We reveal a correspondence between the homological torsion of the Bianchi
groups and new geometric invariants, which are effectively computable thanks to
their action on hyperbolic space. We use it to explicitly compute their
integral group homology and equivariant $K$-homology. By the Baum/Connes
conjecture, which holds for the Bianchi groups, we obtain the $K$-theory of
their reduced $C^*$-algebras in terms of isomorphic images of the computed
$K$-homology. We further find an application to Chen/Ruan orbifold cohomology.
% {\it To cite this article: Alexander D. Rahm, C. R. Acad. Sci. Paris, Ser. I
+++ (2011).

Rahm, Alexander D.

English

arXiv

Note bilingue anglais/francais. We reveal a correspondence between the homological torsion of the Bianchi groups and new geometric invariants, which are effectively computable thanks to their action on hyperbolic space. We use it to explicitly compute their integral group homology and equivariant K-homology. By the Baum/Connes conjecture, which holds for the Bianchi groups, we obtain the K-theory of their reduced C*-algebras in terms of isomorphic images of the computed K-homology. We further find an application to Chen/Ruan orbifold cohomology. ______________________________ Nous mettons en é́vidence une correspondance entre la torsion homologique des groupes de Bianchi et de nouveaux invariants gé́omé́triques, calculables grâce à leur action sur l'espace hyperbolique. Nous l'utilisons pour calculer explicitement leur homologie de groupe à coefficients entiers et leur K-homologie é́quivariante. En consé́quence de la conjecture de Baum/Connes, qui est vé́rifiée pour ces groupes, nous obtenons la K-thé́orie de leurs C*-algèbres ré́duites en termes d'images isomorphes de la K-homologie calculée. Nous trouvons d'ailleurs une application à la cohomologie d'orbi-espace de Chen/Ruan

Rahm, Alexander

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Homology and K-theory of the Bianchi groups

Alexander D. Rahm

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Comptes Rendus Mathematique

We reveal a correspondence between the homological torsion of the Bianchi groups and new geometric invariants, which are effectively computable thanks to their action on hyperbolic space. We use it to explicitly compute their integral group homology and equivariant K-homology. By the Baum/Connes conjecture, which holds for the Bianchi groups, we obtain the K-theory of their reduced C*-algebras in terms of isomorphic images of the computed K-homology. We further find an application to Chen/Ruan orbifold cohomology.peer-reviewe

