Summary: Infinitely many hyperbolic Coxeter groups
through dimension 19
Department of Mathematics
University of Texas at Austin
Austin, TX 78712
We prove the following: there are infinitely many finite-covolume (resp. cocom-
pact) Coxeter groups acting on hyperbolic space Hn for every n 19 (resp.
n 6). When n = 7 or 8, they may be taken to be nonarithmetic. Further-
more, for 2 n 19, with the possible exceptions n = 16 and 17, the number
of essentially distinct Coxeter groups in Hn with noncompact fundamental do-
main of volume V grows at least exponentially with respect to V . The same
result holds for cocompact groups for n 6. The technique is a doubling trick
and variations on it; getting the most out of the method requires some work
with the Leech lattice.
AMS Classification numbers Primary: 20F55