  We study divergence and thickness for general Coxeter groups $W$. We first
characterise linear divergence, and show that if $W$ has superlinear divergence
then its divergence is at least quadratic. We then formulate a computable
combinatorial invariant, hypergraph index, for arbitrary Coxeter systems
$(W,S)$. This generalises Levcovitz's definition for the right-angled case. We
prove that if $(W,S)$ has finite hypergraph index $h$, then $W$ is (strongly
algebraically) thick of order at most $h$, hence has divergence bounded above
by a polynomial of degree $h+1$. We conjecture that these upper bounds on the
order of thickness and divergence are in fact equalities, and we prove our
conjecture for certain families of Coxeter groups. These families are obtained
by a new construction which, given any right-angled Coxeter group, produces
infinitely many examples of non-right-angled Coxeter systems with the same
hypergraph index. Finally, we give an upper bound on the hypergraph index of
any Coxeter system $(W,S)$, and hence on the divergence of $W$, in terms of,
unexpectedly, the topology of its associated Dynkin diagram.
