aa r X i v : . [ m a t h . A T ] J u l BIANCHI’S ADDITIONAL SYMMETRIES
ALEXANDER D. RAHM
Abstract.
In a 2012 note in Comptes Rendus Math´ematique, the author didtry to answer a question of Jean-Pierre Serre; it has recently been announcedthat the scope of that answer needs an adjustment, and the details of thisadjustment are given in the present paper. The original question is the following.Consider the ring of integers O in an imaginary quadratic number field, and theBorel–Serre compactification of the quotient of hyperbolic 3–space by SL ( O ).Consider the map α induced on homology when attaching the boundary intothe Borel–Serre compactification. How can one determine the kernel of α (in degree 1) ? Serre used a global topological argument and obtained the rank of the kernelof α . He added the question what submodule precisely this kernel is. Introduction
As the note “On a question of Serre” [13] by the author had been published pre-maturely, the proof of the key lemma remaining sketchy, the author did contactan expert for the Borel–Serre compactification, Lizhen Ji, for the project of estab-lishing a detailed version of that key proof. Lizhen Ji then found out that thereare cases where that proof does not apply. So it is necessary to correct the scopeof the lemma, which has been announced in Comptes Rendus Math´ematique [14],and is detailed in the present paper.The Bianchi groups, SL ( O ) over the ring O of integers in an imaginary qua-dratic field Q ( √− D ), where D is a square-free positive natural integer, act natu-rally on hyperbolic 3-space H (cf. the monographs [8], [9], [10]). For a subgroupΓ of finite index in SL ( O ), consider the Borel–Serre compactification Γ \ b H ofthe orbit space Γ \H , constructed by Borel and Serre [4], which is a manifold withboundary. In our case, the boundary components of Γ \ b H are disjoint, which allowsfor an explicit description (see the appendix of Serre’s paper [17]). Throughout thispaper, we will exclude the ring O from being the Gaussian integers, in Q ( √− , orthe Eisensteinian integers, in Q ( √− . In those two special cases, the boundaryat the single cusp is a 2–sphere, and these two special cases can easily be treatedseparately by an explicit manual computation. So we are in the setting where eachboundary component of Γ \ b H is a 2-torus T σ which compactifies the cusp σ . Date : July 16, 2020.
Question 1 (Serre [17]) . Consider the map α induced on homology when attachingthe boundary into the Borel–Serre compactification Γ \ b H . How can one determinethe kernel of α (in degree 1) ?Motivation of Question 1 . The search for a set of generators for the kernelof the attaching map has motivations that are indicated already in Serre’s 1970paper. The paper does (after achieving its goal, namely solving the congruencesubgroups problem), describe how to calculate the Abelianisation of the investi-gated arithmetic groups in terms of generators for the kernel of the attaching map.Knowledge about the Abelianisation Γ ab of Γ a (congruence subgroup in a) Bianchigroup, especially in terms of generating matrices, has the following application tothe Langlands programme (we can assume that this application was already re-alised by Serre at that time, because of his subsequent question in the paper).For each weight k of modular forms, there is a coefficient module M k such thatthe Eichler–Shimura–Harder isomorphism identifies the space of weight k modularforms for Γ with the cohomology H (Γ; M k ). The latter can be computed explic-itly as Hom(Γ ab , M k ) when Γ ab is given in terms of generating matrices. Thesecomputations would then not have to pass by an explicit computation of a fun-damental domain like the Bianchi fundamental polyhedron, which is a necessarystep still nowadays [15, 16] for computing the dimension of H (Γ; M k ).The following lemma was the key for the approach to Question 1 pursued in the2012 note. In the remainder of this paper, we consider an imaginary quadraticfield Q ( √− D ) with D a square-free positive natural integer, and we set Γ =SL ( O Q ( √− D ) ). We decompose the 2-torus T σ in the classical way into a 2–cell,two edges and a vertex. Lemma 2.
Let n be the number of prime divisors of D . Let N = 2 n − for D ≡ , N = 2 n for D ≡ or . Then Γ \H admits at least N cusps σ such that the inclusion of T σ into Γ \ b H induces on H ( T σ ) a map of rank . The restriction to N cusps was missing in the 2012 note, and we will give acompleted proof in Section 2, making use of this restriction. For this purpose, weexploit symmetries of the quotient space Γ \H , which Bianchi did find additionallyto the “basic” ones which occur for all studied rings O and which are given bycomplex conjugation and a rotation of order 2 around the origin of the complexplane (see Section 1). Corollary 3.
