Ideal triangle groups, dented tori, and numerical analysis

By Richard Evan Schwartz. 2001

64 citations recorded by the source corpus.

We prove the Goldman-Parker Conjecture: A complex hyperbolic ideal triangle group is discretely embedded in PU(2, 1) if and only if the product of its three standard generators is not elliptic. We also prove that such a group is indiscrete if the product of its three standard generators is elliptic. A novel feature of this paper is that it uses a rigorous computer assisted proof to deal with difficult geometric estimates.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/be9ce810c9088c535f098ac202d6accd34a88d75

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In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

Traces in Complex Hyperbolic Triangle Groups (2004, methodology): “Schwartz proved this conjecture for ð1; 1; 1Þ-groups in [ 13 ], and for ð p1; p2; p3Þgroups with p1, p2, p3 sufficiently large in [16].”
Flexing closed hyperbolic manifolds. (2007, background): “As in [16], once one has succeeded in flexing a representation, it is natural to attempt an understanding of the possible degenerations.”
Real hyperbolic on the outside, complex hyperbolic on the inside (2003, methodology): “It follows from this fact, and from symmetry, that there are real slices S1 , S2, S3 ⊂ ΓΥ, such that S j ∩ Π12 = γ j. Figure 3.2 shows a set homeomorphic to the union Π ∞ ∪ S ∞ 1 ∪ S ∞ ∪ S ∞ . The hexagon represents Π ∞ . The drawing contains the complete bipartite graph K3,3 and hence is not plana”
Spherical CR Geometry and Dehn Surgery (2007, background): “In [9] Schwartz gave a proof of this conjecture that depended on numerical analysis.”