −〉 , we appeal to a topological refor-mulation. Interpreted in a topological setting, the homomorphism can be viewed as lifting thehomeomorphisms to the cover and using the induced action on the first homology group. Letus be more precise. Suppose ˜ ∗ is a point in the fiber over ∗ ∈ Σ for the cover ˜ Σ → Σ . Fora homeomorphism ϕ ∈ Homeo (Σ , ∗ ) , the lift of ϕ is the homeomorphism ˜ ϕ fixing ˜ ∗ such that Π ◦ ˜ ϕ = ϕ ◦ Π where Π : ˜ Σ → Σ is the covering map. The homeomorphism ϕ will lift preciselywhen the induced action of ϕ on π (Σ , ∗ ) preserves R = π ( ˜ Σ , ˜ ∗ ) , i.e. its mapping class is an el-ement of Γ . Moreover, this lifting induces a well-defined homomorphism Γ → Mod ( ˜ Σ , ˜ ∗ ) . Recallthat ˆ R is naturally identified with H ( ˜ Σ , (cid:81) ) . If f ∈ Γ and ˜ f ∈ Mod ( ˜ Σ , ˜ ∗ ) is its lift, then ρ ( f ) = ˜ f ∗ where ˜ f ∗ is the induced action on H ( ˜ Σ , (cid:81) ) . We can now establish the following. Lemma 6.2.
The action of ρ ( f ) preserves the sesquilinear form 〈− , −〉 for all f ∈ Γ .Proof. By the above discussion, ρ ( f ) is equivalent to ˜ f ∗ which is the action induced by theorientation-preserving mapping class ˜ f . Since such a homomorphism preserves 〈− , −〉 Sp , and, inaddition, ρ ( f ) acts by a (cid:81) [ H ] -automorphism by Lemma 6.1, it must preserve 〈− , −〉 .Since the image acts by (cid:81) [ H ] -automorphisms, it decomposes into actions on each isotypiccomponent. Consequently, we obtain representations ρ i = ρ H , p , i : Γ → Aut A i ( M i , 〈− , −〉 ) by projecting the action. Here, we recall a theorem from [ ] which describes the image of Aut ( F n ) for a representationanalogous to ρ . We will use this result to show that the image of the handlebody group maps(virtually) onto the O -points of the Levi factor of a parabolic subgroup of Aut A i ( A g − i , 〈− , −〉 ) .The essential connection between the two is that Map ( H , ∗ ) surjects onto Aut ( F g ) via the actionon its fundamental group.We set up some additional notation and define some new terms. For any group X and sub-group Y , we set Aut ( X | Y ) = { f ∈ Aut ( X ) | f ( Y ) = Y } .Moreover, for Y normal, let Aut t ( X | Y ) be the subgroup of Aut ( X | Y ) consisting of automorphismsacting trivially on X / Y . Note that in this notation Γ =
Aut ( T g ) + ∩ Aut t ( T g | R ) . We have a26omomorphism η = η H , p : Aut t ( F g | S ) → Aut (cid:81) [ H ] (ˆ S ) defined similarly to ρ ; namely restrict theautomorphism to S , project to the abelianization S , and take the induced map on ˆ S = S ⊗ (cid:90) (cid:81) .By Gaschütz’s theorem (Theorem 2.1), ˆ S ∼ = (cid:81) [ H ] g − ⊕ (cid:81) . Consequently, for each i , we obtain,via projection, a representation η i : Aut t ( F g | S ) → Aut A i ( A g − i ) = GL g − ( A opi ) .Recall that A i is simple and isomorphic to Mat m i ( D i ) for some division algebra D i and integer m i .Thus, we can furthermore identify the target space as follows.GL g − ( A opi ) ∼ = GL g − ( Mat m i ( D i ) op ) ∼ = GL ( g − ) m i ( D opi ) The special linear group SL ( g − ) m i ( D opi ) is the group of matrices C satisfying nrd B / L ( C ) = B = End D i ( D m i ( g − ) i ) .Now, suppose K is some number field, and suppose D is a finite-dimensional K -algebra. Thena subring R ⊆ D is an order if it contains a K -basis of D and it is a finitely generated (cid:90) -module.Recall that A i is finite-dimensional over K i , (the τ -fixed subfield of its center), and so D i is afinite-dimensional K i -algebra. The next theorem is a direct consequence of Theorem 5.5 of [ ] . Theorem 6.3. (Grunewald–Lubotzky) Keeping notation as above, if g ≥ , then there exists an order R i of D opi such that the intersection η i ( Aut t ( F g | S )) ∩ SL m i ( g − ) ( R i ) is of finite index in SL m i ( g − ) ( R i ) . Remark 6.4.
Theorem 5.5 in [ ] is proved for g ≥ . However, as explained on page 1592 there,it is also true when g = in many cases, e.g. if m i > . Moreover, even if m i = , the image of η i contains finite index subgroups of the upper and lower unitriangular subgroups of SL m i ( g − ) ( R i ) .The issue is that it is not clear there that these unitriangular subgroups generate SL m i ( g − ) ( R i ) . Butin any event, they generate a Zariski dense subgroup. Here, we will analyze the image of the handlebody subgroup Map ( H , ∗ ) of Mod (Σ , ∗ ) . Thehandlebody group naturally has a homomorphism to Aut ( F g ) via the action on the fundamentalgroup of H . We define the finite index subgroup Λ = Γ ∩ Map ( H , ∗ ) We will show that the image of Λ under ρ i block uppertriangularizes, and furthermore thatthe blocks on the diagonal range over a wide variety of matrices. More precisely, we have thefollowing result. Similar to Section 3, we set C ∗ = ( τ ( c ji )) for any matrix C = ( c i j ) ∈ Mat m ( A opi ) .We define the parabolic subgroup of GP = (cid:168)(cid:130) ( C ∗ ) − E C (cid:140) | C , E ∈ Mat g − ( A opi ) and E ∗ C = C ∗ E (cid:171) .Let pr : P → GL g − ( A opi ) be the homomorphism (cid:130) ( C ∗ ) − E C (cid:140) (cid:55)→ C .27 roposition 6.5. After identifying
Aut A i ( M i ) with GL ( g − ) ( A opi ) via the ordered basis m (cid:48) i ,1 , . . . , m (cid:48) i , g − , m (cid:48)(cid:48) i ,1 , . . . , m (cid:48)(cid:48) i , g − from Lemma 5.5, we have ρ i (Λ) ⊆ P Moreover, if g ≥ , then there is an order R i of D opi such that pr ( ρ i (Λ)) is a finite index subgroupin SL m i ( g − ) ( R i ) . If g = , then pr ( ρ i (Λ)) is Zariski dense in SL m i ( g − ) ( R i ) . The first part of the proposition is essentially equivalent to the following lemma.
Lemma 6.6.
The following is a well-defined commutative diagram. Λ ρ −−−→ Aut (cid:81) [ H ] (ˆ R | ˆ P ) (cid:121) (cid:121) Aut t ( F g | S ) η −−−→ Aut (cid:81) [ H ] (ˆ S ) Proof.
