, rsquare)= { local(a,b,c,d, s_limitSummand, cz_R, cz_W, discriminant, squareRoot)

, With c = 0, we get just the trivial stabilizing matrices (except for m= 1 or 3): */ listput(currentStabilizer

, */ /* c = j +k*w runs through the lattice of integers with 0 < |c| <= 1/r */ j_limit = ceil

, /* j runs from -j_limit through j_limit: */ for (j = -j_limit, j_limit, /* |c| = sqrt

. /*-therefore, 2*j/(m+1) (+,-) 2*sqrt(-j^2*m +(m+1)/(rsquare) )/(m+1) */ discriminant = -j^2*m +

, for (k = floor( 2*j/(m+1) -k_limitSummand), ceil( 2*j/(m+1) + k_limitSummand)

, */ if ( norm(nfbasistoalg(K,c)) <= 1/rsquare && c !=

. /*-print, cz =

. */-cz_r-=-component, , vol.1

, cz_W = component, vol.2

, -r^2|c|^2)/m). */ s_limitSummand = 2*sqrt

, for ( s = floor(cz_W -s_limitSummand), ceil(cz_W + s_limitSummand), discriminant = 1 -rsquare*norm(nfbasistoalg(K,c)) -(m/4)*cz_W^2 +(m/2)*cz_W*s -(m/4)*s^2

. /*-q, cz_R -cz_W/2 +s/2 (+,-) sqrt( discriminant) */ qPlus = cz_R -cz_W/2 +s/2 +squareRoot

(. Finishstabilizer and ). , Finish the computation of the stabilizer of (z,rsquare) in Hyperbolic Space

. Stabilizingmatrix-=-listcreate,

, cz_R = component( cz, vol.1

, cz_W = component( cz, vol.2

, Floege deduces a = conj(d) -2*Re(cz). */ TwoRe_cz = 2*cz_R -cz_W

, the ring of integers: */ if( frac(TwoRe_cz) == 0

/. , Check that |cz-d|^2 +r^2*|c|^2 == 1, */ if( norm(nfbasistoalg(K, cz -d)) +rsquare*norm, p.1

/. ,

, == [0,0]~, /* Check that z == conj(d-cz)(az-b) -r^2*conj(c)a. */ if( z == nfeltmul(K, conjugate(d -cz), nfeltmul(K,a,z) -b) -rsquare*, /* Check that b is in the ring of integers: */ if ( frac(b)

, / if( z != PoincareAction(a,b,c,d,z,rsquare), print("***Error in function FinishStabilizer

. Stabilizingmatrix,

, ***Error in function FinishStabilizer

*. /*-return and }. Stabilizingmatrix,

A. Figure, 8: The subfunction FinishStabilizer of the function computeVertexStabilizer3mod4

, Computation of the identification matrices [a,b; c,d] */ /* which transport the point p=(z,r) into the point p_2 =(Zeta, rho)*/ /* in hyperbolic space, p.2

, local(lefthandside, ThereIsNoRamification, s_limitMinus, s_limitPlus,q, rByRho, rRhoInverse, rcSquare)

, z = component, p.1

, rsquare = component, vol.2

, Zeta = component, vol.1, p.2

, rhosquare = component, vol.2, p.2

, issquare(rsquareByRhosquare, &rByRho)

(. and Z. , Zeta

/. First,

. /*-d-=-g+hw-=-g-+h, /2 +1/2sqrt(-m)) runs through the ring of integers with |d|^2 = r/rho. */ /* |d|^2 = (g -h/2)^2 + m

, */ /* If the above discriminant is zero, run just one case, else two cases for g_+

, for ( Case = ThereIsNoRamification, 1, discriminant = rByRho -m*h^2/4

, if( issquare( discriminant, &squareRoot), g = h/2 +(-1)^Case*squareRoot

/. , Check that |d|^2 = r/rho */ if( (norm(nfbasistoalg(K,d)))^2 == rsquareByRhosquare

, /* check that b is in the ring of integers: */ if

K. , L. , ). ==-rzetabyrhosquare, and /. ,

A. Figure, 10: The function getIdentificationMatrices3mod4

/. Second, The equation (2') gives us $ r^2|c|^2 \leq \frac{r}{\rho} $. */ /* c = j +kw runs through the ring of integers of the number field, verifying */ /* 1/(r*rho) >= normsquare(c) > 0

