Hecke operators in Bredon (co)homology, K-(co)homology and Bianchi groups
HHECKE OPERATORS IN BREDON (CO)HOMOLOGY, K-(CO)HOMOLOGYAND BIANCHI GROUPS
DAVID MU ˜NOZ, JORGE PLAZAS, AND MARIO VEL ´ASQUEZ
Abstract.
In this article we provide a framework for the study of Hecke operators acting on theBredon (co)homology of an arithmetic discrete group. Our main interest lies in the study of Heckeoperators for Bianchi groups. Using the Baum-Connes conjecture, we can transfer computationsin Bredon homology to obtain a Hecke action on the K -theory of the reduced C ∗ -algebra of thegroup. We show the power of this method giving explicit computations for the group SL ( Z [ i ]).In order to carry out these computations we use an Atiyah-Segal type spectral sequence togetherwith the Bredon homology of the classifying space for proper actions. Introduction
Hecke operators play a prominent role in the study of arithmetic groups. The action of Heckeoperators on various cohomology theories associated to arithmetic groups and their symmetricspaces provides an essential tool bridging the analytic and arithmetic aspects of the theory. Animportant class of arithmetic groups that has received a lot of attention recently is that of Bianchigroups. These are groups of the form PSL ( O Q ( √− D ) ) where O Q ( √− D ) is the ring of integers of animaginary quadratic field. For free subgroups of Bianchi groups, Mesland and S¸eng¨un in [MS¸20]have recently defined a Hecke action on K -homology using Kasparov’s bivariant KK -theory. Wetackle the general case by first defining Hecke operators on Bredon (co)homology, allowing us to thentransfer the computations to K -theory in full generality. Computations in Bredon (co)homologycan be carried out using Atiyah-Segal type spectral sequences. We develop the correspondingmachinery which we then apply to explicitly compute the Hecke action on the K -homology of thegroup PSL ( Z [ i ]).The plan of the article is as follows. In section 2 we review the definition of Bredon modulesand Bredon (co)homology. Given a discrete group G , Bredon modules associated to families ofsubgroups of G provide coefficient systems for G -equivariant (co)homology theories. We focusin the case where the coefficient system is given by the representation ring. In this case theBredon (co)homology of a G -CW-complex X is given in terms of the representation rings of its cellstabilizers. In section 3 we define equivariant K-(co)homology in terms of spectra and discuss itsrelation to Bredon (co)homology using spectral sequences. This in turn leads via the Baum-Connesconjecture to a description of the K -theory of the reduced C ∗ -algebra of G and the possibilityof defining Hecke operators at the level of such algebras. In section 4 we review the theory ofHecke algebras and introduce the natural Hecke action on group cohomology. Here we also discussHecke correspondences over a G -space. In section 5 we develop the machinery necessary to defineHecke operators in Bredon (co)homology and transfer these to equivariant K -(co)homology. The Date : January 26, 2021.2010
Mathematics Subject Classification.
Primary 55N25; Secondary 58B34, 46L80, 20C08.
Key words and phrases.
Bianchi groups, Bredon (co)homology, K-theory, Hecke operators. a r X i v : . [ m a t h . K T ] J a n DAVID MU˜NOZ, JORGE PLAZAS, AND MARIO VEL´ASQUEZ core of our treatment lies in identifying the appropriate restriction, corestriction and conjugationmorphisms necessary leading to the action of Hecke correspondences. Section 6 of the article isdevoted to computations for Bianchi groups. Expressing Bianchi groups as amalgamated products,we can carry out the computations in terms of the representation theory of their factors viewedas cell stabilizers on G -spaces. Explicit computations in the case of Hecke operators associated tocongruence subgroups of PSL ( Z [ i ]) of prime level are carried out in full. We close the article insection 7 with a few concluding remarks.2. Bredon (Co)homology
Bredon (Co)homology.
Bredon (co)homology for finite groups was introduced by Bredonin [Bre67a], [Bre67b] in order to provide an appropriate framework for coefficient systems in anequivariant (co)homology theory. The theory can be extended to arbitrary topological groups(cf.[Ill75]). In this section we recall the main aspects of the theory for discrete groups. Ourtreatment follows that of [SG05] and [MV03].We begin by defining the orbit category which is central to Bredon’s definition of (co)homologicalinvariants for spaces with a group action.
Definition 2.1.
Let G be a discrete group and let F be a family of subgroups of G , closed underconjugation and finite intersections. Define the orbit category Or F ( G ) as the category whose objectsare sets of the form G/H with H ∈ F , and whose morphisms are given by G -maps. Notice thatsuch a morphism f g : G/H → G/K is determined by an element gK ∈ G/K with g − Hg ⊂ K , sothat it sends the coset H to the coset gK , i.e. we have an identificationMor Or F ( G ) ( G/H, G/K ) = Maps(
G/H, G/K ) G . When F is the family of all subgroups of G we simply denote Or F ( G ) by Or ( G ).Throughout what follows we fix a choice of a family F of subgroups of G as above.Denote by Ab the category of abelian groups. Definition 2.2.
A covariant (resp. contravariant) Bredon module is a covariant (resp. contravari-ant) functor M : Or F ( G ) −→ Ab . A morphism Ψ : M −→ N between Bredon modules is given by a natural transformation between the corresponding functors.This means that for each H ∈ F there is a morphism of abelian groupsΨ( G/H ) : M ( G/H ) −→ N ( G/H )and for every f g : G/H → G/K we have, in the covariant case, a commutative diagram M ( G/H ) M ( f g ) (cid:47) (cid:47) Ψ( G/H ) (cid:15) (cid:15) M ( G/K ) Ψ( G/K ) (cid:15) (cid:15) N ( G/H ) N ( f g ) (cid:47) (cid:47) N ( G/K )whilst in the cotravariant case the horizontal arrows are reversed.
ECKE OPERATORS ON BIANCHI GROUPS 3 If M and N are both covariant (resp. contravariant) Bredon modules, the group structurein each of the Hom( M ( G/H ) , N ( G/H )) induces an abelian group structure in the set of natu-ral transformations mor(
M, N ). It can be shown moreover that the category of covariant (resp.contravariant) Bredon modules is an Abelian category.If M is a contravariant Bredon module and N is a covariant Bredon module, we define theabelian group M ⊗ F N = (cid:77) H ∈ F M ( G/H ) ⊗ Z N ( G/H ) (cid:30) ∼ where the relation ∼ is generated by M ( f )( m ) ⊗ n − m ⊗ N ( f )( n ) for each f : G/H → G/K , m ∈ M ( G/K ), and n ∈ N ( G/H ).Given a CW-complex Z we denote by C ∗ ( Z ) its cellular chain complex. Let X be G -CW-complex,for each n ≥ C n ( X ) by C n ( X ) : G/H (cid:55)−→ C n ( X H ) , where X H is the subspace of X fixed by the subgroup H . Let { δ α } be the set of n -cells of X , thenthere is an isomorphism C n ( X H ) ∼ = (cid:77) α Z [ δ Hα ] , where δ Hα is the H -fixed point set of δ α ; explicitly, δ Hα is δ α if the cell is fixed by H , and otherwiseis empty, in which case it does not count in the sum. For a morphism f g : G/H → G/K we have f g := C n ( X )( f g ) : C n ( X K ) → C n ( X H ) , δ Kα (cid:55)−→ g · δ Kα =: δ Hα g . For each H ∈ F the usual boundary map ∂ : C n ( X H ) → C n − ( X H ) induces a boundary map ∂ : C n ( X ) −→ C n − ( X ) . If M is a contravariant Bredon module and X is a G -CW-complex we obtain a cochain complexmor( C ∗ ( X ) , M ) . Definition 2.3.