If the class group of O is isomorphic to Z / m Z for some m ∈ N ,then for all cusps σ of Γ \H , the inclusion of T σ into Γ \ b H induces on H ( T σ ) amap of rank .Proof. Let the numbers n and N be as in Lemma 2. By [5, thm. 3.22], ourhypothesis on the class group of O is equivalent to m = ( n − , D ≡ ,n, D ≡ . IANCHI’S ADDITIONAL SYMMETRIES 3
It is well known that each ideal class of O corresponds to one cusp of Γ \H , sothere are exactly N cusps, and Lemma 2 yields the claim. (cid:3) As the class group type assumed in Corollary 3 occurs only finitely manytimes [19], it might seem disappointing that the scope of the theorem deducedfrom it in [13] is much more narrow than it was originally claimed, but this ac-tually means that once that we leave this scope of validity, the topology of theBianchi orbifolds near their boundary is much richer than asserted in [13], and canprovide interesting studies for future generations of mathematicians.1.
Bianchi’s additional symmetries
In the cases Γ = SL ( O ) which he considered, Luigi Bianchi did make an ex-haustive description of the symmetries of the quotient space Γ \H . They are givenby outer automorphisms of Γ, and a subgroup of these automorphisms is, for allstudied rings O , the Klein four-group with generators c and e , where for γ ∈ Γ,the matrix c ( γ ) = γ is the complex conjugate of γ , and e ( γ ) = E · γ · E with E = (cid:18) − (cid:19) ∈ GL ( O ) – see [18, remark below lemma 4.19]. We will considerthis Klein four-group as the “basic” symmetries, and will call “additional” thesymmetries of Γ \H given by outer automorphisms outside this Klein four-group.Throughout this paper, we use the upper-half space model H for hyperbolic 3-space, as it is the one used by Bianchi. As a set, H = { ( z, ζ ) ∈ C × R | ζ > } . Then, the basic symmetries take the form c ( z, ζ ) = ( z, ζ ) and e ( z, ζ ) = ( − z, ζ ).The list of Bianchi’s symmetries presented in this section will allow us to reducethe proof in the following section in each case to the situation at the cusp at infin-ity. As a fundamental domain in hyperbolic space, we make use of the polyhedronwith missing vertices at the cusps, described by Bianchi [2], and which we willcall the Bianchi fundamental polyhedron . It is the intersection of a fundamentaldomain in H for the stabiliser in Γ of the cusp at ∞ (in the shape of a rectan-gular prism) with the set of points of H that are closer to ∞ than to any othercusp, with respect to Mendoza’s distance to cusps [11]. We consider the cellularstructure on Γ \ b H induced by the Bianchi fundamental polyhedron. From Bianchi’sarticles [2, 3], it was not clear that one can always compute the Bianchi fundamen-tal polyhedron, a fact which meanwhile has been established by an algorithm ofRichard G. Swan [18] that has been implemented at least twice ([12], which is astraight implementation of Swan’s algorithm, and [1, 7], which additionally usesideas of Cremona [6]). The reason why in some cases Bianchi did not obtain theBianchi fundamental polyhedron, must be that some “sfere di riflessione impro-pria” did escape him [2, §
20; at the first example, D = 14, it is the hemisphere ofcenter + √− and radius ]. Note that in the upper-half space model, totally ge-odesic planes are modelled by vertical planes, respectively by hemispheres centeredin the boundary, so we will call the latter “reflecting (hemi)spheres” to continue ALEXANDER D. RAHM
Bianchi’s terminology, even though they are to be thought as mirroring planes inhyperbolic space. An improper reflection sphere (“sfera di riflessione impropria”)has its reflection composed with the order-2-rotation around its vertical radius (theone connecting its center with its highest point). Instead of the missing reflectingspheres, Bianchi did find additional symmetries which did allow him to constructa differently shaped but equivalent fundamental domain [3]. These symmetries aregoing to be the foundations of our proof. Since Bianchi’s papers, a cusp is called singular when it is not in the Γ-orbit of ∞ . Theorem 4 (Bianchi) . Let D , n and N be as in Lemma 2. Then there are N − symmetries of the Bianchi fundamental polyhedron, each flipping the cusp ∞ witha different singular cusp, while leaving invariant a hemisphere of radius smallerthan , and preserving the Γ -cell structure. We will call those symmetries
Bianchi’s additional symmetries . We sketch Bianchi’s proof of Theorem 4 in order to fix notation.