, We find the extremal values of j by an argument analogous to the one in the documentation of computeVertexStabilizer3mod4. */ for( j = -jbound, jbound, /* We have |c|^2 = (j -k/2)^2 +m(k/2)^2 = j^2 +((m+1)k^2)/4 -jk, */ /* therefore k_limit^2 + 4/(m+1)*(-j*k_limit +j^2 -1/(r*rho)) = 0. */ if( (m+1)*rRhoInverse >= j^2*m, /* (check discriminant >= 0 ) */ /* Thus let */ k_limitPlus = 2/(m+1)

, k_limitMinus = 4/(m+1)*j -k_limitPlus

, /(m+1)*j -2*sqrt(( (m+1)*rRhoInverse -j^2*m)/(m+1)). */ for ( k = floor(k_limitMinus)

, Decompose cz as cz_real +cz_omegacoeff*w with rational integer coefficients. */ cz = nfeltmul( K, c, z)

, cz_real = component( cz, vol.1

, cz_omegacoeff = component( cz, vol.2

, cz = nfbasistoalg(K, cz

, rcSquare = rsquare*norm(c)

. /*-d-=, verifying */ /* normsquare(cz -d) +r^2*normsquare(c) = r/rho. */ /* We decompose d as q +sw, where q and s are rational integers. */ s_limitPlus = cz_omegacoeff + 2*sqrt, q+s*w runs through the ring of integers of the number field

, s_limitMinus = 2*cz_omegacoeff -s_limitPlus

. /*-=, /m) */ for( s = floor(s_limitMinus), ceil(s_limitPlus), discriminant = rByRho -rcSquare -m/4*(cz_omegacoeff -s)^2

/. , Check that 'discriminant' is a rational square: */ if( type(discriminant) == "t_FRAC" || type(discriminant) == "t_INT

, /* obtain q+ with the positive squareroot */ q = cz_real -cz_omegacoeff/2 +s/2 +squareRoot

, 0, /* Check that q is a rational integer */ vertexTransport = concat( vertexTransport, getRemainingEntries(round(q),s,c,z,cz,Zeta,rsquare,rhosquare, rsquareByRhosquare, rZetaByRhoSquare, rByRho) )

, if( frac(q) == 0, /* Check that q is a rational integer */ vertexTransport = concat( vertexTransport, getRemainingEntries

, ***Error in function getIdentificationMatrices3mod4, on vertexTransport from, length( vertexTransport) != stabilizerCardinal, error(Str

, /* return the list */ vertexTransport }

A. Figure, 11: The function getIdentificationMatrices3mod4

/. Second, / /* c = j +kw runs through the ring of integers of the number field, verifying */ /* 1/(r*rho) >= normsquare(c) = j^2 +mk^2 */ oneByRrho = rByRho/rsquare

, for ( k = -kbound, kbound, jbound = ceil( sqrt( oneByRrho -m*k^2))

. =--jbound,

, != 0 && (norm(c))^2 <= 1/(rsquare*rhosquare), cz = nfeltmul

, cz_real = component( cz, vol.1

, cz_omegacoeff = component( cz, vol.2

, cz = nfbasistoalg(K, cz

. /*-d, =: g+hw runs through the ring of integers of the number field, verifying */ /* normsquare(cz -d) +r^2*normsquare(c) = r/rho. */ hBound = abs(cz_omegacoeff) +sqrt( (rByRho -rsquare*

/. , / for( h = -round(hBound), round( hBound), discriminant = rByRho -m*(cz_omegacoeff-h)^2 -rsquare*norm

/. , Check that 'discriminant' is a rational square: */ if( type(discriminant) == "t_FRAC" || type(discriminant) == "t_INT

, / if( frac(g) == 0, vertexTransport = concat( vertexTransport, getRemainingEntries( round(g),h,c,z,cz,Zeta,rsquare,rhosquare, rsquareByRhosquare, rZetaByRhoSquare, rByRho)); cz_real -squareRoot

, / if( frac(g) == 0, vertexTransport = concat( vertexTransport, getRemainingEntries

, ***Error in function getIdentificationMatrices, on vertexTransport from, length( vertexTransport) != stabilizerCardinal, print

, /* return the list */ vertexTransport }

A. Figure, 14: The function getIdentificationMatrices

B. Allombert, C. Batut, K. Belabas, D. Bernardi, H. Cohen et al., Emmanuel Tollis, Ilya Zakharevitch, and the PARI group, 2010.