Let X be a G -CW-complex and let M be a contravariant Bredon module. Wedefine the n -th Bredon cohomology group of X with coefficients in M as H nG ( X ; M ) = H n (mor( C ∗ ( X ) , M )) . Analogously, if N is a covariant Bredon module and X is a G -CW-complex we obtain a chaincomplex C ∗ ( X ) ⊗ F N. Definition 2.4.
Let X be a G -CW-complex and let N be a covariant Bredon module. We definethe n -th Bredon homology group of X with coefficients in N as H Gn ( X ; N ) = H n ( C ∗ ( X ) ⊗ F N ) . As mentioned above contravariant Bredon modules form an Abelian category. We will now definea class of projective objects which will play an important role in computations.Let K ∈ F . We define the standard projective contravariant Bredon module P K as the functorgiven in objects of Or F ( G ) by P K ( G/H ) = Z [mor( G/H, G/K )] , for H ∈ F , DAVID MU˜NOZ, JORGE PLAZAS, AND MARIO VEL´ASQUEZ and which associated to a morphism f : G/H → G/H , the morphism P K ( f ) : P K ( G/H ) → P K ( G/H ) is given by the linear extension of pre-composing with f .For the Bredon modules P K , an appropriate form of the Yoneda Lemma shows that given acontravariant Bredon module M there is an induced isomorphism of Abelian groupsev K : mor( P K , M ) −→ M ( G/K ) , ϕ (cid:55)−→ ev K ( ϕ ) = ϕ ( G/K )(1) . In a similar manner, if N is a covariant Bredon module, there is an isomorphism P K ⊗ F N ∼ = N ( G/K ) . See [MV03] for more information on these isomorphisms.Let X be a G -CW-complex and, as above, let { δ α } be the set of n -cells of X , and let { e β } be aset of G -representatives of those n -cells; we know that C n ( X H ) ∼ = (cid:77) α Z [ δ Hα ] ∼ = (cid:77) β Z [( G · e β ) H ] . Moreover, if S β is the stabilizer of the cell e β , and the g ’s are taken as representatives in G/S β then there is a ge β fixed by H if and only if g is such that g − Hg ⊂ S β , so we have a bijectivecorrespondence ( G · e β ) H = mor( G/H, G/S β ) . Therefore, we obtain C n ( X H ) ∼ = (cid:77) β Z [mor( G/H, G/S β )] = (cid:77) β P S β ( G/H ) , so, as Bredon modules, C n ( X ) ∼ = (cid:77) β P S β . We have an isomorphism of chain complexesmor G ( C ∗ ( X ) , M ) ∼ = (cid:89) β ∗ mor( P S β ∗ , M ) ∼ = (cid:89) β ∗ M ( G/S β ∗ ) , where { β ∗ } indexes the G -representatives of ∗ -cells. This becomes a direct sum assuming there arefinite representatives for the cells.2.2. Coefficients in the representation ring.
Consider now the family F of finite subgroupsof G . For computations of equivariant K-theory and K-homology we will use the contravariantBredon module R which acts on objects of Or F ( G ) by sending G/H to R ( H ), the representationring of the subgroup H . At the level of morphisms, R acts via the composition of restriction andthe isomorphism given by conjugation, so for any f g : G/H → G/L the morphism R ( f g ) is thecomposition R ( L ) Res Lg − Hg −−−−−−−−→ R ( g − Hg ) ∼ = −−→ R ( H ) . Then, as above, we have an isomorphismmor G ( C n ( X ) , R ) ∼ = (cid:77) α R ( S α ) , ECKE OPERATORS ON BIANCHI GROUPS 5 with the assumption that there are finite orbit representatives of n -cells. Here, the coboundarymap is given by restriction of representations, from the stabilizer of an n -cell to the stabilizer ofthe corresponding ( n + 1)-cell that contains it.Similarly, we can consider R as a covariant Bredon module, setting R ( f g ) to be the composition R ( H ) ∼ = −−→ R ( g − Hg ) Ind Lg − Hg −−−−−−−→ R ( L ) . Then the chain complex C n ( X ) ⊗ F R can be described as C n ( X ) ⊗ F R ∼ = (cid:77) α R ( S α ) , where the boundary map is given by induction of representations.3. Equivariant K-homology and the Baum-Connes conjecture
Spectra and homology theories.Definition 3.1.
A spectrum E = { ( E ( n ) , σ ( n )) | n ∈ Z } is a sequence of pointed spaces { E ( n ) | n ∈ Z } together with pointed maps, called structure maps, σ ( n ) : E ( n ) ∧ S −→ E ( n + 1) . A map of spectra f : E → E (cid:48) is a sequence of maps f ( n ) : E ( n ) → E (cid:48) ( n ) which are compatiblewith the structure maps σ ( n ). The homotopy groups of a spectrum are defined by π n ( E ) := colim k →∞ π n + k ( E ( k )) . Where the n -th structure map of the system π n + k ( E ( k )) is given by the composite π n + k ( E ( k )) S −→ π n + k +1 ( E ( k ) ∧ S ) σ ( k ) ∗ −−−→ π n + k +1 ( E ( k + 1)) , where S denotes the suspension homomorphism.We will denote by SPECTRA the category of spectra.If E is a spectrum, one obtains a (non-equivariant) homology theory H ∗ ( − ; E ) by defining for aCW-pair ( X, A ) and n ∈ Z H n ( X, A ; E ) = π n ( X + ∪ A + cone( A + ) ∧ E ) , where X + is obtained from X by adding a disjoint base point and cone denotes the (reduced)mapping cone. The main property of this homology theory is given by the equality H n ( {•} ; E ) = π n ( E ). The definitions can be extended to an equivariant context using the orbit category. DAVID MU˜NOZ, JORGE PLAZAS, AND MARIO VEL´ASQUEZ Or ( G ) -spaces. Denote by
SPACES the category of topological spaces. Also, recall that if F is the family of all subgroups of G we denote Or F by Or ( G ). In order to ease notation we will attimes use lower case letters to denote objects in Or ( G ). A covariant Or ( G )-space is a covariantfunctor X : Or ( G ) −→ SPACES . Given a G -space X , the fixed point set system of X , denoted by Φ X , is the Or ( G )-space definedby Φ X ( G/H ) := Maps(
G/H, X ) G = X H and if θ : G/H → G/K is a G -map corresponding to gK ∈ ( G/K ) H then when x ∈ X K Φ X ( θ )( x ) := gx ∈ X H . Φ defines a contravariant functor from the category of proper G -spaces to the category of Or ( G )-spaces.Let X and Y be Or ( G )-spaces, we define the space X × Or ( G ) Y := (cid:71) c ∈ Obj( Or ( G )) X ( c ) × Y ( c ) / ∼ (3.2)where ∼ is the equivalence relation generated by ( X ( φ )( x ) , y ) ∼ ( x, Y ( φ )( y )) for all morphisms φ : c → d in Or ( G ) and points x ∈ X ( d ) and y ∈ Y ( c ). Definition 3.3.
