Polyhedrons and PBIBDs from hyperbolic manifolds

TL;DR

This paper explores how quotienting hyperbolic plane and space tilings by group actions produces polyhedra and cellulations with notable symmetries and incidence properties, contributing to geometric and combinatorial understanding.

Contribution

It introduces a method to generate polyhedra and cellulations from hyperbolic tilings using group actions, revealing new symmetric structures and incidence relations.

Findings

Construction of polyhedra with hyperbolic symmetry

Identification of PBIBDs from hyperbolic quotients

New geometric structures with interesting symmetries

Abstract

By taking quotients of a certain tiling of hyperbolic plane / space by certain group actions, we obtain geometric polyhedra / cellulations with interesting symmetries and incidence structure.