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RAHM, Alexander

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C. R. Acad. Sci. Paris, Se´rie I, 2011The´orie des groupes/Group TheoryAlge`bre homologique/Homological AlgebraHomology and K-theory of the Bianchi groupsAlexander D. RahmDepartment of Mathematics, Weizmann Institute of Science, Rehovot, Israel(Transmis le 4 avril 2011)Abstract. We reveal a correspondence between the homological torsion of the Bianchi groupsand new geometric invariants, which are eﬀectively computable thanks to theiraction on hyperbolic space. We use it to explicitly compute their integral grouphomology and equivariant K-homology. By the Baum/Connes conjecture, whichholds for the Bianchi groups, we obtain the K-theory of their reduced C∗-algebrasin terms of isomorphic images of the computed K-homology.We further ﬁnd an application to Chen/Ruan orbifold cohomology.Homologie et K-the´orie des groupes de BianchiRe´sume´. Nous mettons en e´vidence une correspondance entre la torsion homologique desgroupes de Bianchi et de nouveaux invariants ge´ome´triques, calculables graˆce a`leur action sur l’espace hyperbolique. Nous l’utilisons pour calculer explicitementleur homologie de groupe a` coeﬃcients entiers et leur K-homologie e´quivariante.En conse´quence de la conjecture de Baum/Connes, qui est ve´riﬁe´e pour cesgroupes, nous obtenons la K-the´orie de leurs C*-alge`bres re´duites en termesd’images isomorphes de la K-homologie calcule´e. Nous trouvons d’ailleurs uneapplication a` la cohomologie d’orbi-espace de Chen/Ruan.1. Version franc¸aise abre´ge´eNous e´tudions la ge´ome´trie d’une certaine classe de groupes arithme´tiques (les groupesde Bianchi), a` travers une action propre sur un espace contractile. Nous acce´dons a` leurhomologie de groupe et leur K-homologie e´quivariante. En plus de de´tail, conside´rons uncorps de nombres quadratique imaginaire Q(√−m), ou` m est un entier positif ne conte-nant pas de carre´. Soit O−m son anneau d’entiers. Les groupes de Bianchi sont les groupesSL2(O−m). Les groupes de Bianchi peuvent eˆtre conside´re´s cruciaux pour l’e´tude d’uneclasse plus large de groupes, les groupes Kleiniens, qui ont de´ja` e´te´ e´tudie´s par HenriPoincare´ [9]. En fait, chaque groupe Kleinien arithme´tique non-cocompact est commensu-rable avec un groupe de Bianchi [8]. Un e´ventail d’informations sur les groupes de BianchiE-mail address: Alexander.Rahm@Weizmann.ac.ilURL: http://www.wisdom.weizmann.ac.il/~rahm/c�2011 Acade´mie des Sciences12 ALEXANDER D. RAHMpeut eˆtre trouve´ dans les monographies [5,6,8]. Ces groupes agissent d’une manie`re natu-relle sur l’espace hyperbolique a` trois dimensions, qui est isomorphe a` l’espace syme´triquequi leur est associe´. Le noyau de cette action est le centre {±1} des groupes, ce qui rendutile l’e´tude du quotient par le centre, PSL2(O−m), que nous appellerons e´galement ungroupe de Bianchi. En 1892, Luigi Bianchi [2] a calcule´ des domaines fondamentaux pourcette action pour quelques uns de ces groupes. Un tel domaine fondamental est de laforme d’un polye`dre hyperbolique (a` quelques sommets manquants pre`s), et nous l’appel-lerons le polye`dre fondamental de Bianchi. Le calcul du polye`dre fondamental de Bianchia e´te´ imple´mente´ sur ordinateur pour tous les groupes de Bianchi [10]. Les images sousSL2(O−m) des faces de ce polye`dre munissent l’espace hyperbolique d’une structure cellu-laire. Pour mieux observer la ge´ome´trie locale, nous passons au complexe cellulaire raﬃne´,que nous obtenons en subdivisant cette structure cellulaire jusqu’a` ce que les stabilisa-teurs dans SL2(O−m) ﬁxent les cellules point par point. Nous allons exploiter ce complexecellulaire de diﬀe´rentes