A covariant Or ( G )-spectrum is a covariant functor E G : Or ( G ) −→ SPECTRA If E G is a covariant Or ( G )-spectrum and Y is a Or ( G )-space, then one obtains a spectrum Y × Or ( G ) E G . Hence, we can extend E G to a covariant functor E G % : G -CW → SPECTRA ( X, A ) (cid:55)→ Φ( X + ∪ A + cone( A + )) × Or ( G ) E G . Lemma 3.4. If E G is a covariant Or ( G ) -spectrum, then we obtain a G -homology theory H G ∗ ( − ; E G ) defined as H Gn ( X, A ; E G ) = π n ( E G % ( X, A )) satisfying H Gn ( G/H ; E G ) = π n ( E G ( G/H )) for n ∈ Z and H ⊆ G .Proof. See Lemmas 2.3 and 2.5 in [L¨uc18]. (cid:3)
Recall that the reduced group C ∗ -algebra C ∗ r ( G ) is the norm closure of the complex group ring C G embedded into the space B ( L ( G )) of bounded operators L ( G ) → L ( G ) equipped with the supnorm given by the right regular representation. Denote by GROUPOIDS the category of groupoids.There is a covariant functor, respecting equivalences, K top : GROUPOIDS −→ SPECTRA , such that for every group G and all n ∈ Z we have π n ( K top ( G )) ∼ = K n ( C ∗ r ( G )) , where K n ( C ∗ r ( G )) is the topological K-theory of the reduced group C ∗ -algebra C ∗ r ( G ), see [Bla06]. ECKE OPERATORS ON BIANCHI GROUPS 7
Conjecture 3.5 (Baum-Connes Conjecture) . A group G satisfies the Baum-Connes Conjecture ifthe assembly map induced by the projection pr : E G → G/GH Gn ( pr ; K top ) : H Gn ( E G ; K top ) −→ H Gn ( G/G ; K top ) = K n ( C ∗ r ( G )) , is bijective for all n ∈ Z . The homology theory H G ∗ ( − ; K top ) is called G -equivariant K-homology and is also denoted by K Gn ( − ). The map H Gn ( pr ; K top ) is called the assembly map and is also denoted by µ G,n .3.3.
K-theory.
The equivariant cohomology theory associated to the Or ( G )-spectra K top is called equivariant K-theory and is denoted by K nG ( − ). By results of L¨uck and Oliver in [LO01] we knowthat in the case of proper G -actions on G -CW-complexes, K ∗ G ( − ) can be defined in terms of vectorbundles. We summarize the construction in the following lines. Definition 3.6.
For any discrete group G and any finite proper G -CW-complex X , let K G ( X ) = K G ( X ) be the Grothendieck group of the semigroup Vect G ( X ) of isomorphism classes of G -vectorbundles over X together with direct sum. Define K − nG ( X ), for all n >
0, by setting K − nG ( X ) = ker( K G ( X × S n ) incl ∗ −−−−−→ K G ( X ) ) . For any proper G -CW-pair ( X, A ), and n ≥
0, set K − nG ( X, A ) = ker( K − nG ( X ∪ A X ) i ∗ −−−→ K − nG ( X ) ) . And, let K nG ( X ) = K − nG ( X ) and K nG ( X, A ) = K − nG ( X, A ).We have then the following equivalence of equivariant homology theories.
Theorem 3.7.
Let G be a discrete group, ( X, A ) be a proper G -CW-pair and n be an integernumber. There is a natural isomorphism K nG ( X, A ) ∼ = K nG ( X, A ) . Proof.
See [LO01]. (cid:3)
As in the classical case, there is an Atiyah-Hirzebruch type spectral sequence converging to equi-variant K-theory (respectively K -homology) whose E -term is the Bredon cohomology (respectivelyBredon homology) with coefficients in the Bredon module of complex representations R (see [LO01]for details).More concretely, given a discrete group G and any finite dimensional proper G -complex X , theequivariant skeletal filtration of X induces a cohomological spectral sequence with E p, q ( X ) ∼ = H pG ( X ; R ) = ⇒ K ∗ G ( X ) , and a homological spectral sequence with E p, q ( X ) ∼ = H Gp ( X ; R ) = ⇒ K G ∗ ( X ) . Notice that, if dim( X ) = 2 (which is the case when X = E G for a Bianchi group G , cf. below),we have that Bredon cohomology groups (respectively homology groups) are trivial for p >
2, soboth spectral sequences collapse in E . In fact, both spectral sequences induce split short exactsequences (natural in X )(3.8) 0 −→ H G ( X ; R ) −→ K G ( X ) −→ H G ( X ; R ) −→ . DAVID MU˜NOZ, JORGE PLAZAS, AND MARIO VEL´ASQUEZ and(3.9) 0 −→ H G ( X ; R ) −→ K G ( X ) −→ H G ( X ; R ) −→ X , K G ( X ) ∼ = H G ( X ; R ) and K G ( X ) ∼ = H G ( X ; R ). Both the exactsequences and the isomorphisms above are compatible with the induction structure.4. Hecke algebras and Hecke operators
In the case where G is an arithmetic group, given as a discrete subgroup G ⊂ G of a Lie group G , homogeneous G -spaces lead to G -spaces of arithmetic relevance .As many of the arithmetic properties of the group G can be encoded in terms of its Hecke algebra,it will be important for us to study the action of the Hecke algebra on the Bredon cohomology andhomology of these G -spaces. In the following paragraphs we discuss some generalities on Heckealgebras.4.1. Double cosets and Hecke algebras.
Let G be a group. Two subgroups of G are saidto be commensurable if their intersection has finite index in both. Commensurability defines anequivalence relation in the set of subgroups of G . If G and G are subgroups of G that arecommensurable, we use the notation G ∼ G . Let G be a subgroup of G . We define the commensurator of G in G as the subgroupComm G ( G ) = { g ∈ G | G ∼ gGg − } . Note that if G and G are commensurable subgroups of G thenComm G ( G ) = Comm G ( G ) . Example 4.1.
Let G = PGL +2 ( R )and let G be the modular group PSL ( Z ), then the commensurator of G in G isComm G ( G ) = Comm PGL +2 ( R ) (PSL ( Z )) = PGL +2 ( Q ) . Example 4.2.
Let G = PGL ( C )and let Q ( √− D ) ⊂ C be a quadratic imaginary extension of Q . Denote by O Q ( √− D ) the ring ofintegers of Q ( √− D ). If G = Γ D is the corresponding Bianchi groupΓ D = PSL ( O Q ( √− D ) )then the commensurator of G in G isComm G ( G ) = Comm PGL ( C ) (Γ D ) = PGL (cid:16) Q ( √− D ) (cid:17) . Here the Lie group G is the group of real or complex points of an algebraic group defined over Q . ECKE OPERATORS ON BIANCHI GROUPS 9
As above let G be a group and let G be a subgroup of G . Given an element g in Comm G ( G ) weconsider the double coset in G given by GgG.
The left action of G on GgG has a finite number of orbits. To compute this number. let G ( g ) = g − Gg (cid:84) G and notice that the map G −→ GgGγ (cid:55)−→ gγ induces a surjection from G to the quotient G \ GgG with kernel G ( g ) so we have GgG = d (cid:71) i =1 Gα i , where d = [ G : G ( g ) ] . We obtain an analogous decomposition by considering the right action of G on GgG .The above decompositions lead to a natural product of double cosets as follows. If α and β areelements in Comm G ( G ) with GαG = d (cid:71) i =1 Gα i and GβG = e (cid:71) j =1 β j G then we define ( GαG ) · ( GβG ) = (cid:88) c δα,β GδG where c δα,β = number of pairs of indices ( i, j ) such that Gα i β j = Gδ and the expression on the right is viewed as an element of the free abelian group generated bydouble cosets of the form GδG with δ ∈ Comm G ( G ).If ∆ is a subsemigroup of Comm G ( G ) with G ⊆ ∆we denote by A ( G ; ∆)the free abelian group generated by double cosets of the form GgG with g ∈ ∆.The above product of double cosets can be extended linearly to a bilinear map of Z -modules A ( G ; ∆) × A ( G ; ∆) −→ A ( G ; ∆)which is associative. The group A ( G ; ∆) thus becomes a ring which will be called the Hecke algebraof G with respect to ∆. For the above results together with generalities on Hecke algebras seechapter 3 of [Shi94]. In the case ∆ = Comm G ( G ) we will denote A ( G ; ∆) simply by A ( G ). Action of Hecke operators on group cohomology.
