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… 

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References

SHOWING 1-10 OF 19 REFERENCES
Limit points of Kleinian groups and finite sided fundamental polyhedra
Let G be a discrete subgroup of SL(2, C)/{• 1}. Then G operates as a discontinuous group of isometrics on hyperbolic 3-space, which we regard as the open unit ball B a in Euclidean 3-space E a. G
A quadratic parabolic group
When k is a 2-bridge knot with group π K , there are parabolic representations (p-reps) θ: π K → PSL( ): = PSL(2, ). The most obvious problem that this suggests is the determination of a presentation
Discrete parabolic representations of link groups
Let k be a knot of type K and with group π K . Let θ: nK → PSL(ℂ) = PSL (2, ℂ) be a parabolic representation (p-rep) as defined in [14]. We shall call the representation discrete when its image πKθ
The geometry and topology of 3-manifolds
Seven excellent knots
Seven excellent knots," Brown and Thickstun, Low-Dimensional Topology, Vol
  • I, Cambridge University Press,
  • 1982
An elliptical path from parabolic representations to hyperbolic structures
The geometry of discrete groups," in Discrete Groups and Automorphic Functions (W
  • Harvey, ed.), Academic Press, London,
  • 1977
...
1
2
...