manie`res, aﬁn de cerner des aspects diﬀe´rents de la ge´ome´trie deces groupes.Homologie de groupes. Un invariant essentiel des groupes est leur homologie (de´ﬁniepar exemple dans [3]). Nous pouvons la calculer pour les groupes de Bianchi en nous ser-vant du complexe cellulaire raﬃne´. A` ce ﬁn, nous utilisons la suite spectrale e´quivariantede Leray/Serre qui part de l’homologie des stabilisateurs d’un ensemble repre´sentatif decellules, et qui converge vers l’homologie du groupe de Bianchi. Nous pre´cisons dans laproposition 2.1, l’homologie entie`re de PSL2(O−m) dans les cas m = 19, 43, 67 et 163, quiconstituent tous les cas d’anneaux principaux non-Euclidiens. Les cas d’anneaux Eucli-diens sont de´ja` connus de [13]. Des re´sultats re´cents pour des cas de groupe de classed’ide´aux non-trivial se trouvent dans [12]. Nous remarquons que dans les quatre casd’anneaux principaux non-Euclidiens, la torsion dans l’homologie de PSL2(O−m), est dumeˆme type d’isomorphisme. Pour comprendre ceci, nous conside´rons, pour un nombre pre-mier ℓ, le sous-complexe de l’espace d’orbites, compose´ des cellules ayant des stabilisateursqui contiennent des e´le´ments d’ordre ℓ. Nous l’appellerons le sous-complexe de ℓ–torsion.L’e´nonce´ suivant traite la manie`re dans laquelle son type d’home´omorphisme de´terminela suite spectrale e´quivariante. Il est de´montre´ par la re´duction des sous-complexes detorsion eﬀectue´e dans [11]. Cette technique utilise le lemme 2.4 pour cerner le type de sta-bilisateur d’un sommet v qui est adjacent a` exactement deux areˆtes dont les stabilisateursadmettent de la ℓ–torsion. Ensuite, ces deux areˆtes et v sont remplace´s par une seule areˆte.Le the´ore`me 1.2 et quelques informations homologiques sur les groupes ﬁnis en questionsont utilise´s pour ve´riﬁer que les morphismes induits en homologie produisent les meˆmestermes sur la deuxie`me page de la suite spectrale e´quivariante qu’avant le remplacement.The´ore`me 1.1 (cf. theorem 2.2). La partie ℓ–primaire de la deuxie`me page de la suitespectrale e´quivariante convergeante vers l’homologie des groupes de Bianchi de´pend, hors desa ligne infe´rieure, seulement du type d’home´omorphisme du sous-complexe de ℓ–torsion.Dans tous les cas d’anneaux principaux non-Euclidiens, les sous-complexes de 2–torsion,respectivement de 3–torsion, sont home´omorphes, ce qui explique les re´sultats de la propo-sition 2.1. Derrie`re le the´ore`me ci-dessus, il y a la correspondance suivante entre les sous-groupes cycliques non-triviaux des stabilisateurs des sommets, et les lignes ge´ode´siquesHOMOLOGY AND K-THEORY OF THE BIANCHI GROUPS 3autour desquelles ils eﬀectuent une rotation. Il convient d’appeler ces lignes des axes derotation.The´ore`me 1.2 (cf. theorem 2.3). Soit v un sommet quelconque dans l’espace hyperbolique.L’action de son stabilisateur sur l’ensemble des axes de rotation passant par v, induite parl’action du groupe de Bianchi, est e´quivalente a` l’action par conjugaison de ce stabilisateursur ses sous-groupes cycliques non-triviaux.Une e´tude cas par cas [11] pour tous les six types de sous-groupes ﬁnis dans les groupesBianchi nous permet de de´duire du the´ore`me 1.2, le lemme 2.4 utilise´ pour obtenir lethe´ore`me 1.1. Des exemples pour le the´ore`me 1.1 sont donne´s pour la 3–torsion homolo-gique de trente-six groupes de Bianchi sur le tableau 1.K -the´orie. Avec l’information sur l’action des groupes de Bianchi, que nous obtenonsen nous servant des e´nonce´s et me´thodes de´crits ci-dessus, nous pouvons calculer l’ho-mologie de Bredon de groupes de Bianchi, et en de´duire leur K-homologie e´quivariante.Des re´sultats sont pre´sente´s dans le the´ore`me 2.5. En conse´quence de la conjecture deBaum/Connes, qui est