Note that Λ can project to Aut t ( F g | S ) since Map ( H , ∗ ) preserves ker ( φ ) and Γ lies inAut t ( T g | R ) . In addition to commutativity, we have to establish that ρ (Λ) preserves the subspace ˆ P . We translate parts of the diagram into topological language where it becomes the following:Map ( H , ∗ ) ∩ Γ ρ −−−→ Aut (cid:81) [ H ] ( H ( ˜ Σ , (cid:81) ) | ˆ P ) (cid:121) (cid:121) Aut t ( F g | S ) η −−−→ Aut (cid:81) [ H ] ( H ( ˜ H , (cid:81) )) .We prove commutativity in two steps. First, we can lift elements from Map ( H , ∗ ) to Map ( ˜ H , ˜ ∗ ) .and properties of lifts and the fact that S = π ( ˜ H , ˜ ∗ ) implies commutativity in the following dia-gram. Map ( H , ∗ ) ∩ Γ lift −−−→ Map ( ˜ H , ˜ ∗ ) (cid:121) (cid:121) Aut t ( F g | S ) −−−→ Aut ( S ) Since elements of Map ( ˜ H , ˜ ∗ ) preserve ˆ P , the following diagram is well-defined and commu-tative. Map ( ˜ H , ˜ ∗ ) −−−→ Aut ( H ( ˜ Σ , (cid:81) ) | ˆ P ) (cid:121) (cid:121) Aut ( S ) −−−→ Aut ( H ( ˜ H , (cid:81) )) Putting the diagrams together establishes the Lemma.This establishes the block diagonal form claimed in Proposition 6.5. Proving Proposition 6.5is now a matter of combining some of the above results.
Proof of Proposition 6.5.
Let γ ∈ Λ . Lemma 6.6 implies that ρ i ( γ ) preserves M (cid:48) i and so ρ i ( γ ) block diagonalizes. To show that ρ i ( γ ) lies in P as defined, it suffices to show that any element28f G preserving M (cid:48) i has diagonal entries ( C ∗ ) − , C for some matrix C ∈ Mat g − ( A op ) . Notice thatrelative to the basis, (and multiplying in Mat ( − op ) ) 〈 ( a , . . . , a ( g − ) ) , ( b , . . . , b ( g − ) ) 〉 = ( τ ( b ) . . . τ ( b ( g − ) )) (cid:130) I − I (cid:140) a ... a ( g − ) Consequently any matrix (cid:130)
F E C (cid:140) preserving 〈− , −〉 must satisfy (cid:130) F ∗ E ∗ C ∗ (cid:140) (cid:130) I − I (cid:140) (cid:130) F E C (cid:140) = (cid:130) I − I (cid:140) This implies F = ( C ∗ ) − , and C ∗ E = E ∗ C .Theorem 5.1 implies that Λ surjects onto Aut t ( F g | S ) . Lemma 6.6 implies that pr ◦ ρ i = η i ,and so Theorem 6.3 and Remark 6.4 finish the proof.
7. Generating unipotents in the image
Our goal in this section is to prove that the image of ρ H , p , i contains the finite index subgroupsof the unipotent subgroups N + ( O ) and N − ( O ) . We begin in this first subsection by finding twoconcrete examples of unipotents in the image of ρ H , p , i . As before, we fix H and p and drop themfrom the subscripts of ρ and Γ . Let a , b be the simple closed curves on Σ as in Lemmas 5.2 and 5.3. The tightly controlledunipotents are the images of the Dehn twists about a and b which we denote by T a and T b . Lemma 7.1.
Let M (cid:48) i , M (cid:48)(cid:48) i be the modules and m (cid:48) i ,1 , . . . , m (cid:48) i , g − and m (cid:48)(cid:48) i ,1 , . . . , m (cid:48)(cid:48) i , g − their respectivefree A i -bases given by Lemma 5.5. Then, with respect to the ordered basis m (cid:48) i ,1 , . . . , m (cid:48) i , g − , m (cid:48)(cid:48) i ,1 , . . . , m (cid:48)(cid:48) i , g − , ρ i ( T b ) = (cid:130) I e I (cid:140) ρ i ( T − a ) = (cid:130) I e I (cid:140) where e ∈ Mat g − ( A i ) is the matrix with in the upper left corner and in all other entries.Proof. We prove it for T − a ; the proof for T b is similar. Let ϕ be some representative homeomor-phism of T a which fixes ∗ . The lift ˜ ϕ is precisely the twist ϕ on each copy Σ (cid:48) h of Σ . I.e., the liftin Mod ( ˜ Σ , ˜ ∗ ) is simply the composition of twists (cid:81) h ∈ H T a h . Notice that since all the twists havemutually disjoint support, all the T a h commute with each other, so the order of composition isirrelevant. It is a standard fact (see e.g. [
12, Proposition 6.3 ] ) that the action on homology T δ ∗ of a single Dehn twist T δ on a homology class v ∈ H (Σ , (cid:81) ) is T δ ∗ ( v ) = v + 〈 v , [ δ ] 〉 Sp [ δ ] [ δ ] is the homology class of δ .We now simply compute our action. Notice that T a h ∗ ([ a h (cid:48) ]) = [ a h (cid:48) ] for all pairs h (cid:54) = h (cid:48) as a h , a h (cid:48) are disjoint, and so we do not “accumulate” any extra terms in composing. I.e. thecomposition of the T a h ∗ applied to any v ∈ H ( ˜ Σ , (cid:81) ) is (cid:81) h ∈ H T a h ∗ ( v ) = v + (cid:80) h ∈ H 〈 v , [ a h ] 〉 Sp [ a h ]= v + (cid:80) h ∈ H 〈 v , h · [ a ] 〉 Sp h · [ a ]= v + 〈 v , [ a ] 〉 [ a ] By the choice of basis, 〈 v , [ a ] 〉 is − v = b and is 0 when v is any other basis element.Using the inverse Dehn twist then gives us positive 1. N + and N − We again fix some simple component A = A i of (cid:81) [ H ] and the corresponding submodule M = M i of ˆ R with its associated sesquilinear form. Also let ρ = ρ i = ρ H , p , i . Let us now consider thefollowing unipotent subgroups of the corresponding group G = Aut A ( A g − , 〈− , −〉 ) which wecontinue to view in terms of the basis from Lemma 5.5. Let us set B = Mat g − ( A op ) , and for E =( e i j ) ∈ B , define E ∗ to be ( τ ( e ji ) . For any matrix of the form (cid:130) I E I (cid:140) in G , a straightforwardcomputation shows that E ∗ = E . We set N + = (cid:168)(cid:130) I E I (cid:140) ∈ Mat ( B ) | E ∗ = E (cid:171) N − = (cid:168)(cid:130) I E I (cid:140) ∈ Mat ( B ) | E ∗ = E (cid:171) In this section we show that ρ (Γ) contains a lattice in each of these unipotent subgroups.We will produce them by conjugating the elements ρ ( T b ) and ρ ( T − a ) by elements in ρ (Λ) .Recall from Proposition 6.5 that ρ (Λ) consists of elements of the form (cid:130) ( C ∗ ) − E C (cid:140) with E ∗ C = C ∗ E , and note that the inverse of such a matrix is (cid:130) C ∗ E ∗ C − (cid:140) . One can compute that (cid:130) C ∗ E ∗ C − (cid:140) (cid:130) I E (cid:48) I (cid:140) (cid:130) ( C ∗ ) − E C (cid:140) = (cid:130) I C ∗ E (cid:48) C I (cid:140) Recall also from Proposition 6.5 that, if g ≥