Let G be a group and let G be asubgroup of G . As above, let ∆ be a subsemigroup of Comm G ( G ) with G ⊆ ∆. If M is an abeliangroup on which ∆ acts by endomorphisms we can consider M as a G -module. Elements of theHecke algebra A ( G ; ∆) define endomorphisms of the cohomology groups of G with coefficients in M leading to an action of A ( G ; ∆) on H n ( G ; M ). These operators will be called Hecke operatorsassociated to ( G ; ∆). In order to define the Hecke operator corresponding to a double coset GgG with g ∈ ∆ notice that if we have a decomposition GgG = d (cid:71) i =1 Gα i , α i ∈ ∆ , and m ∈ M G is an element of M fixed by G then the element of M given by m | GgG = d (cid:88) i =1 α i m is again fixed by G and independent of the representatives α i so the coset GgG defines a map T g : M G −→ M G . These maps can be linearly extended to define operators associated to elements of A ( G ; ∆): m | ξ = r (cid:88) k =1 c k T g k ( m )for (cid:80) rk =1 c k ( Gg k G ) ∈ A ( G ; ∆) and m ∈ M G . These operators define an action of A ( G ; ∆) on M G .By naturality the action of A ( G ; ∆) on M G extends to an action on the cohomology groups of G with coefficients in M . At the level of group cocycles in the standard complex the action can bedescribed as follows: If g ∈ ∆ GgG = d (cid:71) i =1 Gα i , and for γ ∈ G we denote by σ γg the unique permutation in S d such that G α i γ = G α σ gγ ( i ) . In this way we obtain maps ρ gj : G −→ G for j = 1 , . . . , d where for γ ∈ G the element ρ gj ( γ ) ∈ G is determined by α j γ = ρ gj ( γ ) α σ gγ ( j ) . Now, given a group r -cocycle φ : G × · · · × G −→ M we can take ( T g φ ) ( γ , γ , . . . , γ r ) = d (cid:88) j =1 α − j φ ( ρ gj ( γ ) , ρ gj ( γ ) , . . . , ρ gj ( γ r ))which is again a cocycle, so T g defines a morphism T g : H r ( G ; M ) −→ H r ( G ; M ) ECKE OPERATORS ON BIANCHI GROUPS 11 which can be extended to an action of A ( G ; ∆) on H r ( G ; M ) for r ≥
0, which for r = 0, where wehave H ( G ; M ) = M G , coincides with the action defined above.Further information on this action together with its functorial properties and its relation to theclassical theory of Hecke operators can be found in [KPS81].4.3. Hecke correspondences.
In order to extend the above definition of Hecke operators to theBredon cohomology groups of an arithmetic discrete group G it will be important to understandthe natural action of elements of A ( G ; ∆) on G -spaces and their quotients. The natural frameworkfor this comes from considering elements of the Hecke ring as correspondences defined in terms ofquotients corresponding to double cosets.As in the previous sections let G be a subgroup of a group G . Suppose now that the group G acts on a topological space S and consider the action of the subgroup G on S . We will be interestedin the case where G is a Lie group and S is a homogeneous G -space, also we assume in the followingthat the action of the discrete group G on S satisfies sufficient conditions for the quotient S/G tobe well behaved.Given an element g ∈ Comm G ( G ) consider the groups K = g − Gg ∩ G and g K = G ∩ gGg − . Observe that we have group morphisms K g KG G where the horizontal arrow is a group isomorphism and the vertical ones are inclusions with finiteindex.These morphisms induce maps between the corresponding quotients of S , S/K S/ g KS/G S/G where the horizontal arrow is a homeomorphism and the vertical ones are finite index covers. Thisdiagram determines a correspondence C GgG ⊂ S/G × S/G homeomorphic to S /K . We call this correspondence the Hecke correspondence from
S/G to S/G associated to
GgG . We extend this definition using linearity in order to associate correspondencesto elements of A ( G ; ∆). These can in turn be used to define a Hecke action at the level of sheaveson these spaces and their cohomology by defining operators T g via successive pullbacks and push-forwards along the horizontal maps above. Example 4.3.
The classical example leading to Hecke operators acting on modular forms corre-sponds to G = PGL +2 ( R ) viewed as a group of transformations of the hyperbolic plane S = H . Inthis case if G is a subgroup of G commensurable with PSL ( Z ) the above quotients are modularcurves and the Hecke correspondences coming from them define Hecke operators between spaces ofmodular forms. Example 4.4.
As mentioned at the beginning of this section the main example for us arises from G = PGL ( C )which we will view in what follows as a group of transformations of the hyperbolic 3-space S = H .Hecke correspondences for quotients of H by a Bianchi groupΓ D = PSL ( O Q ( √− D ) )and its subgroups will be used in the following sections in order to define Hecke operators on theBredon cohomology of Bianchi groups.For more on this point of view together with its relation to the theory of automorphic forms see[Har87] and [Har91].5. Hecke operators on Bredon (co)homology and K -theory Restriction and corestriction.
Let us start defining the restriction and corestriction oper-ators in equivariant K-(co)homology and Bredon (co)homology.Let X be a G -CW-complex and let H ⊆ G be a subgroup of finite index, first note that we havenatural isomorphisms given by the induction structure K H ∗ ( X ) ind GH −−−→ K G ∗ ( G/H × X ) ,K ∗ H ( X ) ind GH −−−→ K ∗ G ( G/H × X ) , H H ∗ ( X ; R ) ind GH −−−→ H G ∗ ( G/H × X ; R ) , H ∗ H ( X ; R ) ind GH −−−→ H ∗ G ( G/H × X ; R ) . Define the corestriction morphisms as the compositions K H ∗ ( X ) ind GH −−−→ K G ∗ ( G/H × X ) π ∗ −−→ K G ∗ ( X ) , H H ∗ ( X ; R ) ind GH −−−→ H G ∗ ( G/H × X ; R ) π ∗ −−→ H G ∗ ( X ; R ) ,K ∗ H ( X ) ind GH −−−→ K ∗ G ( G/H × X ) π −−→ K ∗ G ( X ) , H ∗ H ( X ; R ) ind GH −−−→ H ∗ G ( G/H × X ; R ) π −−→ H ∗ G ( X ; R ) , where π : G/H × X → X is the second projection, and π denotes the shriek map that in thiscase can be defined as the composition K ∗ G ( G/H × X ) p ∗ −→ K ∗ G ( C [ G/H ] × X ) T h −−→ K ∗ G ( X ) , where C [ G/H ] denotes the complex vector space generated by
G/H with the canonical G -action, p : C [ G/H ] × X → G/H × X is the natural projection and T h is the Thom isomorphism as in Thm.3.14 in [LO01]. We denote the above compositions by cores GH (or simply cores if H and G are clearfrom the context). This construction can be performed also in the case of Bredon cohomology. Remark 5.1. If G is finite and X = {•} , corestriction corresponds to the usual induction morphismon the representation ring. ECKE OPERATORS ON BIANCHI GROUPS 13
We have also restriction morphisms denoted by res GH (or simply by res if H and G are clear fromthe context) K G ∗ ( X ) res GH −−−→ K H ∗ ( X ) , H G ∗ ( X ; R ) res GH −−−→ H H ∗ ( X ; R ) ,K ∗ G ( X ) res GH −−−→ K ∗ H ( X ) , H ∗ G ( X ; R ) res GH −−−→ H ∗ H ( X ; R )defined in a similar way as corestriction.5.2. Conjugation and Hecke operators.
