Examine the divergence and thickness of the general Coxeter group $W$. We first characterize the linear divergence and show that if $W$ has a superlinear divergence, its divergence is at least quadratic. Next, we formulate a computable combinatorial invariant, the hypergraph index, for any Coxeter system $(W,S)$. This is a generalization of Levcovitz’s definition for the right angle case. If $(W,S)$ has a finite hypergraph index $h$, then $W$ is (strongly algebraically) at most thick of degrees $h$, and thus can be overlaid by a polynomial of degree $h+. We prove that there is a divergence limited to $1. We speculate that these upper bounds on the order of thickness and divergence are in fact equal, and prove this conjecture for a particular family of Coxeter groups. These families are obtained by a new construction that, given any orthogonal Coxetor group, generates infinitely many examples of non-orthogonal Coxetor systems with the same hypergraph index. Finally, we give an upper bound on the hypergraph index of any Coxeter system $(W,S)$, and thus the divergence of $W$, unexpectedly with respect to the topology of the associated Dynkin diagram.
