Volume 14, issue 5 (2014)

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The growth function of Coxeter dominoes and $2$–Salem numbers

Yuriko Umemoto

Algebraic & Geometric Topology 14 (2014) 2721–2746
Abstract

By the results of Cannon, Wagreich and Parry, it is known that the growth rate of a cocompact Coxeter group in ${ℍ}^{2}$ and ${ℍ}^{3}$ is a Salem number. Kerada defined a $j$–Salem number, which is a generalization of Salem numbers. In this paper, we realize infinitely many $2$–Salem numbers as the growth rates of cocompact Coxeter groups in ${ℍ}^{4}$. Our Coxeter polytopes are constructed by successive gluing of Coxeter polytopes, which we call Coxeter dominoes.

Keywords
hyperbolic Coxeter group, growth rate, $2$–Salem number
Mathematical Subject Classification 2010
Primary: 20F55
Secondary: 20F65, 11K16
Publication
Revised: 4 September 2013
Accepted: 10 September 2013
Published: 5 November 2014
Authors
 Yuriko Umemoto Department of Mathematics Osaka City University 3-3-138, Sugimoto, Sumiyoshi-ku Osaka 558-8585 Japan