Small hyperbolic polyhedra

TL;DR

This paper classifies certain 3D hyperbolic polyhedral orbifolds based on their embedded suborbifolds, providing conditions for hyperbolic turnovers and classifying triangle subgroups in specific cases, advancing understanding of their geometric structures.

Contribution

It offers a classification of hyperbolic polyhedral orbifolds without essential 2-suborbifolds and analyzes triangle subgroups, including a conjectural classification and connections to reflection groups.

Findings

Classified hyperbolic polyhedral orbifolds without essential 2-suborbifolds.

Provided a necessary condition for immersed hyperbolic turnovers.

Showed all triangle subgroups of non-orientable reflection groups arise from reflection subgroups.

Abstract

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.