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Cofinite hyperbolic Coxeter groups, minimal growth rate and Pisot numbers

Ruth Kellerhals

Algebraic & Geometric Topology 13 (2013) 1001–1025
Abstract

By a result of R Meyerhoff, it is known that among all cusped hyperbolic 3–orbifolds the quotient of 3 by the tetrahedral Coxeter group (3,3,6) has minimal volume. We prove that the group (3,3,6) has smallest growth rate among all non-cocompact cofinite hyperbolic Coxeter groups, and that it is as such unique. This result extends to three dimensions some work of W Floyd who showed that the Coxeter triangle group (3,) has minimal growth rate among all non-cocompact cofinite planar hyperbolic Coxeter groups. In contrast to Floyd’s result, the growth rate of the tetrahedral group (3,3,6) is not a Pisot number.

Keywords
Hyperbolic Coxeter group, cusp, growth rates, Pisot number
Mathematical Subject Classification 2000
Primary: 20F55
Secondary: 22E40, 51F15
References
Publication
Received: 12 July 2012
Revised: 30 November 2012
Accepted: 5 December 2012
Published: 5 April 2013
Authors
Ruth Kellerhals
Department of Mathematics
University of Fribourg
CH-1700 Fribourg
Switzerland