Small hyperbolic polyhedra

Shawn Rafalski

Vol. 255 (2012), No. 1, 191–240

DOI: 10.2140/pjm.2012.255.191

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 nonfinite, 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 (nonorientable) 3-dimensional hyperbolic reflection group arises from a triangle reflection subgroup.

Keywords

hyperbolic polyhedra, 3-dimensional Coxeter polyhedra, triangle groups, hyperbolic orbifold, polyhedral orbifold, small orbifold, essential suborbifold, hyperbolic turnover

Mathematical Subject Classification 2010

Primary: 52B10, 57M50, 57R18

Milestones

Received: 1 February 2011

Revised: 28 October 2011

Accepted: 7 November 2011

Published: 14 March 2012