A Decomposition Theorem for Higher Rank Coxeter Groups

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 where the groups in question do not necessarily act cocompactly or cofinitely. We then use this result to demonstrate several classes of Coxeter graphs which define hyperbolic Coxeter groups and to produce structure theorems for hyperbolic Coxeter groups.