Séminaire de Mathématique

Bianchi orbifolds and their torsion subcomplexes

by Prof. Alexander RAHM (National University of Ireland at Galway & IHÉS)

Amphithéâtre Léon Motchane (IHES)

Amphithéâtre Léon Motchane


Le Bois Marie 35, route de Chartres 91440 Bures-sur-Yvette
This talk starts with the Bianchi orbifolds, which arise from the action of the Bianchi groups (SL2 matrix groups over imaginary quadratic integers) on hyperbolic 3-space (their associated symmetric space). Studying these orbifolds, one can obtain : - dimensions of spaces of Bianchi modular forms; - Algebraic K-theory of rings of imaginary quadratic integers; - Group (co)homology and equivariant K-homology of the Bianchi groups; - Chen-Ruan orbifold cohomology of the Bianchi orbifolds. The latter cohomology ring is conjectured to be isomorphic to the cohomology ring of a crepant resolution for the orbifold, which makes it interesting for string theory. What makes Bianchi modular forms interesting, is that there are deep number-theoretical reasons for expecting that the Taniyama-Shimura correspondence can extend to the cuspidal Bianchi modular forms, attaching Abelian varieties to them. For the dimension calculations, use is made of the speaker's constructive answer to a question of Jean-Pierre Serre on the Borel-Serre compactification of the Bianchi orbifolds, which had been open for 40 years. This has so far permitted heavy machine calculations joint with M. Haluk Sengun, locating several of the very rare instances of Bianchi modular forms in a large array of dicriminants and weights, such that these forms are not lifts of classical modular forms. Complementary to this, studies of the torsion part of the cohomology of the Bianchi groups have given rise to a new technique (called Torsion Subcomplex Reduction; some procedures of the technique had beforehand been used as ad hoc tricks by Soulé, Mislin and Henn) for computing the Farrell-Tate cohomology of discrete groups acting on suitable cell complexes. This technique has not only already yielded general formulae for the cohomology of the tetrahedral Coxeter groups as well as, above the virtual cohomological dimension, of the Bianchi groups (and at odd torsion, more generally of SL2 groups over arbitrary number fields, in joint work with Matthias Wendt), it also very recently has allowed Wendt to refine the Quillen conjecture. This talk will further discuss the adaptation of this technique to Bredon homology computations, yielding the equivariant K-homology of the Bianchi groups. The Baum-Connes conjecture, which has been proved for large classes of groups, constructs an (iso)morphism from the equivariant K-homology of a group to the K-theory of its reduced C*-algebra. For the Bianchi groups, this yields the isomorphism type of the mentioned operator K-theory, which would be very hard to compute directly.