Hyperbolic tessellations associated to Bianchi groups

TL;DR

This paper explores hyperbolic tessellations derived from Bianchi groups by analyzing the structure of ideal polytopes in 3D hyperbolic space associated with imaginary quadratic fields.

Contribution

It provides explicit computations of the polytope structures for various imaginary quadratic fields, advancing understanding of Bianchi groups and hyperbolic tessellations.

Findings

Explicit polytope structures for multiple imaginary quadratic fields

Enhanced understanding of hyperbolic tessellations related to Bianchi groups

Foundation for further geometric and algebraic investigations

Abstract

Let F/Q be number field. The space of positive definite binary Hermitian forms over F form an open cone in a real vector space. There is a natural decomposition of this cone into subcones, which descend give rise to hyperbolic tessellations of 3-dimensional hyperbolic space by ideal polytopes. We compute the structure of these polytopes for a range of imaginary quadratic fields.