4, for every C in a finite index subgroup of SL m ( g − ) ( R ) ,the image ρ (Λ) contains a matrix of the form (cid:130) ( C ∗ ) − E C (cid:140) (and for g =
3, this is true forevery C in some Zariski dense subgroup of SL m ( g − ) ( R ) ). Hence, our interest is in the set ofmatrices C ∗ EC where E = E ∗ and C ∈ SL m ( g − ) ( R ) .First, we will understand the action of SL ( B ) = { C ∈ B | nrd B / L ( C ) = } on N = { E ∈ B | E ∗ = E } given by C · E = C ∗ EC . Recall that L is the center of A and hence B , and that K isthe field fixed by τ and hence by ∗ . 30 emma 7.2. The above action of SL ( B ) on N is irreducible over K.Proof. Suppose τ (hence ∗ ) is of the first kind, and choose some isomorphism ϕ : B ⊗ K (cid:67) → Mat n ( (cid:67) ) (for appropriate n ) so that ∗ goes over to the involution ν in (1) or (2) of Lemma 3.6.Applying ϕ to N ⊗ K (cid:67) identifies N ⊗ K (cid:67) with N (cid:48) = { E ∈ Mat n ( (cid:67) ) | ν ( E ) = E } . Let ϕ (cid:48) be thecomposition of B → B ⊗ K (cid:67) followed by ϕ . Now, ϕ (cid:48) ( SL ( B )) consists of the K -points of a form ofSL n in SL n ( (cid:67) ) , and so ϕ (cid:48) ( SL ( B )) is Zariski dense in SL n ( (cid:67) ) by [
5, Theorem A ] . (See also [ ] or [ ] .) Thus, if SL n ( (cid:67) ) acts irreducibly on N (cid:48) , so does ϕ (cid:48) ( SL ( B )) , and so SL ( B ) acts irreduciblyon N . We will show that for any nonzero E ∈ N (cid:48) , the SL n ( (cid:67) ) -orbit spans N (cid:48) . Since scaling E doesnot affect the span, it is equivalent to prove this for the GL n ( (cid:67) ) -orbit.In case (1), we have ν ( C ) = C t , and N (cid:48) is the set of symmetric matrices and the action is C · E = C t EC . This action is well-known to be irreducible. In case (2), we have ν ( C ) = J C t J − .Notice that J − = J t = − J . The set N (cid:48) consists of those matrices E such that J E t J − = E whichis equivalent to J − E being skew-symmetric. Then, we have J − ( C · E ) = J − J C t J − EC = C t ( J − E ) C Consequently, this case reduces to the fact that the action E → C t EC on the set of skew-symmetricmatrices E is irreducible. The irreducibility of this action is also well-known.Suppose now that τ is of second kind. We use a proof very similar to the previous cases. ByLemma 3.6, there is an isomorphism ϕ : B ⊗ K (cid:67) → Mat n ( (cid:67) ) × Mat n ( (cid:67) ) (for appropriate n ), sothat ∗ goes over to the involution ν defined by ν ( C , D ) = ( D t , C t ) . Thus, the homomorphism ϕ identifies N ⊗ K (cid:67) with N (cid:48) = { ( E , E t ) ∈ Mat n ( (cid:67) ) × Mat n ( (cid:67) ) } . Let ϕ (cid:48) be the compositionof B → B ⊗ K (cid:67) followed by ϕ . By [
5, Theorem A ] , the image ϕ (cid:48) ( SL ( B )) is Zariski dense inSL n ( (cid:67) ) × SL n ( (cid:67) ) . Just as before, SL ( B ) acts irreducibly on N if SL n ( (cid:67) ) × SL n ( (cid:67) ) acts irreduciblyon N (cid:48) . In this case, the action on N (cid:48) is ( C , D ) · ( E , E t ) = ( C t , D t )( E , E t )( D , C ) = ( C t E D , D t E t C ) .It is clearly irreducible.We can now show that our image contains a finite index subgroup of N + ( O ) = G ( O ) ∩ N + . Lemma 7.3.
The intersection ρ (Λ) ∩ N + ( O ) is of finite index in N + ( O ) .Proof. Let F = ρ (Λ) ∩ G ( O ) . We show that the subgroup of the (abelian) group N + ( O ) generatedby conjugates of (cid:130) I e I (cid:140) by F is of finite index in N + ( O ) . (Note that by construction, thisunipotent matrix lies in ρ (Λ) .) Using Proposition 6.5, this is equivalent to the subgroup of N ( O ) generated by the orbit of e under the action of the finite index subgroup G = pr ( ρ (Λ)) ofSL m ( g − ) ( R ) . By Proposition 6.5, the group G is Zariski dense in SL ( B ) ∼ = SL g − ( A op ) . Conse-quently, since SL ( B ) acts irreducibly by Lemma 7.2, so does G , and thus the group generated bythe G -orbit of e is of finite index in N + ( O ) .Using this lemma, we can now see that ρ (Λ) contains block diagonal matrices. In fact, wecan show the following. 31 emma 7.4. Assume g ≥ . The image ρ (Λ) contains a finite index subgroup of the followinggroup: (cid:168)(cid:130) ( C ∗ ) − C (cid:140) ∈ Mat ( B ) | C ∈ SL m ( g − ) ( R ) (cid:171) where R is the order from Theorem 6.3.Proof. Let F = ρ (Λ) ∩ G ( O ) which necessarily lies in P ( O ) = P ∩ G ( O ) by Proposition 6.5. Recallthat unipotent subgroups have the congruence subgroup property, so, by Lemma 7.3, there issome ideal a of O such that ρ (Λ) ∩ N + ( O ) contains N + ( a ) (see e.g. page 303 of [ ] ). Afterpassing to a finite index subgroup F (cid:48) of F , we can ensure that if (cid:130) ( C ∗ ) − E C (cid:140) ∈ F (cid:48) , then (cid:130) I E I (cid:140) ∈ N + ( a ) and therefore (cid:130) I − C ∗ E I (cid:140) also lies in N + ( a ) . The result then followsfrom Proposition 6.5 and the computation: (cid:130) ( C ∗ ) − E C (cid:140) (cid:130) I − C ∗ E I (cid:140) = (cid:130) ( C ∗ ) − C (cid:140) Remark 7.5.
For g = , a similar argument using Remark 6.4 instead of Proposition 6.5 will showthat ρ (Λ) contains a Zariski dense subgroup of (cid:168)(cid:130) ( C ∗ ) − C (cid:140) ∈ Mat ( B ) | C ∈ SL m ( g − ) ( R ) (cid:171) .The block diagonal matrices act in similar ways on N + ( O ) and N − ( O ) by conjugation. Ap-plying the argument of Lemma 7.3 but using the block diagonal matrices from Lemma 7.4 andRemark 7.5 in place of ρ (Λ) ∩ G ( O ) , we obtain the following lemma. Lemma 7.6.
The intersection ρ (Γ) ∩ N − ( O ) is of finite index in N − ( O ) .