ve´riﬁe´e pour les groupes de Bianchi [7], nous obtenons la K-the´oriedes C∗-alge`bres re´duites des groupes de Bianchi comme images isomorphes.2. IntroductionWe study the geometry of a certain class of arithmetic groups (the Bianchi groups) bymeans of a proper action on a contractible space. This helps to determine their grouphomology and their equivariant K-homology. In more detail, we denote by Q(√−m), withm a square-free positive integer, an imaginary quadratic number ﬁeld, and by O−m itsring of integers. The Bianchi groups are the groups SL2(O−m). The Bianchi groups maybe considered as a key to the study of a larger class of groups, the Kleinian groups, whichdates back to work of Henri Poincare´ [9]. In fact, each non-cocompact arithmetic Kleiniangroup is commensurable with some Bianchi group [8]. A wealth of information on theBianchi groups can be found in the monographs [5, 6, 8]. These groups act in a naturalway on hyperbolic three-space, which is isomorphic to the symmetric space SL2(C)/SU2associated to them. The kernel of this action is the centre {±1} of the groups. Thus it isuseful to study the quotient of a Bianchi group by its centre, namely PSL2(O−m), whichwe also call a Bianchi group. In 1892, Luigi Bianchi [2] computed fundamental domainsfor this action when m = 1, 2, 3, 5, 6, 7, 10, 11, 13, 15 and 19. Such a fundamental domainhas the shape of a hyperbolic polyhedron (up to a missing vertex at certain cusps, whichrepresent the ideal classes of O−m), so we will call it the Bianchi fundamental polyhedron.The computation of the Bianchi fundamental polyhedron has been implemented for allBianchi groups [10]. The images under SL2(O−m) of the facets of this polyhedron equiphyperbolic three-space with a cell structure. In order to view clearly the local geometry,we pass to the reﬁned cell complex, which we obtain by subdiving this cell structure untilthe cell stabilisers ﬁx the cells pointwise. We will see how to exploit this cell complex indiﬀerent ways, in order to see diﬀerent aspects of the geometry of these groups.Group homology. An essential invariant of groups is their homology (deﬁned for in-stance in [3]). We can compute it for the Bianchi groups using the reﬁned cell complexand the equivariant Leray/Serre spectral sequence which starts from the homology of the4 ALEXANDER D. RAHMstabilisers of representatives of the cells, and converges to the group homology of theBianchi groups. We will now state the results for simple integer coeﬃcients in the casesm = 19, 43, 67 and 163, which are the non-Euclidean principal ideal domain cases. Incontrast to these, the Euclidean principal ideal domain cases are already known from [13].For some results in class number 2, see [12]. The virtual cohomological dimension of theBianchi groups is 2. In degrees strictly above 2, we express their homology in terms of thefollowing Poincare´ series at the primes ℓ = 2 and ℓ = 3:P ℓm(t) :=∞�q=3dimFℓ Hq�PSL2�O−m�; Z/ℓZ�tq.These two primes are the only numbers which occur as orders of non-trivial ﬁnite elementsof PSL2(O−m). So it has been shown [11] that the integral homology of these groups is,in all the mentioned degrees, a direct sum of copies of Z/2Z and Z/3Z.Proposition 2.1. The integral homology of PSL2(O−m), for m ∈ {19, 43, 67, 163}, is ofisomorphism type Hq(PSL2(O−m); Z) ∼=�Zβ1−1 ⊕ Z/4Z ⊕ Z/2Z⊕ Z/3Z, q = 2,Zβ1 , q = 1,wherem 19 43 67 163β1 1 2 3 7gives the Betti number β1, and is in all higher degrees a directsum of copies of Z/2Z and Z/3Z, with the number of copies speciﬁed by the Poincare´ seriesP 2m(t) =−t3(t3−2t2+2t−3)(t−1)2(t2+t+1)and P 3m(t) =−t3(t2−t+2)(t−1)(t2+1).We remark that in these four cases, the torsion in the integral homology of PSL2(O−m)is of the same isomorphism type. To