Now suppose that the discrete group G is given as asubgroup of a Lie group G . Given an element g ∈ Comm G ( G ) there are conjugation morphisms K H ∗ ( X ) Ad g −−→ K gHg − ∗ ( X ) , H H ∗ ( X ; R ) Ad g −−→ H gHg − ∗ ( X ; R ) ,K ∗ H ( X ) Ad g −−→ K ∗ gHg − ( X ) , H ∗ H ( X ; R ) Ad g −−→ H ∗ gHg − ( X ; R ) , induced from the conjugation morphism from Or ( H ) to Or ( gHg − ). Definition 5.2.
Let G be a discrete subgroup of a Lie group G and let X be a G -CW-complex.Given an element g ∈ Comm G ( G ) and a finite index subgroup H ⊆ G we define the Hecke operatorassociated to ( G, H, g, X ) as the composition K Gn ( X ) res (cid:47) (cid:47) K Hn ( X ) Ad g (cid:47) (cid:47) K gHg − n ( X ) cores (cid:47) (cid:47) K Gn ( X ) . We will denote this operator by T g,X when G and H are clear from the context.Similarly, for Bredon homology, let X be a proper G -CW-complex, we denote by T g,X to theHecke operator associated to ( G, H, g, X ) defined as the composition H Gn ( X ; R ) res (cid:47) (cid:47) H Hn ( X ; R ) Ad g (cid:47) (cid:47) H gHg − n ( X ; R ) cores (cid:47) (cid:47) H Gn ( X ; R ) . Because all of these morphisms are defined by maps of spectra, they are natural on X , then wehave the following commutative diagram. K Gn (E G ) µ G,n (cid:15) (cid:15) res (cid:47) (cid:47) K Hn (E G ) µ H,n (cid:15) (cid:15) Ad g (cid:47) (cid:47) K gHg − n (E G ) µ gHg − ,n (cid:15) (cid:15) cores (cid:47) (cid:47) K Gn (E G ) µ G,n (cid:15) (cid:15) K n ( C ∗ r ( G )) res (cid:47) (cid:47) K n ( C ∗ r ( H )) Ad g (cid:47) (cid:47) K n ( C ∗ r ( gHg − )) cores (cid:47) (cid:47) K n ( C ∗ r ( G ))Thus, in the case that the group G satisfies the Baum-Connes conjecture, we can compute theHecke operators defined on the K -theory of the reduced C ∗ -algebra of G using the Hecke operatorsdefined on the left hand side of the Baum-Connes conjecture, the main point being that these aremore suitable for computations. Summarizing we have the following. Theorem 5.3.
There is a commutative diagram K Gn ( E G ) µ G,n (cid:15) (cid:15) T g,EG (cid:47) (cid:47) K Gn ( E G ) µ G,n (cid:15) (cid:15) K n ( C ∗ r ( G )) T g, {•} (cid:47) (cid:47) K n ( C ∗ r ( G )) . We also have a relation between T g,X and T g,X . Theorem 5.4.
Let G be a discrete group such that E G is a G -CW-complex of dimension at most2. We have commutative diagrams (cid:47) (cid:47) H G ( E G ; R ) (cid:47) (cid:47) T g,EG (cid:15) (cid:15) K G ( E G ) (cid:47) (cid:47) T g,EG (cid:15) (cid:15) H G ( E G ; R ) (cid:47) (cid:47) T g,EG (cid:15) (cid:15) (cid:47) (cid:47) H G ( E G ; R ) (cid:47) (cid:47) K G ( E G ) (cid:47) (cid:47) H G ( E G ; R ) (cid:47) (cid:47) and K G ( E G ) ∼ = (cid:15) (cid:15) T g,EG (cid:47) (cid:47) K G ( E G ) ∼ = (cid:15) (cid:15) H G ( E G ; R ) T g,EG (cid:47) (cid:47) H G ( E G ; R ) . Proof.
Since E G and G/H × E G are proper G -CW-complexes of dimension at most 2, we havenatural short exact sequences as in 3.9 and the natural identification K G ( EG ) ∼ = H G ( EG ; R ). (cid:3) We conclude this section explaining how to compute T g,X directly from the Bredon chain complex.First note that if C n ( X ) ⊗ F R ∼ = (cid:77) α R ( S α ) , then C n ( G/H × X ) ⊗ F R ∼ = (cid:77) α (cid:77) χ ∈ G/H R ( χHχ − ∩ S α ) . The morphism res : H Gn ( X ; R ) −→ H Gn ( G/H × X ; R ) ∼ = H Hn ( X ; R )is induced by the restriction morphism on the representation ring (cid:77) α R ( S α ) −→ (cid:77) α (cid:77) χ ∈ G/H R ( χHχ − ∩ S α ) . The morphism Ad g : H Hn ( X ; R ) −→ H gHg − n ( X ; R )is induced by the isomorphism (cid:77) α (cid:77) χ ∈ G/H R ( χHχ − ∩ S α ) −→ (cid:77) α (cid:77) χ ∈ G/gHg − R ( χgHg − χ − ∩ S α ) . ECKE OPERATORS ON BIANCHI GROUPS 15
Finally, the morphismcores : H gHg − n ( X ; R ) ∼ = H Gn ( G/gHg − × X ; R ) −→ H Gn ( X ; R )is induced by the induction morphism on the representation ring (cid:77) α (cid:77) χ ∈ G/gHg − R ( χgHg − χ − ∩ S α ) −→ (cid:77) α R ( S α ) . For Bredon cohomology the Hecke operator can be described in a similar way.As we will see in the next section, these three morphisms can also be computed directly in theBredon homology of X as an H -space.6. Computations for Bianchi groups
In this section we specialize to the case where G is a Bianchi group. We use Thm. 5.3 and Thm.5.4 in order to explicitly compute the action of the operators T g, {•} for the group Γ = PSL ( Z [ i ])for elements g ∈ PGL ( O Q ( i ) ) associated to primes in Z [ i ]. As will be seen below, these techniquesapply for other Bianchi groups as well. Definition 6.1.
Let D be a positive square-free integer, and let O D = O Q ( √− D ) be the ring ofintegers of the imaginary quadratic extension Q ( √− D ). The Bianchi group associated to D isdefined as Γ D = PSL ( O D ) = SL ( O Q ( √− D ) ) / {± I } . We can describe the rings O D explicitly in terms of the discriminant of the quadratic field Q ( √− D ). Let δ = √− D and η = (1 + δ ), we have O D = Z [ δ ] for D ≡ , , and O D = Z [ η ] for D ≡ . For a proof see [Art11, Chapter 13].For D = 1 , , , ,
11 the ring O D is an Euclidean domain. For these values of D we will referto the corresponding Bianchi groups Γ D as Euclidean Bianchi groups .In general, except from D = 3, Bianchi groups can be described as amalgamated products. Theamalgam decompositions for the Euclidean Bianchi groups are described below. Proposition 6.2.