8. The image of the mapping class group
We are now ready to finish the proof of our main technical result Theorem 1.6, and then todeduce Theorem 1.8 which outlines the general procedure of obtaining arithmetic quotients ofthe mapping class group. In the next section, we will apply the procedure to specific examples of H and r to deduce Theorem 1.2, its corollaries, and Theorem 1.3. First, let
Γ = Γ H , p , ρ = ρ H , p , G = G H , i , and G = G H , i . We can now use the results of Lemmas 7.3,7.4, 7.6 to get the following. Let G be the algebraic group Aut A ( A g − , 〈− , −〉 ) . It is a reductivegroup but not necessarily semisimple. Let G be the elements of reduced norm 1 over L . Thissubgroup G is semisimple.First, let us see why a finite index subgroup of Γ must map into G ( O ) = Aut O ( O g − , 〈− , −〉 ) .Recall that ρ is originally defined by the action of Γ < Aut ( T g ) + on R = R / [ R , R ] which is induced32y the action of Γ on R < T g . After tensoring R by (cid:81) we obtain ˆ R ∼ = (cid:81) [ H ] g − ⊕ (cid:81) and the actionof Γ projects to an action of A g − preserving 〈− , −〉 . Since Γ preserves R ⊂ ˆ R , it also preserves R ∩ A g − which is a lattice in the K -vector space A g − . Thus, some finite index subgroup of Γ maps into G ( O ) .Recall that P is the parabolic subgroup of G which consists of those elements preservingthe submodule M (cid:48) in the notation of Section 5. Let P = P ∩ G . Then P is the normalizerof the unipotent subgroup N + . The group P is a semi-direct product of its Levi factor L andits unipotent radical, which is equal to N + . The Levi factor L is isomorphic to GL g − ( A op ) ∼ = GL ( g − ) m ( D op ) . Let T be the maximal K -split torus of L given by diagonal matrices with entriesin K ; in particular, T is isomorphic to ( K ∗ ) m ( g − ) . Viewing L as block diagonal matrices, T is alsoa maximal K -split torus for G . I.e. the following is a maximal K -split torus for G : { (cid:130) ( C ∗ ) − C (cid:140) | C ∈ T } .(See classification in [
43, Table II ] .) Let Φ be a K -root system with respect to this torus. Then forevery α ∈ Φ , the corresponding root subgroup lies either in N + , N − , or L .Now, when g ≥
4, Lemmas 7.3, 7.4, 7.6 imply, therefore, that for every α ∈ Φ , the intersectionof ρ (Γ) with the root subgroup is commensurable to the integral points of the K -root subgroup.Recall also that unipotent subgroups have an affirmative answer to the congruence subgroupproblem [ ] , and hence a finite index subgroup of the root subgroup contains a congruencesubgroup of it. For g =
3, we can still get the same conclusion even though Lemma 7.4 is notvalid, as explained in Remark 6.4.We are now in a position to use Theorem 1.2 of [ ] which asserts that exactly in such asituation the group generated by congruence subgroups of U α as α ranges over Φ is an arithmeticgroup, i.e. of finite index in G ( O ) . (The theorem requires that the (cid:81) -rank of G is at least 2,and this is always true when g ≥ ρ (Γ) contains a finite index subgroupof G ( O ) . By Proposition 3.9, G ( O ) is of finite index in G ( O ) , and so ρ (Γ) and G ( O ) arecommensurable. We have so far only proven results for Mod (Σ , ∗ ) . We now show how the homomorphism ρ induces a map on Mod (Σ) . Let ε : Mod (Σ , ∗ ) → Mod (Σ) be the natural map that forgets the fixedpoint. The following proposition says that after passing to a finite index subgroup, ρ factorsthrough ε . Proposition 8.1.
Let p : T g → H be a φ -redundant homomorphism and ρ H , p : Γ H , p → Aut (cid:81) [ H ] (ˆ R , 〈− , −〉 ) the corresponding homomorphism. Then, there is a finite index subgroup Γ (cid:48) H , p < Γ H , p so that ρ H , p | Γ (cid:48) H , p factors through ε .Proof. We recall the Birman Exact Sequence for Mod (Σ , ∗ ) : ( [
4, 12 ] )1 → T g c → Mod (Σ , ∗ ) ε → Mod (Σ) → T g maps to the “point-pushing” mapping classes and each c ( α ) for α ∈ T g acts as conjuga-tion by α on the fundamental group. For ρ to factor through ε on some subgroup Γ (cid:48) < Mod (Σ , ∗ ) ,we need ε (Γ (cid:48) ∩ c ( T g )) to act trivially on R / [ R , R ] . Hence, it suffices that Γ (cid:48) ∩ c ( T g ) ⊆ c ( R ) .33et α ∈ T g and suppose c ( α ) ∈ Γ H , p . Since R is of finite index, there is some n such that α n ∈ R , and so c ( α ) n acts trivially on R / [ R , R ] by conjugation. Thus, ρ (Γ H , p ∩ c ( T g )) consistsentirely of torsion elements. The image ρ (Γ H , p ) lies in Aut ( R , 〈− , −〉 Sp ) which is isomorphic toSp ( + ( g − ) | H | , (cid:90) ) and contains some finite index torsion-free subgroup G . Thus, Γ (cid:48) H , p = ρ − ( G ) ∩ Γ H , p is the required subgroup. Theorem 1.6 and Section 8.2 establish Theorem 1.8 which presents a procedure to obtain arith-metic quotients and which we summarize now. This theorem will be convenient for use in Section9 and possibly the future.Let Σ g be a closed surface, let H be some finite group generated by d ( H ) < g generators, andlet A be a simple component of (cid:81) [ H ] . This is the required input for the theorem. By Lemma 3.2,the standard involution τ on (cid:81) [ H ] defined by h (cid:55)→ h − restricts to an involution on A which, byabuse of notation, we will also call τ .Set M = A g − with free basis x , . . . , x g − , y , . . . y g − and let 〈− , −〉 be the skew-Hermitianform sesquilinear relative to τ such that 〈 x i , y j 〉 = δ i j , 〈 y i , x j 〉 = − δ i j , 〈 x i , x j 〉 =
0, and 〈 y i , y j 〉 =
0. Let G = Aut A ( A g − , 〈− , −〉 ) , and set Ω H , A = G ( O ) = G ∩ Aut O ( O g − , 〈− , −〉 ) where O is theorder in A which is the image of (cid:90) [ H ] .To obtain our representation, we need some φ -redundant homomorphism p : T g → H . Let φ : T g → F g be the map on fundamental groups induced by the inclusion Σ g (cid:44) → H g . Since d ( H ) < g , there is some epimorphism p (cid:48) : F g → H which maps at least one free generator to thetrivial element. The composition p = p (cid:48) ◦ φ is φ -redundant. Recall that R is the kernel of p and R = R / [ R , R ] . From the discussion in Section 6, the following subgroup of Mod (Σ g , ∗ ) ∼ = Aut ( T g ) + Γ = { f ∈ Aut ( T g ) + | p ◦ f = p } has a well-defined action on R , hence on ˆ R ∼ = (cid:81) [ H ] g − ⊕ (cid:81) , and hence on A g − . This induces ahomomorphism Γ → G by Lemmas 6.1 and 6.2. By Theorem 1.6, there is a finite index subgroup Γ (cid:48) < Γ and a homomorphism ρ : Γ (cid:48) → Ω H , A with finite index image. By Lemma 8.1, there is afinite index subgroup Γ H , A < Mod (Σ g ) and a representation ρ H , A : Γ H , A → Ω H , A with finite indeximage. This establishes the theorem. ρ H , r Notice that in the above Ω H , r = Ω H , A is uniquely determined by H and r , but the representation ρ H , r = ρ H , A depends on the choice of the surjective representation p : T g → H . One wayto get other epimorphisms p is via the action of the group Aut ( T g ) × Aut ( H ) by ( ψ , ϕ ) · p = ϕ ◦ p ◦ ψ − . The corresponding induced representations ρ are “equivalent” in the followingsense. Let inn ( ψ ) : Aut ( T g ) → Aut ( T g ) denote the inner automorphism inn ( ψ )( f ) = ψ f ψ − .The proof of the following lemma is elementary, so we omit it. Lemma 8.2.