understand this, we consider, for a prime ℓ, the sub-complex of the orbit space consisting of the cells with elements of order ℓ in their stabiliser.We call it the ℓ–torsion subcomplex. The following statement on how its homeomorphismtype determines the equivariant spectral sequence is proven by the reduction of the torsionsubcomplex carried out in [11]. This technique uses lemma 2.4 to determine the possibletype of stabiliser of a vertex v with exactly two adjacent edges which have ℓ–torsion intheir stabilisers. Then these two edges, together with v, are replaced by a single edge; andtheorem 2.3 as well as some homological information about the ﬁnite groups in questionare used to check that the induced morphisms on homology produce the same terms onthe second page of the equivariant spectral sequence as before the replacement.Theorem 2.2. The ℓ–primary part of the second page of the equivariant spectral sequenceconverging to the integral homology of PSL2(O−m) depends outside the bottom row onlyon the homeomorphism type of the ℓ–torsion subcomplex.Examples for this theorem are given for the prime ℓ = 3 and thirty-six Bianchi groupsin ﬁgure 1 (for ℓ = 2, see [11]). In all the non-Euclidean principal ideal domain cases, the2–torsion, and respectively 3–torsion subcomplexes are homeomorphic, which explains theresults in proposition 2.1. Underlying theorem 2.2, there is the following correspondencebetween the non-trivial cyclic subgroups of the vertex stabilisers and the geodesic linesaround which they eﬀect a rotation, and which we shall call rotation axes.HOMOLOGY AND K-THEORY OF THE BIANCHI GROUPS 5Figure 1. Results for the 3–torsion homology, expressed in P ℓm(t)m specifying the Bianchi group3-torsion subcomplex,homeomorphism typeP 3m(t)2, 5, 6, 10, 11, 15, 22, 29, 34, 35,46, 51, 58, 87, 95, 115, 123, 155,159, 187, 191, 235, 267�−2t3t−17, 19, 43, 67, 139, 151, 163� �−t3(t2−t+2)(t−1)(t2+1)13, 37, 91, 403, 427� � � �2�−t3(t2−t+2)(t−1)(t2+1)�39� � �−2t3t−1 +−t3(t2−t+2)(t−1)(t2+1)Theorem 2.3 ( [11]). For any vertex v in hyperbolic space, the action of its stabiliser onthe set of rotation axes passing through v, induced by the action of the Bianchi group, isequivalent to the conjugation action of this stabiliser on its non-trivial cyclic subgroups.A case by case study [11] for all the six types of ﬁnite sub-groups in the Bianchi groupsallows us to deduce from theorem 2.3 the following lemma, which is useful in order toobtain theorem 2.2.Lemma 2.4. Let v be a vertex in the reﬁned cell complex. Then the number n of orbits ofedges in the reﬁned cell complex adjacent to v, with stabiliser in PSL2(O−m) isomorphicto Z/ℓZ, is given as follows for ℓ = 2 and ℓ = 3.Isomorphism type of the vertex stabiliser {1} Z/2Z Z/3Z Z/2Z× Z/2Z S3 A4n for ℓ = 2 0 2 0 3 2 1n for ℓ = 3 0 0 2 0 1 2.Here we have written S3 for the symmetric group on three letters and A4 for thealternating group on four letters. Note that we obtain the same table for SL2(O−m) afterreplacing the edge and vertex stabiliser types by their pre-images under the projectionSL2(O−m) → PSL2(O−m), which are respectively: Z/2Z, Z/4Z, Z/6Z, the 8-elementsquaternion group, the 12-elements binary dihedral group and the binary tetrahedral group.K -theory. With the above information about the action of the Bianchi groups, we canfurther compute the Bredon homology of the Bianchi groups, from which we can deducetheir equivariant K-homology. The results of the computations [11] are the following.Theorem 2.5. Let β1 be the Betti number speciﬁed in proposition 2.1. For O−m principal,the equivariant K-homology of Γ := PSL2(O−m) is isomorphic tom = 1 m = 2 m = 3 m = 7 m = 11 m ∈ {19, 43, 67, 163}KΓ0 (EΓ) Z6 Z5 ⊕ Z/2Z Z5 ⊕ Z/2Z Z3 Z4 ⊕ Z/2Z Zβ1−1 ⊕ Z3 ⊕ Z/2ZKΓ1 (EΓ) Z Z3 0 Z3 Z3 Z⊕ Zβ1 .6 ALEXANDER D. RAHMThe remainder of the equivariantK-homology of Γ is given by Bott 2-periodicity. By theBaum/Connes conjecture, which holds for the Bianchi groups [7], we obtain the K-theoryof the reduced C∗-algebras of the Bianchi groups as isomorphic images.Complex orbifolds. The information we have concerning the action of the Bianchigroups on real hyperbolic three-space provides explicit orbifold structures. These orb-ifolds serve as models in Cosmology [1].We can complexify these orbifolds. In [11], the product structure on their Chen/Ruanorbifold cohomology has been determined for all Bianchi groups; and an algorithm hasbeen given to compute the underlying vector space structure. This is a step towardschecking Ruan’s cohomological crepant resolution conjecture [4], which is of importancein Mathematical Physics and is completely open in 3 complex dimensions outside theglobal quotient case, hence these orbifolds are interesting test cases.Acknowledgements. The author thanks Stephen Gelbart for support and encouragement.He thanks Philippe Elbaz-Vincent, Louis Funar and Anthony Joseph for a careful lecture of the manuscript.And he would like to thank again all the people acknowledged in [11].References[1] Ralf Aurich, Frank Steiner, and Holger Then, Numerical computation of Maass waveforms and an ap-plication to cosmology, Contribution to the Proceedings of the ”International School on MathematicalAspects of Quantum Chaos II”, to appear in Lecture Notes in Physics (Springer) (2004).[2] Luigi Bianchi, Sui gruppi di sostituzioni lineari con coeﬃcienti appartenenti a corpi quadratici im-maginarˆı, Math. Ann. 40 (1892), no. 3, 332–412. MR1510727[3] Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag,1982. MR672956 (83k:20002), Zbl 0584.20036[4] Weimin Chen and Yongbin Ruan, A new cohomology theory of orbifold, Comm. Math. Phys. 248(2004), no. 1, 1–31. MR2104605 (2005j:57036), Zbl 1063.53091[5] Ju¨rgen Elstrodt, Fritz Grunewald, and Jens Mennicke, Groups acting on hyperbolic space, SpringerMonographs in Mathematics, Springer-Verlag, Berlin, 1998. MR1483315 (98g:11058)[6] Benjamin Fine, Algebraic theory of the Bianchi groups, Monographs and Textbooks in Pure andApplied Mathematics, vol. 129, Marcel Dekker Inc., New York, 1989. MR1010229 (90h:20002)[7] Pierre Julg and Gennadi Kasparov, Operator K-theory for the group SU(n, 1), Vol. 463. MR1332908(96g:19006)[8] Colin Maclachlan and Alan W. Reid, The arithmetic of hyperbolic 3-manifolds, Graduate Texts inMathematics, vol. 219, Springer-Verlag, New York, 2003. MR1937957 (2004i:57021)[9] Henri Poincare´, Me´moire, Acta Math. 3 (1966), no. 1, 49–92. Les groupes kleine´ens. MR1554613[10] Alexander D. Rahm, Bianchi.gp, Open source program (GNU general public license), 2010. Avail-able at http://tel.archives-ouvertes.fr/tel-00526976/, this program computes a fundamentaldomain for the Bianchi groups in hyperbolic 3-space, the associated quotient space and essentialinformation about the integral homology of the Bianchi groups.[11] , (Co)homologies and K-theory of Bianchi groups using computational geometric models, Ph.D.Thesis, Institut Fourier, Universite´ de Grenoble et Universita¨t Go¨ttingen, soutenue le 15 octobre 2010.http://tel.archives-ouvertes.fr/tel-00526976/.[12] Alexander D. Rahm and Mathias Fuchs, The integral homology of PSL2 of imaginary quadratic integerswith non-trivial class group, J. Pure and Applied Algebra 215 (2011), 1443–1472.[13] Joachim Schwermer and Karen Vogtmann, The integral homology of SL2 and PSL2 of Euclidean imag-inary quadratic integers, Comment. Math. Helv. 58 (1983), no. 4, 573–598. MR728453 (86d:11046),Zbl 0545.20031

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