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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
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 Hn for every n 19 (resp. n 6). When n = 7 or 8, they may be taken to be nonarithmetic. Furthermore, for 2 n 19, with the possible exceptions n = 16 and 17, the number of essentially distinct Coxeter groups in Hn with noncompact fundamental domain 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.

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