Suppose p : T g → H is some epimorphism and p = ϕ p ψ − for some ( ψ , ϕ ) ∈ Aut ( T g ) × Aut ( H ) . Let R i = ker ( p i ) ,let ˆ R i = ( R i / [ R i , R i ]) ⊗ (cid:90) (cid:81) , and let ρ H , p i : Γ H , p i → Aut (cid:81) [ H ] (ˆ R i , 〈− , −〉 ) be the corresponding representation as presented in Section 6. Then, there is an isomorphism ε : Aut (cid:81) [ H ] (ˆ R ) → Aut (cid:81) [ H ] (ˆ R ) such that ρ H , p = ε ◦ ρ H , p ◦ inn ( ψ ) . T g → H are φ -redundant, so a natural question is: how transitivelydoes Aut ( T g ) × Aut ( H ) act on them? Suppose p , p : T g → H are respectively φ -redundant and φ -redundant, and p (cid:48) , p (cid:48) : F g → H are the respective induced maps. Recall from the introductionand Section 5 that all maps φ : T g → F g are induced by some identification Σ ∼ = ∂ H . This factcombined with Theorem 5.1 implies that p and p are are in the same Aut ( T g ) × Aut ( H ) -orbit if p (cid:48) and p (cid:48) are in the same Aut ( F g ) × Aut ( H ) -orbit.The question then reduces to the transitivity of the Aut ( F g ) × Aut ( H ) action on redundanthomomorphisms F g → H . In [
27, Conjecture 6.3 ] , it was conjectured that this action is transitive.This is known to be true when H is solvable [ ] and when H is simple [
10, 14 ] (see also [ ] ). If the conjecture is true, then our homomorphisms ρ H , r depend only on ( H , r ) up to the above equivalence.A more general problem is to understand the Aut ( T g ) × Aut ( H ) -orbits of all epimorphisms,Epi ( T g , H ) , not just those which are φ -redundant. One obstruction to transitivity is H ( H , (cid:90) ) .Given an epimorphism T g → H , there is an induced map H ( T g , (cid:90) ) → H ( H , (cid:90) ) which specifiesa well-defined element of H ( H , (cid:90) ) / Out ( H ) up to the Aut ( T g ) × Aut ( H ) -action. Those epimor-phisms T g → H factoring through F g and, in particular, φ -redundant homomorphisms inducethe (Out ( H ) -orbit of the) trivial element in H ( H , (cid:90) ) / Out ( H ) . Theorem 6.20 of [ ] shows thatfor any fixed finite group H and sufficiently large g , the Aut ( T g ) × Aut ( H ) -orbits of Epi ( T g , H ) are in bijective correspondence with H ( H , (cid:90) ) / Out ( H ) . We make the following conjecture. Conjecture.
If H is finite and g > d ( H ) , then the Aut ( T g ) × Aut ( H ) -orbits of Epi ( T g , H ) are inone-to-one correspondence with H ( H , (cid:90) ) / Out ( H ) . Note that for any surjective homomorphism p : T g → H and irreducible (cid:81) -representation r of H , there is an associated representation ρ H , r even if p is not φ -redundant. The assumption that p is φ -redundant is only necessary for us to establish that the image is arithmetic. If the aboveconjecture is true, then, up to the natural equivalence, ρ H , r is uniquely determined by the pair ( H , r ) and an element of H ( H , (cid:90) ) / Out ( H ) . ( T ) and Theorem 1.6 for the case g = g =
2, allthe virtual arithmetic quotients we obtain for Mod (Σ g ) are lattices in higher-rank semisimpleLie groups (whose simple factors are of (cid:82) -rank ≥
2) and hence have Kazhdan’s property ( T ) .We do not know if this is the case for the analogous virtual quotient of Aut ( F n ) constructed in [ ] . There it is possible that the image has an infinite abelian quotient. See [ ] for a detaileddiscussion. As promised in the introduction, we show that the groups in the Johnson filtration are “verydifferent” from the new Torelli groups we define here. Precisely, we prove the following.
Proposition 8.3.
Assume g ≥ . Let H be a nontrivial finite group, r a nontrivial irreducible (cid:81) -representation of H, and let Γ = Γ H , r and ρ = ρ H , r be as in Theorem 1.8. Let T = T H , r = ker ( ρ H , r ) and I any element in the Johnson filtration of Mod (Σ) . Then I · T is of finite index in Mod (Σ) . ρ H , r ( I ) has finite index image in Ω H , r when H is nontrivial. Thus, our main results imply also that the classical Torelli group and thesubgroups in the Johnson filtration have a rich collection of arithmetic quotients. Proof.
First, suppose that I is the classical Torelli group, i.e. the kernel of the standard homo-morphism Mod (Σ) → Sp ( g , (cid:90) ) . Let I (cid:48) = I ∩ Γ . Then I (cid:48) and T are both normal in Γ . Consider I · T / I which is a normal subgroup of Γ / I , a finite index subgroup of Sp ( g , (cid:90) ) . By Margulis’normal subgroup theorem [
30, Theorem (4’) ] , it is either finite or of finite index. In the lattercase, it follows that I · T is of finite index in Mod (Σ) .Suppose then that I · T / I is finite. Then ˆ T = I ∩ T is of finite index in T . Consider now ˆ T I / ˆ T in Γ / ˆ T . The latter group is commensurable to a higher rank lattice, so as above ˆ T I / ˆ T is finite orof finite index. If the latter holds, we are done. If the former holds, then I ∩ ˆ T is of finite indexin I , and so T and I are commensurable, but that is impossible.Now, let I be any group in the Johnson filtration and let I be the classical Torelli group.Let I (cid:48) = Γ ∩ I and I (cid:48) = Γ ∩ I . The quotient I / I is nilpotent, and thus I (cid:48) / I (cid:48) is nilpotent. Let Π : Γ → Γ / T be the quotient map. The above two paragraphs show that Π( I (cid:48) ) is of finite indexin Γ / T . Thus, Π( I (cid:48) ) / Π( I (cid:48) ) is a nilpotent quotient of a higher rank lattice and hence finite. Thisimplies that I · T is of finite index in Mod (Σ) .
