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# The Minimal Growth Rate of Cocompact Coxeter Groups in Hyperbolic 3-space

Published:2013-02-13
Printed: Apr 2014
• Ruth Kellerhals,
Department of Mathematics, University of Fribourg, Fribourg Pérolles, Switzerland
• Alexander Kolpakov,
Department of Mathematics, University of Fribourg, Fribourg Pérolles, Switzerland
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## Abstract

Due to work of W. Parry it is known that the growth rate of a hyperbolic Coxeter group acting cocompactly on ${\mathbb H^3}$ is a Salem number. This being the arithmetic situation, we prove that the simplex group (3,5,3) has smallest growth rate among all cocompact hyperbolic Coxeter groups, and that it is as such unique. Our approach provides a different proof for the analog situation in ${\mathbb H^2}$ where E. Hironaka identified Lehmer's number as the minimal growth rate among all cocompact planar hyperbolic Coxeter groups and showed that it is (uniquely) achieved by the Coxeter triangle group (3,7).
 Keywords: hyperbolic Coxeter group, growth rate, Salem number
 MSC Classifications: 20F55 - Reflection and Coxeter groups [See also 22E40, 51F15] 22E40 - Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx] 51F15 - Reflection groups, reflection geometries [See also 20H10, 20H15; for Coxeter groups, see 20F55]

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