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On the Cohomological Crepant Resolution Conjecture for the complexified Bianchi orbifolds ; Rahm, Alexander in Algebraic and Geometric Topology (in press) We give formulae for the Chen--Ruan orbifold cohomology for the orbifolds given by a Bianchi group acting on complex hyperbolic 3-space. The Bianchi groups are the arithmetic groups PSL_2(A), where A is ... [more ▼] We give formulae for the Chen--Ruan orbifold cohomology for the orbifolds given by a Bianchi group acting on complex hyperbolic 3-space. The Bianchi groups are the arithmetic groups PSL_2(A), where A is the ring of integers in an imaginary quadratic number field. The underlying real orbifolds which help us in our study, given by the action of a Bianchi group on real hyperbolic 3-space (which is a model for its classifying space for proper actions), have applications in physics. We then prove that, for any such orbifold, its Chen-Ruan orbifold cohomology ring is isomorphic to the usual cohomology ring of any crepant resolution of its coarse moduli space. By vanishing of the quantum corrections, we show that this result fits in with Ruan's Cohomological Crepant Resolution Conjecture. [less ▲] Detailed reference viewed: 88 (6 UL)The mod 2 cohomology rings of congruence subgroups in the Bianchi groups ; ; Rahm, Alexander E-print/Working paper (in press) We provide new tools for the calculation of the torsion in the cohomology of congruence subgroups in the Bianchi groups : An algorithm for finding particularly useful fundamental domains, and an analysis ... [more ▼] We provide new tools for the calculation of the torsion in the cohomology of congruence subgroups in the Bianchi groups : An algorithm for finding particularly useful fundamental domains, and an analysis of the equivariant spectral sequence combined with torsion subcomplex reduction. [less ▲] Detailed reference viewed: 58 (1 UL)The Farrell--Tate and Bredon homology for PSL_4(Z) via cell subdivisions ; Rahm, Alexander ; in Journal of Pure and Applied Algebra (2019), 223(7), 2872-2888 We provide some new computations of Farrell–Tate and Bredon (co)homology for arithmetic groups. For calculations of Farrell–Tate or Bredon homology, one needs cell complexes where cell stabilizers fix ... [more ▼] We provide some new computations of Farrell–Tate and Bredon (co)homology for arithmetic groups. For calculations of Farrell–Tate or Bredon homology, one needs cell complexes where cell stabilizers fix their cells pointwise. We provide two algorithms computing an efficient subdivision of a complex to achieve this rigidity property. Applying these algorithms to available cell complexes for PSL_4(Z) provides computations of Farrell–Tate cohomology for small primes as well as the Bredon homology for the classifying spaces of proper actions with coefficients in the complex representation ring. [less ▲] Detailed reference viewed: 134 (15 UL)Bianchi's additional symmetries Rahm, Alexander E-print/Working paper (2019) Detailed reference viewed: 20 (1 UL)Genuine Bianchi modular forms of higher level, at varying weight and discriminant Rahm, Alexander ; Tsaknias, Panagiotis in Journal de Théorie des Nombres de Bordeaux (2019), 31(1), 27-48 Bianchi modular forms are automorphic forms over an imaginary quadratic field, associated to a Bianchi group. Those of the cuspidal Bianchi modular forms which are relatively well understood, namely ... [more ▼] Bianchi modular forms are automorphic forms over an imaginary quadratic field, associated to a Bianchi group. Those of the cuspidal Bianchi modular forms which are relatively well understood, namely (twists of) base-change forms and CM-forms, are what we call non-genuine forms; the remaining forms are what we call genuine. In a preceding paper by Rahm and Şengün, an extreme paucity of genuine forms has been reported, but those and other computations were restricted to level One. In this paper, we are extending the formulas for the non-genuine Bianchi modular forms to higher levels, and we are able to spot the first, rare instances of genuine forms at higher level and higher weight. [less ▲] Detailed reference viewed: 168 (16 UL)Simulation study of the electrical tunneling network conductivity of suspensions of hard spherocylinders ; ; Rahm, Alexander et al in Physical Review. E. (2018), 98 Using Monte Carlo simulations, we investigate the electrical conductivity of networks of hard rods with aspect ratios 10 and 20 as a function of the volume fraction for two tunneling conductance models ... [more ▼] Using Monte Carlo simulations, we investigate the electrical conductivity of networks of hard rods with aspect ratios 10 and 20 as a function of the volume fraction for two tunneling conductance models. For a simple, orientationally independent tunneling model, we observe nonmonotonic behavior of the bulk conductivity as a function of volume fraction at the isotropic-nematic transition. However, this effect is lost if one allows for anisotropic tunneling. The relative conductivity enhancement increases exponentially with volume fraction in the nematic phase. Moreover, we observe that the orientational ordering of the rods in the nematic phase induces an anisotropy in the conductivity, i.e., enhanced values in the direction of the nematic director field. We also compute the mesh number of the Kirchhoff network, which turns out to be a simple alternative to the computationally expensive