Volume 10, issue 2 (2006)

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Infinitely many hyperbolic Coxeter groups through dimension 19

Daniel Allcock

Geometry & Topology 10 (2006) 737–758
 arXiv: 0903.0138
Abstract

We prove the following: there are infinitely many finite-covolume (resp. cocompact) Coxeter groups acting on hyperbolic space ${H}^{n}$ for every $n\le 19$ (resp. $n\le 6$). When $n=7$ or $8$, they may be taken to be nonarithmetic. Furthermore, for $2\le n\le 19$, with the possible exceptions $n=16$ and $17$, the number of essentially distinct Coxeter groups in ${H}^{n}$ with noncompact fundamental domain of volume$\le V$ grows at least exponentially with respect to $V$. The same result holds for cocompact groups for $n\le 6$. The technique is a doubling trick and variations on it; getting the most out of the method requires some work with the Leech lattice.

Keywords
Coxeter group, Coxeter polyhedron, Leech lattice, redoublable polyhedon
Mathematical Subject Classification 2000
Primary: 20F55
Secondary: 51M20, 51M10