A. Ash, Deformation retracts with lowest possible dimension of arithmetic quotients of selfadjoint homogeneous cones, Math. Ann, vol.225, issue.1, p.343, 1977.

R. Aurich, F. Steiner, and H. Then, Numerical computation of Maass waveforms and an application to cosmology, Contribution to the Proceedings of the "International School on Mathematical Aspects of Quantum Chaos II, 2004.

P. Baum, A. Connes, and N. Higson, Classifying space for proper actions and K-theory of group C * -algebras, C * -algebras, Contemp. Math, vol.167, pp.240-291, 1993.

K. Belabas and H. Gangl, Generators and relations for K2OF , K-Theory, vol.31, p.11154, 2004.

E. Berkove, The integral cohomology of the Bianchi groups, Trans. Amer. Math. Soc, vol.358, issue.3, p.2237880, 2006.

, The mod-2 cohomology of the Bianchi groups, Trans. Amer. Math. Soc, vol.352, issue.10, p.11043, 2000.

L. Bianchi, Sui gruppi di sostituzioni lineari con coefficienti appartenenti a corpi quadratici immaginarî, Math. Ann, vol.40, issue.3, pp.332-412, 1892.

S. Boissière, E. Mann, and F. Perroni, The cohomological crepant resolution conjecture for P(1, 3, 4, 4), Internat. J. Math, vol.20, issue.6, pp.791-801, 2009.

R. Martin, A. Bridson, and . Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, vol.319, p.53038, 1999.

K. S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol.87, 1982.

W. Chen and Y. Ruan, A new cohomology theory of orbifold, Comm. Math. Phys, vol.248, issue.1, p.57036, 2004.

P. Cherix, M. Cowling, P. Jolissaint, P. Julg, and A. Valette, Gromov's a-T-menability. MR1852148, vol.197, p.22007, 2001.

P. Cherix, F. Martin, and A. Valette, Ergodic Theory Dynam, Spaces with measured walls, the Haagerup property and property (T), vol.24, p.22006, 2004.

J. E. Cremona, Hyperbolic tessellations, modular symbols, and elliptic curves over complex quadratic fields, Compositio Math, vol.51, issue.3, pp.275-324, 1984.

J. Elstrodt, F. Grunewald, and J. Mennicke, PSL(2) over imaginary quadratic integers, Arithmetic Conference, vol.94, p.10051, 1981.

, Groups acting on hyperbolic space, MR1483315 (98g:11058), 1998.

B. Fantechi and L. Göttsche, Orbifold cohomology for global quotients, Duke Math. J, vol.117, issue.2, p.14062, 2003.

B. Fine, Algebraic theory of the Bianchi groups, Monographs and Textbooks in Pure and Applied Mathematics, vol.129, 1989.

D. Flöge, Zur Struktur der PSL2über einigen imaginär-quadratischen Zahlringen, MR704107 (85c:11043), Zbl 0498, vol.183, pp.255-279, 1983.

, Dissertation: Zur Struktur der PSL2über einigen imaginär-quadratischen Zahlringen, Johann-Wolfgang-Goethe-Universität, 1980.

P. E. Gunnells, Modular symbols for Q-rank one groups and Vorono? reduction, J. Number Theory, vol.75, issue.2, p.11084, 1999.

N. Higson and G. Kasparov, E-theory and KK-theory for groups which act properly and isometrically on Hilbert space, Invent. Math, vol.144, issue.1, p.19005, 2001.

G. Humbert, Sur la réduction des formes d'Hermite dans un corps quadratique imaginaire, C. R. Acad. Sci. Paris, vol.16, pp.189-196, 1915.

P. Julg and G. Kasparov, Operator K-theory for the group SU(n, 1), J. Reine Angew. Math, vol.463, pp.99-152, 1995.

C. Maclachlan and A. W. Reid, The arithmetic of hyperbolic 3-manifolds, Graduate Texts in Mathematics, vol.219, p.57021, 2003.

E. R. Mendoza, Cohomology of PGL2 over imaginary quadratic integers, Bonner Mathematische Schriften, Mathematisches Institut, vol.128, p.22012, 1979.

G. Mislin and A. Valette, Proper group actions and the Baum-Connes conjecture, Advanced Courses in Mathematics. CRM Barcelona, 2003.