With D a square-free positive natural integer satisfying either • Case A: D ≡ D admits a prime divisor m < D ; • or Case B: D ≡ D > M σ = q − Dm −√ m √ m q − Dm in Case A; M σ = √− D +1 √ D − − √ √ √− D − √ ! in Case B,where we choose m to be the smallest prime divisor of D . The action on theboundary of hyperbolic space is given by classical M¨obius transformations, (cid:18) a bc d (cid:19) · z = az + bcz + d . Therefore, Bianchi’s additional symmetry flips the cusp ∞ with the cusp σ = √− Dm in Case A; σ = √− D +12 in Case B. .We can check that the cusp σ is singular by checking that the ideal I σ is notprincipal, where I σ = ( √− D, m ) in Case A; I σ = ( √− D + 1 ,
2) in Case B.We assume the contrary and lead it to a contradiction: Suppose that there are c ∈ I σ , a, b ∈ O with ac = √− D , bc = m in Case A; ac = √− D + 1, bc = 2 in Case B.We easily show that c is in the Z -module with generators the two generators of I σ (we might say that “ c admits coefficients in Z ”) when we usein Case A, that m is a divisor of D ; in Case B, that D ≡ IANCHI’S ADDITIONAL SYMMETRIES 5Type σ r σ M σ ConditionsI a m + a √− Dcm c √ m a √ m + a q − Dm b √ mc √ m a q − Dm − a √ m D ≡ , Dm quadratic residue mod m,ma + Dm a + mbc = 1II a D − a mc √− DcD c q Dm a √ m + a q − Dm b q − Dm c q − Dm a √ m − a q − Dm D ≡ ,m quadratic residue mod Dm ,ma + Dm a + Dm bc = 1III a + a √− D c c √ a + a √− D √ b √ c √ a √− D − a √ ! D ≡ ,a + Da + 4 bc = 2IV a D − a √− D cD c √ D a + a √− D √ b √ √− Dc √ √− D a − a √− D √ ! D ≡ D,a + Da + 4 Dbc = 2V a m + a √− D cm c √ m a √ m + a q − Dm √ b √ mc √ m − a √ m + a q − Dm √ D ≡ Dm have the samequadratic character mod m,ma + Dm a + 4 mbc = 2VI a D + a m √− D cD c q Dm a √ m + a q − Dm √ b q − Dm c q − Dm a √ m − a q − Dm √ D ≡ m have the samequadratic character mod Dm ,ma + Dm a + 4 Dm bc = 2VII a m + a √− D cm c √ m a √ m + a q − Dm b √ mc √ m − a √ m + a q − Dm D ≡ Dm quadratic residue mod m,ma + Dm a + 4 mbc = 4VIII a D + a m √− D cD c q Dm a √ m + a q − Dm b q − Dm c q − Dm a √ m − a q − Dm D ≡ m quadratic residue mod Dm ,ma + Dm a + 4 Dm bc = 4 Table 1.
Bianchi’s additional symmetries [3, § III]. The parameters a , a , b, c ∈ Z and the divisor m of D have to satisfy the specifiedconditions. Examples for Type I are m = 2 , a = 0 , a = c = 1,( D, b ) ∈ { (6 , − , (10 , − , (22 , − , (58 , − } ; examples for TypeIII are a = a = c = 1, ( D, b ) ∈ { (5 , − , (13 , − , (37 , − } .solve for the coefficients of c and arrive at the desired contradiction. Therefore, thecusp σ is singular. As we have chosen m to be the smallest prime dividing D , thecusp σ is a vertex of the Bianchi fundamental polyhedron. This entails that thematrix M σ of Bianchi’s additional symmetry flips the Bianchi fundamental poly-hedron onto itself, interchanging two isometric halves of it which are separated bya plane equidistant to the cusps ∞ and σ , with respect to Mendoza’s distance tocusps [11]. The Bianchi group Γ is a normal subgroup of index 2 in the group h Γ , M σ i , whence this symmetry preserves the Γ-cell structure. The other symme-tries are obtained similarly [3], and we collect them in Table 1. The conditions inthe table are implied by det( M σ ) = 1. Then we read off from the table that theradius r σ of the reflecting sphere is smaller than 1, and that σ is a singular cusp(its numerator and denominator generate a non-principal ideal). (cid:3) ALEXANDER D. RAHM Proof of Lemma 2
We will use Bianchi’s additional symmetries for decomposing the Bianchi funda-mental polyhedron into two isometric parts with the cutting plane between thembeing equidistant between ∞ and the singular cusp at which T σ is located. Proof of lemma 2.
