Non-crystallographic tail-triangle C-groups of rank 4 and interlacing number 2

TL;DR

This paper investigates non-crystallographic rank 4 Coxeter groups using modular reduction over primes in the golden ratio's quadratic integer ring, proving they form C-groups and classifying their reflection group structures.

Contribution

It introduces a novel application of modular reduction to non-crystallographic Coxeter groups of rank 4 and classifies the resulting finite reflection groups.

Findings

All reduced groups are proven to be C-groups.

Classification of reduced groups as reflection groups over finite fields.

Applicable for primes in the quadratic integer ring .

Abstract

This work applies the modular reduction technique to the Coxeter group of rank 4 having a star diagram with labels 5, 3, and $k = 3, 4, 5, \text{ or } 6$. As moduli, we use the primes in the quadratic integer ring $\mathbb{Z}[\tau]$, where $\tau = \frac{1 + \sqrt{5}}{2}$, the golden ratio. We prove that each reduced group is a C-group, regardless of the prime used in the reduction. We also classify each reduced group as a reflection group over a finite field, whenever applicable.