9. Finite groups and arithmetic quotients
As explained in the introduction, our main result gives us a way to associate a virtual epimor-phism ρ from Mod (Σ) to an arithmetic group Ω from the choice of a finite group H and irre-ducible (cid:81) -representation r or, equivalently, a simple component A of (cid:81) [ H ] . In this section, wepresent various kinds of arithmetic groups which arise as we vary H and r .In the first section, we prove Corollary 1.4. While this is formally a corollary of Theorem 1.2as stated, our actual proof must proceed by deducing the corollary directly. Then, for genus 2, thevarious examples in Theorem 1.2 actually follow from the “corollary”. Note, then, that for genus2, we do not obtain these virtual arithmetic quotients of Mod (Σ ) by various representations ρ H , p . In the next few sections, we prove Theorem 1.2 by listing the finite groups H and irre-ducible representations r , establishing the requisite properties and determining the arithmeticgroup Ω H , A with the use of results in Section 4 when necessary. We then prove Corollary 1.5using example (a) of Theorem 1.2. In the final section, we discuss more generally the questionof what simple (cid:81) -algebras with involution can arise as factors of (cid:81) [ H ] and deduce Theorem 1.3. g = H = (cid:90) / (cid:90) and we let p : T → (cid:90) / (cid:90) be some surjective φ -redundant map. Here, (cid:81) [ H ] ∼ = (cid:81) ⊕ A where A ∼ = (cid:81) as a (cid:81) -algebra and where as a (cid:81) [ H ] module, the nontrivial elementof H acts on A by multiplication by −
1. Note that the standard involution τ on (cid:81) [ H ] is trivial,and hence its restriction to A is trivial. Thus, Aut ( A g − , 〈− , −〉 ) = Sp ( (cid:81) ) = SL ( (cid:81) ) . FromLemma 7.1, it follows directly that ρ (Γ) contains the generating elements (cid:130) (cid:140) (cid:130) (cid:140) .36oreover, as ρ (Γ H , p ) preserves R ⊆ ˆ R , there is some finite index subgroup of Γ H , p whose imageis Sp ( (cid:90) ) .Because of the corollary, Mod (Σ ) virtually surjects onto a finitely generated free group, andthus onto all free groups of finite rank. Since all the arithmetic groups in Theorem 1.2 are finitelygenerated, Mod (Σ ) virtually surjects onto them. In this section, we describe an irreducible representation for the symmetric group and alternatinggroup. Let Sym ( m + ) denote the symmetric group on m + ( m + ) the alternatinggroup on m + ( m + ) on (cid:81) m + acting by the permutation matrices. The vector w = (
1, 1, . . . , 1 ) is a fixed vector,but Sym ( m + ) acts irreducibly on the orthogonal complement V = w ⊥ . The representation r : Sym ( m + ) → GL ( V ) is irreducible and moreover, the induced homomorphism ˜ r : (cid:81) [ Sym ( m + )] → End (cid:81) ( V ) is surjective, and so the corresponding simple factor in (cid:81) [ Sym ( m + )] is A ∼ = Mat m ( (cid:81) ) . Since the center is L = (cid:81) , the fixed field K under τ is K = L = (cid:81) . Since A ⊗ K (cid:82) = A ⊗ (cid:81) (cid:82) ∼ = Mat m ( (cid:82) ) , by Proposition 4.3, τ is of first kind and orthogonal type.For m ≥
3, the restriction r : Alt ( m + ) → GL ( V ) is irreducible, so A = Mat m ( (cid:81) ) is a simplecomponent of (cid:81) [ Alt ( m + )] also. For the same reasons as above, the induced involution τ on A is of first kind and orthogonal type. For all m ≥
2, a 3-cycle γ and m -cycle δ for m odd (resp. m + δ for m even) generate Alt ( m + ) ; moreover, we can choose δ , γ such that Alt ( m + ) is generated by all elements of the form δ k γδ − k . In this section, we establish example (a) of Theorem 1.2 for genus g ≥
3. Here and in thefollowing three sections, we use our main result, Theorem 1.8, and so here, and in the followingsections, we generally present the following: • the finite group H and a demonstration that d ( H ) ≤ < g • an irreducible representation r of H and / or the corresponding simple factor A of (cid:81) [ H ] • an identification of a finite index subgroup of the arithmetic group Ω H , A .For example (a), let H = Sym ( m + ) . It is well-known that Sym ( m + ) is generated by twoelements. We take r to be the standard representation, so the corresponding simple factor A isMat m ( (cid:81) ) as explained in Section 9.2.We now describe Ω H , A . Let G = Aut A ( A g − , 〈− , −〉 ) , and let σ be the involution of End A ( A g − ) associated to 〈− , −〉 . Since τ is of orthogonal type, the involution σ associated to 〈− , −〉 is of symplectic type by Lemma 3.5. The endomorphism ring End A ( A g − ) is isomorphic toMat ( g − ) ( A op ) ∼ = Mat m ( g − ) ( (cid:81) ) . Over fields K of characteristic 0, all symplectic involutions ofMat ( g − ) ( K ) are equivalent, and so there is an isomorphism ϕ : End A ( A g − ) → Mat m ( g − ) ( (cid:81) ) sending σ to the standard symplectic involution. The intersection of ϕ ( G ( O )) and Sp ( m ( g − ) , (cid:90) ) is of finite index in the latter. By Theorem 1.8, there is a virtual epimorphism of Mod (Σ g ) onto Ω H , A = G ( O ) which, up to finite index, is isomorphic to Sp ( m ( g − ) , (cid:90) ) . This establishesexample (a) in Theorem 1.2. 37 .4. Example (b) of Theorem 1.2 The required finite group will be H = Dih ( n ) for m = H = Alt ( m + ) × Dih ( n ) for m ≥ ( n ) is the dihedral group of order 2 n which has presentationDih ( n ) = 〈 x , y | x n = y = y x y − = x − 〉 For m =
2, we require a group containing Dih ( n ) which we will describe below. (We remarkthat for g ≥
4, one can use H = Sym ( m + ) × Dih ( n ) for all m ; however this choice is notsuitable for g = n even.)We first establish (b) in the case of m ≥
3. The group H is generated by ( γ , y ) and ( δ , x ) forthe following reasons. Both ( γ , 1 ) and ( y ) are powers of ( γ , y ) . By conjugating ( γ , 1 ) by powersof ( δ , x ) , we obtain all elements of the form ( δ k γδ − k , 1 ) . From this, we generate Alt ( m + ) × ( γ , y ) and ( δ , x ) generate H .We next describe an irreducible representation of Dih ( n ) . Let ζ be a primitive n th root ofunity, and let α → α denote the order 2 automorphism of (cid:81) ( ζ ) defined by ζ (cid:55)→ ζ − . Set B = { (cid:130) α ββ α (cid:140) | α , β ∈ (cid:81) ( ζ ) } .The group ring (cid:81) [ H ] surjects onto the (cid:81) -algebra B via the homomorphism ˜ s defined by˜ s ( x ) = (cid:130) ζ ζ (cid:140) ˜ s ( y ) = (cid:130) (cid:140) .The center L of B is equal to the subfield of (cid:81) ( ζ ) invariant under , i.e. L = (cid:81) ( ζ ) + . Recall thatthe induced involution, call it ν , on B is the unique involution satisfying ν ( ˜ s ( z )) = ˜ s ( z ) − for all z ∈ Dih ( n ) . It follows that the involution is the one given by conjugate transpose of the matrix,and so ν is of first kind. Moreover, dim L ( B ) = ν -fixed vector subspace is 3-dimensional,so ν is of orthogonal type.We can describe B further. Recall that B is a matrix ring over a division algebra with center L .Notice, however, that B is not a division algebra, and since it has dimension 4 over L , it followsthat B ∼ = Mat ( L ) .We now describe the simple factor A of (cid:81) [ H ] and the involution τ . Let ˜ s (cid:48) : (cid:81) [ Alt ( m + )] → Mat m ( (cid:81) ) be the standard representation from Section 9.2, and let ν (cid:48) be the involution ofMat m ( (cid:81) ) induced by the standard involution of (cid:81) [ Alt ( m + )] . Set A = Mat m ( (cid:81) ) ⊗ (cid:81) B . Thereis a surjective homomorphism ˜ r : (cid:81) [ H ] → A defined by ˜ r ( z , w ) = ˜ s (cid:48) ( z ) ⊗ ˜ s ( w ) on ( z , w ) ∈ H = Alt ( m + ) × Dih ( n ) . The involution τ of A induced by the standard involution of (cid:81) [ H ] satisfies τ ( x ⊗ b ) = ν (cid:48) ( x ) ⊗ ν ( b ) .It is straightforward to check that τ is of first kind. Let V + , V − (resp. W + , W − ) be the +
