Reflective and quasi-reflective Bianchi groups

TL;DR

This paper provides a complete classification of reflective and quasi-reflective subgroups within Bianchi groups, which are important in hyperbolic geometry and group theory.

Contribution

It offers the first comprehensive classification of reflective and quasi-reflective Bianchi groups and their extensions, advancing understanding in hyperbolic reflection groups.

Findings

Complete classification of reflective Bianchi groups

Identification of quasi-reflective subgroups

Extensions of Bianchi groups analyzed

Abstract

A discrete subgroup of the group of isometries of the hyperbolic space is called reflective if up to a finite index it is generated by reflections in hyperplanes. The main result of this paper is a complete classification of the reflective (and quasi-reflective) subgroups among the Bianchi groups and their extensions.