Half-turn linked pairs of isometries of hyperbolic 4-space

TL;DR

This paper develops a comprehensive geometric theory for factorizing isometries of hyperbolic 4-space, focusing on linked pairs expressed via involutions, and provides conditions and constructions for such pairs.

Contribution

It introduces a new geometric framework involving hyperbolic pencils and twisting planes, and characterizes when pairs of isometries are linked by half-turns in hyperbolic 4-space.

Findings

Complete characterization of linked pairs of isometries

Conditions for factorization via half-turns

Constructed examples of linked pairs in 4D hyperbolic space

Abstract

In this paper we develop a complete theory of factorization for isometries of hyperbolic 4-space. Of special interest is the case where a pair of isometries is linked, that is, when a pair of isometries can be expressed each as compositions of two involutions, one of which is common to both isometries. Here we develop a new theory of hyperbolic pencils and twisting planes involving a new geometric construction, their half-turn banks. This enables us to give complete results about each of the pair-types of isometries and their simultaneous factorization by half-turns. That is, we provide geometric conditions for each such pair to be linked by half-turns. The main result gives a necessary and sufficient condition for any given pair of isometries to be linked. We also provide a procedure for constructing a half-turn linked pair of isometries of $\mathbb{H}^4$ that do not restrict to lower dimensions, yielding an example that gives a negative answer to a question raised by Ara Basmajian and Karan Puri.