1- and − ν (cid:48) (resp. ν ). Then, the + τ is V + ⊗ W + + V − ⊗ W − , and,after counting dimension, one finds that this eigenspace is more than half the dimension of A .Consequently, τ is of orthogonal type.In total, there is a surjective homomorphism s : (cid:81) [ H ] → A ∼ = Mat m ( B ) ∼ = Mat m ( L ) withinduced orthogonal involution on Mat m ( L ) . Arguing as in the Section 9.2, we find that, up to38nite index, Ω H , A = G ( O ) is isomorphic to Sp ( · ( m )( g − ) , O ) = Sp ( m ( g − ) , O ) . Thus,there is a virtual epimorphism Mod (Σ g ) onto Sp ( m ( g − ) , O ) .For m =
1, the simple factor of (cid:81) [ H ] is A = B , and clearly d ( H ) =
2. The analysis aboveapplies mutatis mutandis to establish that G ( O ) , up to finite index, is Sp ( ( g − ) , O ) and thereis a virtual epimorphism Mod (Σ g ) → Sp ( ( g − ) , O ) .Now suppose m =
2. We construct H as a subgroup of the units in Mat ( B ) . Namely, let X = ˜ s ( x ) ∈ B and Y = ˜ s ( y ) ∈ B , and set H = (cid:168)(cid:130) X k Z Z (cid:140) , (cid:130) X k + ZZ (cid:140) ∈ Mat ( B ) | Z ∈ s ( Dih ( n )) , k ∈ (cid:90) (cid:171) This is a group which is finite and generated by the following two elements (cid:130) Y Y (cid:140) (cid:130) XI (cid:140) .The inclusion H → Mat ( B ) induces a homomorphism (cid:81) [ H ] → Mat ( B ) def = A which is surjective.The induced involution τ satisfies τ (( b i j )) = ( ν ( b ji )) . Using arguments similar to the above,we find that G ( O ) is, up to finite index, Sp ( ( g − ) , O ) and there is a virtual epimorphismMod (Σ g ) → Sp ( ( g − ) , O ) . Here, we use the finite group H = Sym ( m + ) × Cyc ( n ) for m ≥ H = Cyc ( n ) for m = ( n ) = (cid:90) / n (cid:90) . We first describe an irreducible representation of Cyc ( n ) . Namely, there is anirreducible representation Cyc ( n ) → (cid:81) ( ζ ) × sending the generator of Cyc ( n ) to ζ , a primitive n throot of unity. The corresponding map (cid:81) [ Cyc ( n )] → (cid:81) ( ζ ) is surjective. The induced involution, ν on L = (cid:81) ( ζ ) is the unique one sending ζ to ζ − which is an involution of the second kind withfixed field K = (cid:81) ( ζ ) + .Suppose now that m ≥
2. The group Sym ( m + ) × Cyc ( n ) has generators ( γ , 0 ) and ( δ , 1 ) where γ is a transposition and δ is an m + ( γ , 0 ) by ( δ , 1 ) generateSym ( m + ) ×
0, and from there it is obvious that H is generated by both elements.Via an argument similar to the one used in Section 9.4, we have a surjective homomorphism (cid:81) [ H ] → Mat m ( (cid:81) ) ⊗ (cid:81) ( ζ ) ∼ = Mat m ( (cid:81) ( ζ )) = A . Since the center is L = (cid:81) ( ζ ) , the induced involu-tion on A is of second kind by Proposition 4.3. The τ -fixed subfield K is (cid:81) ( ζ ) + .It only remains to describe Ω H , A . From the description of 〈− , −〉 , it is clear that a matrix E isthe adjoint of C ∈ Mat g − ( A op ) if and only if E ∗ (cid:130) I − I (cid:140) = (cid:130) I − I (cid:140) C where E ∗ is the map defined by ( e i j ) ∗ = ( τ ( e ji )) . Thus, the adjoint involution for 〈− , −〉 is σ ( C ) = (cid:130) I − I (cid:140) − C ∗ (cid:130) I − I (cid:140) .Since Mat g − ( A op ) ∼ = Mat m ( g − ) ( (cid:81) ( ζ )) , the involution σ is also the adjoint involution for anondegenerate (cid:81) ( ζ ) -valued Hermitian form 〈− , −〉 on (cid:81) ( ζ ) m ( g − ) ( [
23, Theorem 4.2 ] ). From39he fact that 〈− , −〉 has a maximal isotropic submodule, one can deduce that 〈− , −〉 has amaximal isotropic subspace (i.e. of (cid:81) ( ζ ) -dimension m ( g − ) ) as well. All Hermitian formson (cid:81) ( ζ ) m ( g − ) with a maximal isotropic subspace are equivalent (e.g. one can apply the samearguments as in Lemma 5.5), and so the form 〈− , −〉 is equivalent to the form defining U ( m ( g − ) , m ( g − ) , (cid:81) ( ζ )) .Thus, there is an isomorphism from a finite index subgroup of Ω H , A = G ( O ) to a finite indexsubgroup of SU ( m ( g − ) , m ( g − ) , O ) where O is the ring of integers in (cid:81) ( ζ ) . By Theorem 1.8,there is a virtual epimorphism from Mod (Σ g ) onto SU ( m ( g − ) , m ( g − ) , O ) .For the case m =
1, a simplified version of the above argument applies, and so there is avirtual epimorphism Mod (Σ g ) onto SU (( g − ) , ( g − ) , O ) . For these examples, we use the finite group H = Alt ( m + ) × Dic ( n ) for m ≥ ( n ) for m = ( n ) is the dicyclic group of order 4 n . The group Dic ( n ) has the followingpresentation: Dic ( n ) = 〈 x , y | x n = y = x n , y − x y = x − 〉 .For m =
2, we use a group containing Dic ( n ) which we will describe below and which has aconstruction similar to that for example (b).We begin with the case m ≥
3. The group H = Alt ( m + ) × Dic ( n ) is generated by thetwo elements ( γ , y ) and ( δ , x ) . This follows verbatim from the same argument for Alt ( m + ) × Dih ( n ) .The group ring (cid:81) [ Dic ( n )] has a representation onto a division algebra D defined as follows.Let ζ be the primitive 2 n th root of unity, and let denote the unique order 2 automorphism of (cid:81) ( ζ ) sending ζ (cid:55)→ ζ − . The division algebra is D = (cid:168)(cid:130) α β − β α (cid:140) ∈ Mat ( (cid:81) ( ζ )) | α , β ∈ (cid:81) ( ζ ) (cid:171) .The center L of D is (cid:81) ( ζ ) + , and the epimorphism ˜ s : (cid:81) [ Dic ( n )] → D is defined on the generatorsby ˜ s ( x ) = (cid:130) ζ ζ (cid:140) ˜ s ( y ) = (cid:130) − (cid:140) The induced involution, call it ν , on D is that involution satisfying ν ( ˜ s ( z )) = ˜ s ( z ) − for all z ∈ Dic ( n ) . Thus, in terms of the matrices, ν is conjugate transpose, and it is of first kind (on D ).The fixed subspace of ν has L -dimension 1 and D has L -dimension 4, so ν is of symplectic type.We describe the simple factor A of (cid:81) [ H ] . Let ˜ s (cid:48) : (cid:81) [ Alt ( m + )] → Mat m ( (cid:81) ) be the homo-morphism from Section 9.2, and let ν (cid:48) be the involution of Mat m ( (cid:81) ) induced by the standardinvolution of (cid:81) [ Alt ( m + )] . Let A = Mat m ( (cid:81) ) ⊗ (cid:81) D ∼ = Mat m ( D ) . There is a surjective homomor-phism ˜ r : (cid:81) [ H ] → A defined by ˜ r ( z , w ) = ˜ s (cid:48) ( z ) ⊗ ˜ s ( w ) on ( z , w ) ∈ H . The induced involution, τ , on A is equal to ν (cid:48) ⊗ ν . By an argument similar to the one in Section 9.4, one can show τ is of first kind and symplectic type. By Theorem 1.7, G = Aut A ( A g − , 〈− , −〉 ) is the L -points( K = L here) of an L -defined algebraic group whose real form is SO ( m ( g − ) , 2 m ( g − ) , (cid:82) ) .Thus, there is a virtual epimorphism of Mod (Σ g ) onto Ω H , A = G ( O ) , an arithmetic subgroup ofSO ( m ( g − ) , 2 m ( g − ) , (cid:82) ) . 40or m =
