Hyperbolic Tessellations Associated to Bianchi Groups

  title={Hyperbolic Tessellations Associated to Bianchi Groups},
  author={Dan Yasaki},
  • D. Yasaki
  • Published in ANTS 12 August 2009
  • Mathematics
Let F/ℚ be a number field. The space of positive definite binary Hermitian forms over F form an open cone in a real vector space. There is a natural decomposition of this cone into subcones. In the case of an imaginary quadratic field these subcones descend to hyperbolic space to give rise to tessellations of 3-dimensional hyperbolic space by ideal polytopes. We compute the structure of these polytopes for a range of imaginary quadratic fields. 
Hyperbolic tessellations and generators of K_3 for imaginary quadratic fields
We develop methods for constructing explicit generating elements, modulo torsion, of the K_3-groups of imaginary quadratic number fields. These methods are based on either tessellations of hyperbolic
The image of the Borel-Serre bordification in algebraic K-theory
We give a method for constructing explicit non-trivial elements in the third K-group (modulo torsion) of an imaginary quadratic number field. These arise from the relative homology of the map
The Bianchi groups are the groups (P)SL2 over a ring of integers in an imaginary quadratic number field. We reveal a correspondence between the homological torsion of the Bianchi groups and new
Non‐integrality of some Steinberg modules
We prove that the Steinberg module of the special linear group of a quadratic imaginary number ring which is not Euclidean is not generated by integral apartment classes. Assuming the generalized
Perfect lattices over imaginary quadratic number fields
An adaptation of Voronoi theory for imaginary quadratic number fields of class number greater than 1 is presented and an application of the algorithm which allows to determine generators of the general linear group of an $\O_K$-lattice is presented.
Bounds on entries in Bianchi group generators
Upper and lower bounds are given for the maximum Euclidean curvature among faces in Bianchi’s fundamental polyhedron for PSL2(O) in the upper-half space model of hyperbolic space, where O is an
Higher torsion in the Abelianization of the full Bianchi groups
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
Arithmetic Aspects of Bianchi Groups
We discuss several arithmetic aspects of Bianchi groups, especially from a computational point of view. In particular, we consider computing the homology of Bianchi groups together with the Hecke
Arithmetic Aspects of Bianchi Groups
We discuss several arithmetic aspects of Bianchi groups, especially from a computational point of view. In particular, we consider computing the homology of Bianchi groups together with the Hecke
The homological torsion of PSL_2 of the imaginary quadratic integers
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


Binary Hermitian forms over a cyclotomic field
Weight Reduction for Mod l Bianchi Modular Forms
Hecke Operators and Hilbert Modular Forms
A technique to compute the action of the Hecke operators on the cuspidal cohomology H3(Γ;C) of GL2(O) for F real quadratic, which contains cuspid Hilbert modularforms of parallel weight 2.
Modular Symbols for Q-Rank One Groups and Voronı Reduction
Abstract LetGbe a reductive algebraic group of Q -rank one associated to a self-adjoint homogeneous cone defined over Q , and letΓ⊂Gbe a torsion-free arithmetic subgroup. Letdbe the cohomological
Periods of Cusp forms and elliptic curves over imaginary quadratic fields
In this paper we explore the arithmetic correspondence between, on the one hand, (isogeny classes of) elliptic curves E defined over an imaginary quadratic field K of class number one, and on the
The Mod-2 cohomology of the Bianchi groups
The Bianchi groups are a family of discrete subgroups of PSL2 (C) which have group theoretic descriptions as amalgamated products and HNN extensions. Using Bass-Serre theory, we show how the
The Cohomology of Lattices in SL(2, ℂ)
In the case Γ = SL(2,O) for the ring of integers O in an imaginary quadratic number field, the theory of lifting is made explicit and lower bounds linear in n are obtained and two instances with nonlifted classes in the cohomology are discovered.
The cohomology of lattices in SL(2,C)
This paper contains both theoretical results and experimental data on the behavior of the dimensions of the cohomology spaces H^1(G,E_n), where Gamma is a lattice in SL(2,C) and E_n is one of the
The integral cohomology of the Bianchi groups
We calculate the integral cohomology ring structure for various members of the Bianchi group family. The main tools we use are the Bockstein spectral sequence and a long exact sequence derived from