Small hyperbolic polyhedra

Details Small hyperbolic polyhedra Small hyperbolic polyhedra

2011-02-01

arxiv.org/abs/1102.0322v4

Mathematics

Geometric Topology

49 pages, 42 figures. Version to appear, incorporates final feedback from the referee

Scientific paper

We classify the 3-dimensional hyperbolic polyhedral orbifolds that contain no embedded essential 2-suborbifolds, up to decomposition along embedded hyperbolic triangle orbifolds (turnovers). We give a necessary condition for a 3-dimensional hyperbolic polyhedral orbifold to contain an immersed (singular) hyperbolic turnover, we classify the triangle subgroups of the fundamental groups of orientable 3-dimensional hyperbolic tetrahedral orbifolds in the case when all of the vertices of the tetrahedra are non-finite, and we provide a conjectural classification of all the triangle subgroups of the fundamental groups of orientable 3-dimensional hyperbolic polyhedral orbifolds. Finally, we show that any triangle subgroup of a (non-orientable) 3-dimensional hyperbolic reflection group arises from a triangle reflection subgroup.

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