H. Poincaré, A. Mémoire, and . Math, Les groupes kleinéens. MR1554613, vol.3, pp.49-92, 1883.

A. D. Rahm and M. Fuchs, The integral homology of PSL2 of imaginary quadratic integers with non-trivial class group, Journal of Pure and Applied Algebra, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00370722

A. D. Rahm, . Bianchi, and . Gp, , 2010.

J. G. Ratcliffe, Foundations of hyperbolic manifolds, vol.149, p.57029, 2006.

R. Riley, Applications of a computer implementation of Poincaré's theorem on fundamental polyhedra, POINCARE, A library of FORTRAN routines, vol.40, 1983.

J. Rubén, B. Sánchez-garcía, and K. Sl, J. Pure Appl. Algebra, vol.3, issue.5, 2008.

, Equivariant K-homology for some Coxeter groups, J. Lond. Math. Soc, issue.2, p.19006, 2007.

J. Schwermer and K. Vogtmann, The integral homology of SL2 and PSL2 of Euclidean imaginary quadratic integers, Comment. Math. Helv, vol.58, issue.4, 1983.

R. G. Swan, Generators and relations for certain special linear groups, Advances in Math, vol.6, 1741.

A. Valette, From notes taken by Indira Chatterji, Introduction to the Baum-Connes conjecture, 2002.

K. Vogtmann, Rational homology of Bianchi groups, Math. Ann, vol.272, issue.3, p.22025, 1985.

C. and T. C. Wall, Resolutions for extensions of groups, Proc. Cambridge Philos. Soc, vol.57, pp.251-255, 1961.

J. Weeks, SnapPea, program for the study of hyperbolic 3-manifolds

D. Yasaki, Hyperbolic tessellations associated to Bianchi groups, Algorithmic Number Theory. 9th International Symposium, 2010.

E. Zaslow, Topological orbifold models and quantum cohomology rings, Comm. Math. Phys, vol.156, issue.2, p.32045, 1993.
DOI : 10.1007/bf02098485

URL : http://arxiv.org/pdf/hep-th/9211119

A. Rahm@ujf-grenoble,

. Résumé, Ces groupes agissent d'une manière naturelle sur l'espace hyperboliqueà 3 dimensions. Ils constituent une clef pour l'étude d'une classe plus large de groupes, les groupes Kleiniens,étudiés depuis Poincaré. En fait, chaque groupe Kleinien arithmétique non-cocompact est commensurable avec un des groupes de Bianchi. L'auteur a implémentéà l'ordinateur, le calcul d'un domaine fondamental pour ces groupes. En calculant les stabilisateurs et identifications sur ce domaine fondamental, nous obtenons une structure explicite d'orbi-espace. Nous nous en servons pourétudier des aspects différents de la géométrie des groupes de Bianchi. D'abord, nous calculons l'homologie de groupeà coefficients entiers,à l'aide de la suite spectraleéquivariante de Leray/Serre. Ensuite, nous calculons l'homologie de Bredon de groupes de Bianchi, de laquelle nous déduisons leur K-homologieéquivariante. Par la conjecture de Baum/Connes, qui est vérifiée par nos groupes, nous obtenons la K-théorie des C*-algèbres réduites de nos groupes. Finalement, nous complexifions nos orbi-espaces

. Mots-clés, These groups act in a natural way on hyperbolic three-space. The Bianchi groups are a key to the study of a larger class of groups, the Kleinian groups, which dates back to works of Poincaré. In fact, each non-cocompact arithmetic Kleinian group is commensurable with some Bianchi group. The author has implemented the computation of a fundamental domain for the Bianchi groups. By computing the stabilisers and identifications on this fundamental domain, we obtain an explicit orbifold structure. We use it to study different aspects of the geometry of our groups. Firstly, we compute group homology with integer coefficients, using the equivariant Leray/Serre spectral sequence. Secondly, we compute the Bredon homology of the Bianchi groups, from which we deduce their equivariant K-homology. By the Baum/Connes conjecture, which is verified by the Bianchi groups, we obtain the K-theory of the reduced C*-algebras of the Bianchi groups, Homologie de groupes arithmétiques, théorie des nombres, espace classifiant pour actions propres, topologie algébrique, cohomologie d'orbifold de Chen/Ruan, conjecture de Baum/Connes. Abstract. This thesis consists of the study of the geometry of a certain class of arithmetic groups

. Keywords, Homology of arithmetic groups, number theory, classifying space for proper actions, algebraic topology, Chen/Ruan orbifold cohomology