  We prove the following: there are infinitely many finite-covolume (resp.
cocompact) Coxeter groups acting on hyperbolic space H^n for every n < 20
(resp. n < 7). When n=7 or 8, they may be taken to be nonarithmetic.
Furthermore, for 1 < n < 20, with the possible exceptions n=16 and 17, the
number of essentially distinct Coxeter groups in H^n with noncompact
fundamental domain of volume less than or equal to V grows at least
exponentially with respect to V. The same result holds for cocompact groups for
n < 7. The technique is a doubling trick and variations on it; getting the most
out of the method requires some work with the Leech lattice.