Consider, in the boundary of the Bianchi fundamental poly-hedron, the fundamental rectangle F for the action of the cusp stabiliser on theplane joined to H at the cusp ∞ . There is a sequence of rectangles in H obtainedas translates of F orthogonal to all the geodesic arcs emanating from the cusp ∞ .This way, the portion of the fundamental polyhedron which touches the cusp ∞ , islocally homeomorphic to the Cartesian product of a geodesic arc with a translateof F .The boundaries of these translates are subject to the same identifications by Γas the boundary of F . Namely, denote by t ( F ) one of the translates of F ; anddenote by γ x and γ y two generators, up to the − F . We can choose γ x = (cid:18) (cid:19) and γ y = (cid:18) ω (cid:19) , with ω = √− D for D ≡ ω = √− D +12 for D ≡ γ x and γ y identify the opposite edges of t ( F ) and make the quotient of t ( F )into a torus. So in the quotient space by the action of Γ, the image of T ∞ iswrapped into a sequence of layers of tori. And therefore in turn, the 3-dimensionalinterior of the Bianchi fundamental polyhedron is wrapped around the image of T ∞ along the entire surface of the latter, such that there is a neighbourhood inside Γ \H which is homeomorphic to the Cartesian product of a 2-torus and an openinterval. Hence, there is, in a suitable ambient space, a neighbourhood of the imageof T ∞ that is homeomorphic to Euclidean 3–space with the interior of a solid torusremoved. Now, considering the cell structure of the torus, we see that preciselyone of the loops generating the fundamental group of T ∞ can be contracted in theinterior of the quotient of the Bianchi fundamental polyhedron — namely, the loopgiven by the edge that has its endpoints identified by γ x . The other loop (the oneobtained by the identification by γ y ) is nontrivially linked with the removed solidtorus. This entails that it cannot be unlinked when moving it in the Borel–Serrecompactification of Γ \H , and thus remains uncontractible there. For each idealclass that is represented by a singular cusp σ which is subject to one of Bianchi’sadditional symmetries, we observe the following. The radius r σ of the reflectingsphere is smaller than 1 (cf. Table 1), and hence the above described contractionhappens within the half touching the cusp ∞ . By our Γ-cell structure preservingisometry, we get a copy of this contraction. The rˆoles of γ x and γ y are then playedby M σ γ x M − σ and M σ γ y M − σ with M σ specified in Table 1. Hence the inclusionof T σ into Γ \ b H makes exactly one of the edges of T σ become the boundary ofa 2–chain. The number of such cusps σ is given by Theorem 4. And Serre’s IANCHI’S ADDITIONAL SYMMETRIES 7 theorem about the rank of the map α [17, th´eor`eme 7] allows us to conclude thatno nontrivial linear combination of these loops can be homologous to zero. (cid:3) Note that only with Bianchi’s additional symmetries, we get more than onelinked loop. Sufficiently many of these symmetries are available only for theBianchi orbifolds specified in Corollary 3, and in general we just have the onelinked loop at the torus at infinity, which was already observed in Serre’s originalpaper.
Acknowledgements.
The author would like to thank Lizhen Ji for a fruitfulcorrespondence which laid the foundations for the present paper. He would liketo heartily thank the anonymous referee for the many very knowledgeable andhelpful suggestions made on various aspects. He is grateful to the University ofLuxembourg for funding his research through Gabor Wiese’s AMFOR grant.
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URL : http://gaati.org/rahm/ E-mail address ::