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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.

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