Alexander D. Rahm

The integral homology of

Abstract We show that a cellular complex defined by Floge allows us to determine the integral homology of the Bianchi groups PS L 2 ( O − m ) , where O − m is the ring of integers of an imaginary quadratic number field Q [ − m ] for a square-free natural number m . In the cases of nontrivial class group, we handle the difficulties arising from the cusps associated to the nontrivial ideal classes of O − m . We use this to compute the integral homology of PS L 2 ( O − m ) in the cases m = 5 , 6 , 10 , 13 and 15 , which previously was known only in the cases m = 1 , 2 , 3 , 7 and 11 with trivial class group.

PSL_2

Denote by Q(sqrt{-m}), with m a square-free positive integer, an imaginary quadratic number field, and by A its ring of integers. The Bianchi groups are the groups SL_2(A). 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 expose a novel technique, the torsion subcomplex reduction, to obtain these invariants. We use it to explicitly compute the integral group homology of the Bianchi groups. Furthermore, this correspondence facilitates the computation of the equivariant K-homology of the Bianchi groups. By the Baum/Connes conjecture, which is verified by the Bianchi groups, we obtain the K-theory of their reduced C*-algebras in terms of isomorphic images of their equivariant K-homology.

of imaginary quadratic integers with nontrivial class group

Abstract 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.

The homological torsion of PSL_2 of the imaginary quadratic integers

Consider the Bianchi groups, namely the SL_2 groups over rings of imaginary quadratic integers. In the literature, there has been so far no example of p-torsion in the integral homology of the full Bianchi groups, for p a prime greater than the order of elements of finite order in the Bianchi group, which is at most 6. However, extending the scope of the computations, we can observe examples of torsion in the integral homology of the quotient space, at prime numbers as high as for instance p = 80737 at the discriminant -1747.

Homology and

In this paper, we present the outcome of vast computer calculations, locating several of the very rare instances of level one cuspidal Bianchi modular forms that are not lifts of elliptic modular forms.

K

We introduce a method to explicitly determine the Farrell-Tate cohomology of discrete groups. We apply this method to the Coxeter triangle and tetrahedral groups as well as to the Bianchi groups, i.e. PSL_2 over the ring of integers in an imaginary quadratic number field. We show that the Farrell-Tate cohomology of the Bianchi groups is completely determined by the numbers of conjugacy classes of finite subgroups. In fact, our access to Farrell-Tate cohomology allows us to detach the information about it from geometric models for the Bianchi groups and to express it only with the group structure. Formulae for the numbers of conjugacy classes of finite subgroups in the Bianchi groups have been determined in a thesis of Kramer, in terms of elementary number-theoretic information on the ring of integers. An evaluation of these formulae for a large number of Bianchi groups is provided numerically in the appendix. Our new insights about the homological torsion allow us to give a conceptual description of the cohomology ring structure of the Bianchi groups.

-theory of the Bianchi groups

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. As a result we can promote previous rational computations to integral computations. Our proof relies on a new efficient algebraic criterion for checking torsion-freeness of K-theory groups, which could be applied to many other classes of groups.