We have Γ ∼ = (cid:0) A ∗ C S (cid:1) ∗ PSL ( Z ) (cid:0) S ∗ C D (cid:1) ;Γ ∼ = G ∗ ( Z ∗ C ) G , where G is the HNN extension of C × C associating two generators and G is the HNN extensionof A associating two -cycles; Γ ∼ = (cid:0) Z ∗ C (cid:1) ∗ ( Z ∗ C ∗ C ) G , where G is the HNN extension of S ∗ C S associating a -cycle with itself; and Γ ∼ = (cid:0) Z ∗ C (cid:1) ∗ ( Z ∗ C ∗ C ) G, where G is the HNN extension of A ∗ C A associating a -cycle with itself. For a proof of the above facts the reader may consult [Fin89].We will now focus on Γ and the explicit computation of Hecke operators associated to it. The group Γ = PSL ( Z [ i ]) . From the above proposition we have an isomorphismΓ ∼ = (cid:0) A ∗ C S (cid:1) ∗ PSL ( Z ) (cid:0) S (cid:48) ∗ C D (cid:1) , (where for a group G the notation G (cid:48) means just an isomorphic copy of G ) with PSL ( Z ) = C (cid:48) ∗ C (cid:48) and the intersections A ∩ S (cid:48) = C (cid:48) , A ∩ D = { } , S ∩ S (cid:48) = { } , and S ∩ D = C (cid:48) . In fact, wecan obtain the presentationΓ = (cid:104) a , b , c , d | a = b = c = d = ( ac ) = ( ad ) = ( bd ) = ( bc ) = 1 (cid:105) , with A = (cid:104) a , c (cid:105) , S = (cid:104) a , d (cid:105) , D = (cid:104) b , d (cid:105) , and S (cid:48) = (cid:104) b , c (cid:105) , so that C = (cid:104) a (cid:105) , C = (cid:104) b (cid:105) , C (cid:48) = (cid:104) c (cid:105) ,and C (cid:48) = (cid:104) d (cid:105) , and explicit matrices that represent the generators, namely a = (cid:18) ii (cid:19) , b = (cid:18) ii (cid:19) , c = (cid:18) − (cid:19) , and d = (cid:18) −
11 0 (cid:19) . Classifying space for proper actions.
Using the above presentation and explicit generators,we can construct a 2-dimensional Γ -CW-complex X , which is a model for E Γ . Let X (0) = Γ /A × { p } (cid:71) Γ /S × { q } (cid:71) Γ /D × { r } (cid:71) Γ /S (cid:48) × { s } , where each D (point) has been labeled with a letter. The 1-skeleton is obtained from the pushoutΓ /C × S (cid:70) Γ /C (cid:48) × S (cid:70) Γ /C × S (cid:70) Γ /C (cid:48) × S X (0) Γ /C × D (cid:70) Γ /C (cid:48) × D (cid:70) Γ /C × D (cid:70) Γ /C (cid:48) × D X (1) ϕ inclusion so that X (1) is the union of X (0) and many copies of D , identifying the image by ϕ and theinclusion, respectively, of many copies of S . Writing each copy of S as two ordered points {− , } and denoting a point in X (0) just as the coset, the map ϕ is defined as follows. For any γ ∈ Γ , ϕ : γC × {− , } (cid:55)→ { γA , γS } ,γC (cid:48) × {− , } (cid:55)→ { γS , γD } ,γC × {− , } (cid:55)→ { γD , γS (cid:48) } ,γC (cid:48) × {− , } (cid:55)→ { γS (cid:48) , γA } . This means that we will add a segment between two points whenever their corresponding cosetsintersect as a coset of any of the cyclic groups in Γ . Take P , Q , R , S as the trivial cosets of A , S , D , S (cid:48) , respectively. The space X (1) would begin to look like this: ECKE OPERATORS ON BIANCHI GROUPS 17 • P • Q • S • R • Q • Q • Q • P (cid:48) • S (cid:48) • S (cid:48) • S (cid:48) • P • R • R • Q (cid:48) • R (cid:48) • R (cid:48) • S The lines
P Q i , i = 1 , ,
3, come from the cosets c C , c C , and ac C , respectively. There areno more cosets of S connected to A . It continues similarly.Finally, we add a 2-cell, filling the square:Γ / { } × S X (1) Γ / { } × D X (2) = X .
This space is proper since all the isotropy groups are finite groups, and this is because X can bethought as the space obtained from a square by the action of Γ with the isotropy groups showedbelow. • A • S • S (cid:48) • D C C (cid:48) C (cid:48) C Also, X is indeed a model for E Γ , since every fixed space X H , H finite subgroup of Γ , isweakly contractible.An alternative description of this space can be found in [Fl¨o83] and [Rah10].6.3. Bredon cohomology for Γ . In order to compute the Hecke operators T g, {•} we start com-puting the Bredon (co)homology groups of Γ with coefficients in the representation ring. We willlater use these groups together with Thm. 5.4 to obtain computations in equivariant K-homology. The Bredon cochain complex with coefficients in the representation ring for the group Γ andthe space X = E Γ has the form0 −→ (cid:77) α R ( S α ) d −−→ (cid:77) α R ( S α ) d −−→ (cid:77) α R ( S α ) −→ , where the sum runs over representatives of n -cells, the S α are the corresponding stabilizers, andthe differentials are given by restriction of representations. We know that R ( A ) ∼ = Z , R ( S ) ∼ = R ( S (cid:48) ) ∼ = Z , R ( D ) ∼ = Z , and R ( C n ) ∼ = Z n , so the cochain complex becomes0 −→ Z d −−−−−→ Z d −−−−−→ Z −→ . Here, d is represented by the matrix ( 1 1 1 1 1 1 1 1 1 1 ) , of rank 1, and d by the matrix − − − − − − − − − − − − − − − − −
10 1 0 1 0 0 − , of rank 8 (blank spaces stand for blocks of zeros).We obtain H n Γ ( X ; R ) ∼ = Z , n = 0; Z , n = 1;0 , n ≥ . Bredon homology is computed in a similar way. We have H Γ n ( X ; R ) ∼ = Z , n = 0; Z , n = 1;0 , n ≥ . Hecke correspondences for prime level congruence subgroups.
We study now Heckeoperators coming from congruence subgroups associated to primes in Z [ i ]. We begin by reviewingthese first.The group of units of the ring of Gaussian integers Z [ i ] is given by { , − , i, − i } . Any primeideal in Z [ i ] is of the form ( π ) where π is an irreducible element of Z [ i ]. Up to units, these primeelements are the following: • π = 1 + i , • π = p for a prime p in Z with p ≡ • π = a + ib , with a + b = p for a prime p in Z with p ≡ ECKE OPERATORS ON BIANCHI GROUPS 19
Notice that (2) = (1 + i ) and that this is the only prime which ramifies in Z [ i ]. This leads toexceptional behavior of the prime 1 + i . One instance of this exceptional behavior will be seen inthe computations for the classifying spaces for proper actions and the isotropy groups arising in thecorresponding Hecke operators.Fix now a prime p in Z [ i ]. Let g be the class of (cid:18) p
00 1 (cid:19) in PGL ( Q ( i )) = Comm PGL ( C ) (Γ ).As above, cf. 4.3, in order to define the Hecke operator T g associated to the double coset Γ g Γ ,we have to consider the Hecke correspondence determined by the congruence subgroup K = Γ ∩ g − Γ g. We start describing explicitly this subgroup. For this, notice that if γ is the class of a matrix (cid:18) a bc d (cid:19) in PGL ( C ) we have g − γg = (cid:18) /p
00 1 (cid:19) (cid:18) a bc d (cid:19) (cid:18) p
00 1 (cid:19) = (cid:18) a b/ppc d (cid:19) . This means that (the class of) any matrix in K will be of this form. Then K = (cid:26)(cid:18) a bc d (cid:19) ∈ Γ : c ∈ p · Z [ i ] (cid:27) , so K is the congruence subgroup K = (cid:26) γ ∈ Γ : γ ≡ (cid:18) · · · (cid:19) mod p (cid:27) . To compute the index of K in Γ we use the following lemma. Here the notation (cid:101) PSL ( F ) standsfor SL ( F ) / {± I } (usually without the tilde it would be the quotient by all the center of the group). Lemma 6.3.
