Cofinite hyperbolic Coxeter groups, minimal growth rate and Pisot numbers

Kellerhals, Ruth¹

¹Department of Mathematics, University of Fribourg, CH-1700 Fribourg, Switzerland

Algebraic and Geometric Topology, Tome 13 (2013) no. 2, pp. 1001-1025

Résumé

By a result of R Meyerhoff, it is known that among all cusped hyperbolic 3–orbifolds the quotient of ℍ3 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.

Détail

DOI : 10.2140/agt.2013.13.1001

Keywords: Hyperbolic Coxeter group, cusp, growth rates, Pisot number

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