conductivity of large systems in order to get a qualitative estimate. [less ▲] Detailed reference viewed: 49 (4 UL)Equivariant K-homology for hyperbolic reflection groups ; ; Rahm, Alexander et al in The Quarterly Journal of Mathematics (2018), 69(4), 1475-1505 We compute the equivariant K-homology of the classifying space for proper actions, for cocompact 3-dimensional hyperbolic reflection groups. This coincides with the topological K-theory of the reduced C ... [more ▼] We compute the equivariant K-homology of the classifying space for proper actions, for cocompact 3-dimensional hyperbolic reflection groups. This coincides with the topological K-theory of the reduced C*-algebra associated to the group, via the Baum-Connes conjecture. We show that, for any such reflection group, the associated K-theory groups are torsion-free. This means that we can complete previous computations with rational coefficients to get results with integral coefficients. On the way, we establish an efficient criterion for checking torsion-freeness of K-theory groups, which can be applied far beyond the scope of the present paper. [less ▲] Detailed reference viewed: 55 (6 UL)On Farrell-Tate cohomology of SL_2 over S-integers Rahm, Alexander ; in Journal of Algebra (2018), 512 In this paper, we provide number-theoretic formulas for Farrell–Tate cohomology for SL_2 over rings of S-integers in number fields satisfying a weak regularity assumption. These formulas describe group ... [more ▼] In this paper, we provide number-theoretic formulas for Farrell–Tate cohomology for SL_2 over rings of S-integers in number fields satisfying a weak regularity assumption. These formulas describe group cohomology above the virtual cohomological dimension, and can be used to study some questions in homology of linear groups. We expose three applications, to (I) detection questions for the Quillen conjecture, (II) the existence of transfers for the Friedlander–Milnor conjecture, (III) cohomology of SL_2 over number fields. [less ▲] Detailed reference viewed: 81 (6 UL)Bounds for the mod 2 cohomology of GL_2(Z[sqrt(-2)][1/2]) ; Rahm, Alexander E-print/Working paper (2017) Detailed reference viewed: 21 (0 UL)Report on the Oberwolfach Workshop "Computations in the Cohomology of Arithmetic Groups" Rahm, Alexander in Oberwolfach Reports (2016, November) Explicit calculations play an important role in the theoretical development of the cohomology of groups and its applications. It is becoming more common for such calculations to be derived with the aid of ... [more ▼] Explicit calculations play an important role in the theoretical development of the cohomology of groups and its applications. It is becoming more common for such calculations to be derived with the aid of a computer. This mini-workshop assembled together experts on a diverse range of computational techniques relevant to calculations in the cohomology of arithmetic groups and applications in algebraic K-theory and number theory with a view to extending the scope of computer aided calculations in this area. [less ▲] Detailed reference viewed: 109 (0 UL)The mod 2 cohomology rings of SL_2 of the imaginary quadratic integers Rahm, Alexander ; in Journal of Pure and Applied Algebra (2016), 220 Detailed reference viewed: 48 (2 UL)On the equivariant K-homology of PSL_2 of the imaginary quadratic integers Rahm, Alexander in Annales de l'Institut Fourier (2016), 66(4), 1667-1689 We establish formulae for the part due to torsion of the equivariant K-homology of all the Bianchi groups (PSL_2 of the imaginary quadratic integers), in terms of elementary number-theoretic quantities ... [more ▼] We establish formulae for the part due to torsion of the equivariant K-homology of all the Bianchi groups (PSL_2 of the imaginary quadratic integers), in terms of elementary number-theoretic quantities. To achieve this, we introduce a novel technique in the computation of Bredon homology: representation ring splitting, which allows us to adapt the recent technique of torsion subcomplex reduction from group homology to Bredon homology. [less ▲] Detailed reference viewed: 70 (4 UL)Complexifiable characteristic classes Rahm, Alexander in Journal of Homotopy & Related Structures (2015), 10(3), 537--548 Detailed reference viewed: 51 (5 UL)A refinement of a conjecture of Quillen Rahm, Alexander ; in Comptes Rendus. Mathématique (2015), 353(9), 779--784 Detailed reference viewed: 89 (7 UL)Accessing the cohomology of discrete groups above their virtual cohomological dimension Rahm, Alexander in J. Algebra (2014), 404 Detailed reference viewed: 85 (0 UL)The subgroup measuring the defect of the abelianization of $ SL_2(\Bbb Z[i])$ Rahm, Alexander in Journal of Homotopy & Related Structures (2014), 9(2), 257--262 Detailed reference viewed: 27 (2 UL)Higher torsion in the Abelianization of the full Bianchi groups Rahm, Alexander in LMS J. Comput. Math. (2013), 16 Detailed reference viewed: 51 (0 UL)The homological torsion of $PSL_2$ of the imaginary quadratic integers Rahm, Alexander in Trans. Amer. Math. Soc. (2013), 365(3), 1603--1635 Detailed reference viewed: 63 (0 UL)On level one cuspidal Bianchi modular forms Rahm, Alexander ; in LMS J. Comput. Math. (2013), 16 Detailed reference viewed: 67 (0 UL)On a question of Serre Rahm, Alexander in Comptes Rendus. Mathématique (2012), 350(15-16), 741--744 Detailed reference viewed: 71 (4 UL) |
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