For a field with q elements, F q , the order of the group (cid:101) PSL ( F q ) is q ( q − if q iseven and q ( q − / if q is odd.Proof. The group SL ( F q ) is the kernel of the surjective homomorphismdet : GL ( F q ) −→ F ∗ q , so | SL ( F q ) | = | GL ( F q ) | / | F ∗ q | = | GL ( F q ) | / ( q − . The order of GL ( F q ) is equal to the number of bases for F q over F q (there is a non-singularmatrix for every pair of linearly independent vectors in F q ), which is equal to the number of non-zero vectors in F q times the number of vectors which are not a multiple of the first one, that is( q − q − q ).Then, | SL ( F q ) | = q ( q − (cid:101) PSL ( F q ) = SL ( F q ) / {± I } , we divide by 2 when q isodd and we do not when q is even, because the characteristic of F q is 2, so I = − I . (cid:3) It is known that the quotient Z [ i ] /p is a field and is isomorphic to F | p | , where | p | is the norm of p in Z [ i ] (the square of its absolute value as a complex number). Consider the surjective homomorphism π : Γ = PSL ( Z [ i ]) −→ (cid:101) PSL ( Z [ i ] /p ) . The kernel of π is the group of matrices that are the identity modulo p ; the index of this subgroupin Γ is equal to the size of (cid:101) PSL ( F | p | ). Since Ker( π ) is contained in the group K , we have(Γ : K ) = (Γ : Ker( π ))( K : Ker( π )) = | (cid:101) PSL ( F | p | ) | ( K : Ker( π )) . Besides, the index ( K : Ker( π )) is equal to the size of the quotient group K /
Ker( π ) ∼ = (cid:26)(cid:18) a b a − (cid:19) ∈ (cid:101) PSL ( F | p | ) (cid:27) , and the size of this group is | p | ( | p | − | p | is even, or | p | ( | p | − /
2, if | p | is odd. (As before, wedo not divide by 2 when | p | is even because F | p | has characteristic 2.)Thus, we obtain that (Γ : K ) = | p | ( | p | − | p | ( | p | −
1) = | p | + 1 . Furthermore, we can give (left and right) coset representatives for Γ modulo K . There are | p | cosets represented by the matrices γ z = (cid:18) z (cid:19) , with z as representatives of Z [ i ] /p ∼ = F | p | , and the last is given by the matrix σ = (cid:18) −
11 0 (cid:19) .6.4.1.
Bredon (co)homology of the congruence subgroup K . Now, we will compute the Bredon(co)homology of E K . First, note that since K is a subgroup of Γ , we can think of X = E Γ as a model for EK . Then, we need K -orbit representatives for n -cells in X . We can start from theright coset partition Γ = (cid:71) Kγ ∈ K \ Γ Kγ.
Note that for any cell e ⊂ X , the Γ -orbit of e splits into the union of some K -orbits,Γ · e = (cid:91) Kγ ∈ K \ Γ Kγ · e, and, after omitting repetitions, the union would be disjoint (apart from the boundaries). To countthese repetitions, it is sufficient to find if there exists any k ∈ K such that γ − k γ (cid:48) ∈ Stab Γ ( e ) for two distinct representatives γ , γ (cid:48) of K \ Γ , in which case we would know that the K -orbits of γe and γ (cid:48) e are the same.In the following paragraphs we focus on the case p = 1 + i , carrying out in full the correspondingcomputations. For other primes, an algorithm in GAP has been developed by the first author (cf.[Mu˜n20]). We have that (Γ : K ) = | i | + 1 = 3 andΓ = Kγ (cid:116) Kγ (cid:116) Kσ = K (cid:18) (cid:19) (cid:116) K (cid:18) (cid:19) (cid:116) K (cid:18) −
11 0 (cid:19) . ECKE OPERATORS ON BIANCHI GROUPS 21
First we search for repeated K -orbits. These are all we need: γ − (cid:18) − i ii − (cid:19) γ = (cid:18) ii (cid:19) = a , σ − (cid:18) i − i (cid:19) γ = (cid:18) ii (cid:19) = a ,γ − (cid:18) −
10 1 (cid:19) γ = (cid:18) −
11 1 (cid:19) = c , σ − (cid:18) (cid:19) γ = (cid:18) − (cid:19) = c ,σ − (cid:18) i − i (cid:19) γ = (cid:18) ii (cid:19) = b , σ − (cid:18) (cid:19) γ = (cid:18) −
11 0 (cid:19) = d ,γ − (cid:18) i − i (cid:19) γ = (cid:18) i − i − i (cid:19) = a c , σ − (cid:18) i i i (cid:19) γ = (cid:18) i − i − i (cid:19) = a c ,γ − (cid:18) i
01 + i − i (cid:19) γ = (cid:18) i − i (cid:19) = ad , σ − (cid:18) − i ii − (cid:19) γ = (cid:18) i − i (cid:19) = a d ,γ − (cid:18) − i i (cid:19) γ = (cid:18) − i i i (cid:19) = bc , σ − (cid:18) − i i i (cid:19) γ = (cid:18) i i − i (cid:19) = bc . For 0-cells we haveΓ · P = K · P, Γ · Q = K · Q, Γ · S = K · S, andΓ · R = K · R (cid:116) Kγ · R = K · R (cid:116) K · (cid:101) R, with (cid:101) R = γ R . For 1-cells we haveΓ · P Q = K · P Q, Γ · QR = K · QR (cid:116) K · Q (cid:101) R, Γ · RS = K · RS (cid:116) K · (cid:101) RS, and Γ · SP = K · SP, with Q (cid:101) R = γ QR and (cid:101) RS = γ RS . Finally, the orbit of the 2-cell is not repeated, so there arethree 2-cells in the quotient X/K . Let E be the 2-cell P QRS .The quotient space
X/K would look like this: • P • Q • S • R • (cid:101) R With two 2-cells E and σE with the sameboundary, and one other 2-cell γ E . The stabilizer of each orbit representative is the intersection between the stabilizer in Γ and thesubgroup K , so Stab K ( P ) = A ∩ K = (cid:104) ac , ca , ac a (cid:105) ∼ = D , Stab K ( Q ) = S ∩ K = (cid:104) a d (cid:105) ∼ = C , Stab K ( R ) = D ∩ K = (cid:104) bd (cid:105) ∼ = C , Stab K ( S ) = S (cid:48) ∩ K = (cid:104) bc (cid:105) ∼ = C , and Stab K ( (cid:101) R ) = Stab Γ ( γ R ) ∩ K = (cid:0) γ Stab Γ ( R ) γ − (cid:1) ∩ K = (cid:0) γ (cid:104) b , d , bd (cid:105) γ − (cid:1) ∩ K = γ (cid:104) b , d , bd (cid:105) γ − ∼ = D . The remaining stabilizers are trivial, except the following two.Stab K ( (cid:101) RS ) = (cid:0) γ Stab Γ ( RS ) γ − (cid:1) ∩ K = (cid:104) γ b γ − (cid:105) ∩ K ∼ = C Stab K ( Q (cid:101) R ) = (cid:0) γ Stab Γ ( QR ) γ − (cid:1) ∩ K = (cid:104) γ d γ − (cid:105) ∩ K ∼ = C The cochain complex becomes0 −→ Z = Z d −−−−−→ Z = Z d −−−−−→ Z −→ . Here, d is represented by the matrix P Q R (cid:101)
R SP Q − − − − QR − − − Q (cid:101) R − RS − − − − (cid:101) RS − − SP − − , of rank 6, and d by the matrix P Q QR Q (cid:101)
R RS (cid:101)
RS SPE σE γ E , of rank 2.We obtain H nK ( X ; R ) ∼ = Z , n = 0;0 , n = 1 , n > Z , n = 2 . As before, the homology is computed in a similar way. We have H Kn ( X ; R ) ∼ = Z , n = 0;0 , n = 1 , n > Z , n = 2 . The Hecke operator T g in Bredon homology via the subgroup K . Restriction.