1, it is clear that H = Dih ( n ) is generated by two elements, and the above argu-ments apply mutatis mutandis to show that there is a virtual epimorphism of Mod (Σ g ) onto anarithmetic subgroup of SO ( ( g − ) , 2 ( g − ) , (cid:82) ) .For m =
2, we construct H as a subgroup of the units of Mat ( D ) . Namely, let X = s ( x ) ∈ D and Y = s ( y ) ∈ D , and set H = (cid:168)(cid:130) X k Z Z (cid:140) , (cid:130) X k + ZZ (cid:140) ∈ Mat ( D ) | Z ∈ s ( Dic ( n )) , k ∈ (cid:90) (cid:171) .Like the group H for m = H here is generated by two elements, and theinclusion H → Mat ( D ) induces a homomorphism (cid:81) [ H ] → Mat ( D ) def = A which is surjective.The induced involution τ satisfies τ (( b i j )) = ( ν ( b ji )) , and so τ is of symplectic type. Usingarguments similar to the above, we find a virtual epimorphism onto an arithmetic subgroup ofSO ( ( g − ) , 4 ( g − ) , (cid:82) ) . This follows rather quickly from example (a) of Theorem 1.2. A finite index subgroup of Sp ( m ( g − ) , (cid:90) ) is mapped onto Sp ( m ( g − ) , (cid:90) / p (cid:90) ) for almost all primes p . The latter contains SL ( m ( g − ) , (cid:90) / p (cid:90) ) and hence contains Sym ( m ( g − ) − ) . Now, every finite group G embeds in Sym ( m ( g − ) − ) for sufficiently large m . Thus, Mod (Σ g ) has a finite index subgroup which is mapped onto G . (cid:81) [ H ] in general From Theorem 1.8, given a finite group H with d ( H ) < g and a simple factor A of (cid:81) [ H ] , thereis a corresponding arithmetic group Ω H , A which is a virtual quotient of Mod (Σ g ) . It is thereforeof interest to understand what kinds of algebras with involution ( A , τ ) are obtained in this way.This seems to be an open problem in general.We start with some basic properties of the pairs ( A , τ ) . Recall that A is isomorphic to Mat m ( D ) for some integer m and some division algebra D with a number field L as center. Recall fromLemma 4.2 that L must be a subfield of a cyclotomic field. This is equivalent to L being an abelianextension of (cid:81) . Recall, moreover, from Proposition 4.3 that L is totally real if and only if τ is offirst kind and L is totally imaginary if and only if τ is of second kind.We will not say much about the involution τ . However, in the case where τ is of first kind,the division algebra D is considerably restricted. The following proposition is essentially theBrauer-Speiser Theorem, and the proof we give is basically that given by Fields in [ ] . Recallthat the degree d of a division algebra D is the integer satisfying d = dim L ( D ) where L is thecenter of D . Proposition 9.1.
If A = Mat m ( D ) has an involution of the first kind, then the degree of D is at most , i.e. D is the field L or a quaternion algebra over L.Proof. If τ is an involution of the first kind, then τ is an isomorphism A ∼ = A op as L -algebras. Thus A ⊗ L A ∼ = A ⊗ L A op ∼ = Mat n ( L ) where n = dim L ( A ) [
11, Proposition 3.12 ] . Thus, the class definedby A in the Brauer group of L has order 1 or 2. (See below for the definition of the Brauer groupand [ ] for more details.) For number fields L , the exponent of Mat m ( D ) in the Brauer groupof L is equal to the degree of D [
40, Theorem 32.19 ] .41et us consider, now, the simpler question of which algebras A appear as simple componentsof (cid:81) [ H ] or what division algebras D appear in such an A ∼ = Mat m ( D ) regardless of m . Noticethat, by using Sym ( (cid:96) + ) and its standard representation, given any algebra A = Mat m ( D ) whichis a simple part of (cid:81) [ H ] , we can also obtain Mat (cid:96) m ( D ) as a simple part of (cid:81) [ H × Sym ( (cid:96) + )] .One can show that this increases the requisite number of generators of the finite group by atmost 1 except when H is trivial. For D = L a field, the algebra Mat m ( D ) is a simple componentof (cid:81) [ H ] for some H precisely when L is a subfield of a cyclotomic field (See e.g. [
34, Theorem3.2 ] ). This leads to the proof of Theorem 1.3. Proof of Theorem 1.3.
We use Theorem 1.8 and follow a similar program as for Theorem 1.2.From the above discussion, it follows that for some s L ∈ (cid:78) , there is a finite group H such thatMat s L ( L ) is a simple component of (cid:81) [ H ] . Consequently, using arguments similar to those forTheorem 1.2, Mat ms L ( L ) is a simple part of (cid:81) [ H × Sym ( m + )] . Let N = d ( H × Sym ( m + )) + L is a totally real field. By Proposition 4.3, τ is of first kind and K = L . By thesame proposition, since Mat ms L ( L ) ⊗ K (cid:82) = Mat ms L ( K ) ⊗ K (cid:82) = Mat ms L ( (cid:82) ) , the involution τ is oforthogonal type. Arguing as for example (b) of Theorem 1.2, we find a virtual epimorphism ofMod (Σ g ) onto Sp ( ms L ( g − ) , O ) where O is the ring of integers in L .Suppose L is a totally imaginary field. By Proposition 4.3, τ is of second kind. Arguing asfor example (c) of Theorem 1.2, we find a virtual epimorphism of Mod (Σ g ) onto SU ( ms L ( g − ) , ms L ( g − ) , O ) where O is the ring of integers in L .If we ask only what division algebras D appear regardless of m , then we can appeal to resultson the Schur subgroup, S ( L ) , of the Brauer group, B ( L ) , for a finite abelian extension L of (cid:81) .The elements of the Brauer group are equivalence classes of central simple L -algebras where A and A are equivalent if they are isomorphic, respectively, to Mat m ( D ) and Mat m ( D ) for somedivision algebra D . The group operation is ⊗ L . The Schur group consists of those classes [ A ] where A is a simple component of some L [ H ] . In the case where L is a finite abelian extension of (cid:81) , the Schur subgroup is also the set of those classes [ A ] where A is a simple component of some (cid:81) [ H ] . (This seems to be well-known among experts, but is also a consequence of the stronger [
34, Theorem 3.2 ] .)We mention some of the easier-to-state theorems in the literature. One basic question thatcan be answered is what possible degree d of a division algebra D is possible. I.e. for what d isdim L ( D ) = d for some division algebra D in the Schur group of L ? It follows from [ ] that, for L containing a primitive d th root of unity, there are infinitely many elements [ D ] in S ( L ) where D has degree d . In particular, in combination with our theorems, this implies that for each of theseinfinitely many division algebras D , some m and sufficiently large g (depending on D ), there is avirtual epimorphism of Mod (Σ g ) onto the arithmetic group G ( O ) , i.e. the elements of reducednorm ! in Aut O ( O g − , 〈− , −〉 ) where O is an order in Mat m ( D ) .In the opposite direction, a result due to Schacher and Fein (independently) tells us that forinfinitely many division algebras D are unattainable regardless of m [ ] . In the special case of L = (cid:81) , it has been shown that S ( (cid:81) ) and B ( (cid:81) ) are equal when restricting to classes [ D ] where thedegree of D is 2 [
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