A decomposition theorem for higher rank Coxeter groups

TL;DR

This paper proves a decomposition theorem for higher rank Coxeter groups, showing their defining graphs can be partitioned into subgraphs corresponding to hyperbolic or higher rank groups, and uses this to identify classes of hyperbolic Coxeter groups.

Contribution

It introduces a novel decomposition theorem for higher rank Coxeter groups and applies it to classify certain Coxeter graphs as hyperbolic.

Findings

Coxeter graphs decompose into subgraphs defining hyperbolic or higher rank groups

Identification of classes of Coxeter graphs that define hyperbolic groups

New structural insights into higher rank Coxeter groups

Abstract

In this paper, we show that any Coxeter graph which defines a higher rank Coxeter group must have disjoint induced subgraphs each of which defines a hyperbolic or higher rank Coxeter group. We then use this result to demonstrate several classes of Coxeter graphs which define hyperbolic Coxeter groups.