# Hyperbolic Tessellations Associated to Bianchi Groups

@inproceedings{Yasaki2010HyperbolicTA, title={Hyperbolic Tessellations Associated to Bianchi Groups}, author={Dan Yasaki}, booktitle={ANTS}, year={2010} }

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.

## 34 Citations

Hyperbolic tessellations and generators of K_3 for imaginary quadratic fields

- Mathematics
- 2019

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

- Mathematics
- 2014

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…

OF PSL2 OF THE IMAGINARY QUADRATIC INTEGERS

- Mathematics
- 2011

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

- MathematicsJournal of Topology
- 2020

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

- Mathematics, Computer ScienceMath. Comput.
- 2015

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

- Mathematics
- 2021

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

- Mathematics
- 2013

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

- Mathematics
- 2012

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

- Mathematics
- 2014

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

- Mathematics
- 2011

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…

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