Normalizers of intermediate congruence subgroups of the Hecke subgroups

The Riemann Zeta-Function and Hecke Congruence Subgroups. II

which was scantly touched on in [9]. We shall proceed with an arbitrary A to a considerable extent but later restrict ourselves to the situation where αn is supported by the set of square-free integers. This is solely to avoid certain technical complexities pertaining to Kloosterman sums associated with Hecke congruence subgroups which do not appear particularly worth dealing with thoroughly, f...

Normalizers of Parabolic Subgroups of Coxeter Groups

We improve a bound of Borcherds on the virtual cohomological dimension of the non-reflection part of the normalizer of a parabolic subgroup of a Coxeter group. Our bound is in terms of the types of the components of the corresponding Coxeter subdiagram rather than the number of nodes. A consequence is an extension of Brink’s result that the non-reflection part of a reflection centralizer is fre...

Counting Congruence Subgroups

Abstract. Let Γ denote the modular group SL(2, Z) and Cn(Γ) the number of congruence subgroups of Γ of index at most n. We prove that lim n→∞ log Cn(Γ) (log n)2/ log log n = 3−2 √ 2 4 . We also present a very general conjecture giving an asymptotic estimate for Cn(Γ) for general arithmetic groups. The lower bound of the conjecture is proved modulo the generalized Riemann hypothesis for Artin-He...

Hecke Eigenforms in the Cohomology of Congruence Subgroups of SL(3, ZZ)

Stability in the homology of congruence subgroups

The homology groups of many natural sequences of groups {Gn}n=1 (e.g. general linear groups, mapping class groups, etc.) stabilize as n→ ∞. Indeed, there is a well-known machine for proving such results that goes back to early work of Quillen. Church and Farb discovered that many sequences of groups whose homology groups do not stabilize in the classical sense actually stabilize in some sense a...