As we explained above, the morphism at the level of chain complexes is givenby restriction of representations of the isotropy groups of each cell. We obtain the following com-mutative diagram.
ECKE OPERATORS ON BIANCHI GROUPS 23 Z Z Z Z Z Z d Γ d Γ d K d K f f f where f is represented by the matrix P Q R S P Q R (cid:101) R S , of rank 10, and f by P Q QR RS SPP Q QR Q (cid:101) R RS (cid:101) RS SP , of rank 6.Using these matrices, the restriction morphismres : H Γ ( X ; R ) ∼ = Z −→ H K ( X ; R ) ∼ = Z
84 DAVID MU˜NOZ, JORGE PLAZAS, AND MARIO VEL´ASQUEZ can be represented by the matrix(6.4) − −
10 0 0 0 1 00 0 0 0 0 1 . Corestriction.
Let g K = gKg − . As before, the morphism in the chain complexes is givenby induction of representations of the isotropy groups of each cell. We have the morphism at thelevel of Bredon chain complexes as follows:0 Z Z Z Z Z Z d Γ d Γ d g K d g K g g g where the g i are the transposed matrices of the f i .In a similar way, the morphismcores : H g K ( X ; R ) ∼ = Z −→ H Γ ( X ; R ) ∼ = Z can be represented the matrix(6.5) − . Ad g . For the conjugation morphism we have0 Z Z Z Z Z Z d K d K d g K d g K h h h where the h i are the identity (the conjugation is not changing the order of the representations). ECKE OPERATORS ON BIANCHI GROUPS 25
Hecke operators.
Finally, we can compute the Hecke operators on the K-homology of theC*-algebra C ∗ r (Γ ). Theorem 6.6.
We have K n ( C ∗ r (Γ )) ∼ = (cid:40) Z , n = 0; Z , n = 1 . The Hecke operator T g, {•} : K n ( C ∗ r (Γ )) −→ K n ( C ∗ r (Γ )) for n = 0 is given by the matrix − −
10 1 1 1 − and is zero for n = 1 .Proof. First note that as Γ satisfies the Baum-Connes conjecture, the discussion at the end ofSection 3 implies K ( C ∗ r (Γ )) ∼ = H Γ (EΓ ; R ) ∼ = Z and K ( C ∗ r (Γ )) ∼ = H Γ (EΓ ; R ) ∼ = Z . Then, Theorems 5.3 and 5.4 imply that T g, {•} can be represented the same as the Hecke operatoron Bredon homology. For n = 0, it is the product of matrices 6.4 and 6.5; for n = 1 it is trivial,since it factors through H K (E K ; R ) = 0. (cid:3) Concluding remarks
These computations are a first step towards developing a general algorithm for working withHecke operators in Bredon cohomology. The implementation of such algorithm in GAP is part offorthcoming work. As mentioned above, an initial GAP algorithm to compute E Γ /K for subgroups K associated to other primes p in Z [ i ] can be found in [Mu˜n20].Relations of our work with arithmetic aspects of the theory involving automorphic forms are stillto be studied. We expect that using the above set up the study of such relations will lead to fruitfuldevelopments of the theory. References [Art11] M. Artin.
Algebra . Pearson Prentice Hall, 2011.[Bla06] B. Blackadar.
Operator algebras , volume 122 of
Encyclopaedia of Mathematical Sciences . Springer-Verlag,Berlin, 2006. Theory of C ∗ -algebras and von Neumann algebras, Operator Algebras and Non-commutativeGeometry, III.[Bre67a] Glen E. Bredon. Equivariant cohomology theories. Bull. Amer. Math. Soc. , 73:266–268, 1967.[Bre67b] Glen E. Bredon.
Equivariant cohomology theories . Lecture Notes in Mathematics, No. 34. Springer-Verlag,Berlin-New York, 1967.[Fin89] Benjamin Fine.
Algebraic theory of the Bianchi groups , volume 129 of
Monographs and Textbooks in Pureand Applied Mathematics . Marcel Dekker, Inc., New York, 1989.[Fl¨o83] Dieter Fl¨oge. Zur Struktur der PSL ¨uber einigen imagin¨ar-quadratischen Zahlringen. Math. Z. , 183(2):255–279, 1983.[Har87] G. Harder. Eisenstein cohomology of arithmetic groups. The case GL . Invent. Math. , 89(1):37–118, 1987. [Har91] G¨unter Harder. Eisenstein cohomology of arithmetic groups and its applications to number theory. In
Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) , pages 779–790.Math. Soc. Japan, Tokyo, 1991.[Ill75] S¨oren Illman. Equivariant singular homology and cohomology. I.
Mem. Amer. Math. Soc. , 1(issue 2,156):ii+74, 1975.[KPS81] Michio Kuga, Walter Parry, and Chih Han Sah. Group cohomology and Hecke operators. In
Manifoldsand Lie groups (Notre Dame, Ind., 1980) , volume 14 of
Progr. Math. , pages 223–266. Birkh¨auser, Boston,Mass., 1981.[LO01] Wolfgang L¨uck and Bob Oliver. The completion theorem in K -theory for proper actions of a discrete group. Topology , 40(3):585–616, 2001.[L¨uc18] Wolfgang L¨uck. Assembly maps. arXiv e-prints , page arXiv:1805.00226, May 2018.[MS¸20] Bram Mesland and Mehmet Haluk S¸eng¨un. Hecke operators in KK -theory and the K -homology of Bianchigroups. J. Noncommut. Geom. , 14(1):125–189, 2020.[Mu˜n20] David Mu˜noz. Hecke Operators in K-theory of Bianchi Groups. Master’s thesis, Pontificia UniversidadJaveriana, January 2020.[MV03] Guido Mislin and Alain Valette.
Proper group actions and the Baum-Connes conjecture . Advanced Coursesin Mathematics. CRM Barcelona. Birkh¨auser Verlag, Basel, 2003.[Rah10] Alexander Rahm. (Co)homologies and K-theory of Bianchi groups using computational geometric models .Theses, Universit´e Joseph-Fourier - Grenoble I, October 2010.[SG05] Rub´en Jos´e S´anchez-Garc´ıa.
Equivariant K-homology of the classifying space for proper actions . ProQuestLLC, Ann Arbor, MI, 2005. Thesis (Ph.D.)–University of Southampton (United Kingdom).[Shi94] Goro Shimura.
Introduction to the arithmetic theory of automorphic functions , volume 11 of
Publicationsof the Mathematical Society of Japan . Princeton University Press, Princeton, NJ, 1994. Reprint of the 1971original, Kanˆo Memorial Lectures, 1.
Departamento de Matem´aticas, Pontificia Universidad Javeriana, Bogot´a D.C, Colombia
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