Hyperbolic triangular prisms with one ideal vertex

TL;DR

This paper classifies all five-sided hyperbolic triangular prisms with one ideal vertex, providing explicit geometric descriptions and matrix generators for their reflection groups in the upper half-space model.

Contribution

It offers a complete classification of hyperbolic triangular prisms with one ideal vertex and explicit constructions of their symmetry groups.

Findings

Complete classification of five-sided hyperbolic triangular prisms with one ideal vertex.

Explicit geometric descriptions in the upper half-space model.

Matrix generators for the associated reflection groups.

Abstract

In this paper, we classify all of the five-sided three-dimensional hyperbolic polyhedra with one ideal vertex, which have the shape of a triangular prism. We show how to find each such polyhedron in the upper half-space model by considering lines and circles in the plane. Finally, we give matrix generators in $\mathrm{PSL}_2(\mathbb C)$ for the orientation-preserving subgroup of each corresponding reflection group.