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# Growth Rates of 3-dimensional Hyperbolic Coxeter Groups are Perron Numbers

Published:2017-12-05
Printed: Jun 2018
• Tomoshige Yukita,
Department of Mathematics, School of Education, Waseda University, Tokyo, Japan
 Format: LaTeX MathJax PDF

## Abstract

In this paper we consider the growth rates of 3-dimensional hyperbolic Coxeter polyhedra with at least one dihedral angle of the form $\frac{\pi}{k}$ for an integer $k\geq{7}$. Combining a classical result by Parry with a previous result of ours, we prove that the growth rates of 3-dimensional hyperbolic Coxeter groups are Perron numbers.
 Keywords: Coxeter group, growth function, growth rate, Perron number
 MSC Classifications: 20F55 - Reflection and Coxeter groups [See also 22E40, 51F15] 20F65 - Geometric group theory [See also 05C25, 20E08, 57Mxx]

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