Infinitely many hyperbolic Coxeter groups through dimension 19

Allcock, Daniel

doi:10.2140/gt.2006.10.737

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Infinitely many hyperbolic Coxeter groups through dimension 19

Daniel Allcock

Geometry & Topology 10 (2006) 737–758

DOI: 10.2140/gt.2006.10.737

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 $n \leq 19$ (resp. $n \leq 6$ ). When $n = 7$ or $8$ , they may be taken to be nonarithmetic. Furthermore, for $2 \leq n \leq 19$ , with the possible exceptions $n = 16$ and $17$ , the number of essentially distinct Coxeter groups in $H^{n}$ with noncompact fundamental domain of volume $\leq V$ grows at least exponentially with respect to $V$ . The same result holds for cocompact groups for $n \leq 6$ . 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

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Publication

Received: 28 April 2005

Revised: 8 August 2005

Accepted: 16 January 2006

Published: 11 July 2006

Proposed: Martin Bridson

Seconded: Benson Farb, Walter Neumann

Authors

Daniel Allcock
Department of Mathematics University of Texas at Austin Austin, TX 78712 USA
http://www.math.utexas.edu/~allcock

Volume 10, issue 2 (2006)