# Applications of a computer implementation of Poincaré’s theorem on fundamental polyhedra

@article{Riley1983ApplicationsOA, title={Applications of a computer implementation of Poincar{\'e}’s theorem on fundamental polyhedra}, author={Robert F. Riley}, journal={Mathematics of Computation}, year={1983}, volume={40}, pages={607-632} }

PoincarCs Theorem asserts that a group F of isometries of hyperbolic space H is discrete if its generators act suitably on the boundary of some polyhedron in H, and when this happens a presentation of F can be derived from this action. We explain methods for deducing the precise hypotheses of the theorem from calculation in F when F is "algorithmi- cally defined", and we describe a file of Fortran programs that use these methods for groups F acting on the upper half space model of hyperbolic 3…

## 72 Citations

The orbit space of a Kleinian group: Riley’s modest example

- Mathematics
- 1983

In the previous paper, Robert Riley [4] and his computer file Poincare found a fundamental domain for the action of a discrete group G of isometries of hyperbolic space HI generated by three…

(Co)homologies and K-theory of Bianchi groups using computational geometric models

- Mathematics
- 2010

This thesis consists of the study of the geometry of a certain class of arithmetic groups, by means of a proper action on a contractible space. We will explicitly compute their group homology, and…

Poincar\'e Bisectors in Hyperbolic Spaces

- Mathematics
- 2012

We determine explicit formulas for the bisectors used in constructing a Dirichlet fundamental domain in hyperbolic two and three space. They are compared with the isometric spheres employed in the…

APPLYING POINCARE'S POLYHEDRON THEOREM TO GROUPS OF HYPERBOLIC ISOMETRIES

- Mathematics
- 2009

We present a computer algorithm which conrms that an approxi- mately discete subgroup of PSL(2;C) is in fact discrete. The algorithm proceeds by constructing the Dirichlet domain of the subgroup in H…

Hyperbolic Structures of Arithmetic Type on Some Link Complements

- Mathematics
- 1983

-d) u {00} c= CP = dH, and each a ePSL2(0d) determines an edge which is the geodesic in H having endpoints a//? and y/d. The tesselations 5^ and ^ are by regular ideal octahedra and tetrahedra,…

Discrete isometry groups of symmetric spaces

- Mathematics
- 2017

Author(s): Kapovich, Michael; Leeb, Bernhard | Abstract: This survey is based on a series of lectures that we gave at MSRI in Spring 2015 and on a series of papers, mostly written jointly with Joan…

Subgroups of Bianchi groups and arithmetic quotients of hyperbolic 3-space

- Mathematics
- 1993

Let O be the ring of integers in an imaginary quadratic number-field. The group PSL 2 (O) acts discontinuously on hyperbolic 3-space H. If Γ ≤ PSL 2 (O) is a torsionfree subgroup of finite index then…

From the Poincaré Theorem to generators of the unit group of integral group rings of finite groups

- MathematicsMath. Comput.
- 2015

We give an algorithm to determine finitely many generators for a subgroup of finite index in the unit group of an integral group ring $\mathbb{Z} G$ of a finite nilpotent group $G$, this provided the…

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…

Almost all Bianchi groups have free, non-cyclic quotients

- Mathematics
- 1992

Let d be a square-free positive integer and let O (= O d ) be the ring of integers of the imaginary quadratic number field ℚ(- d ). The groups PSL 2 (O) are called the Bianchi groups after Luigi…

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