2016•arXiv (Cornell University)Open access

On the growth rates of cofinite 3-dimensional Coxeter groups some of whose dihedral angles are $\frac{\pi}{m}$ for $m\geq{7}$

Tomoshige Yukita

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Abstract

We studied the growth rates of cofinite 3-dimensional hyperbolic Coxeter groups whose dihedral angles are of the form $\frac{\pi}{m}$ for $m=2,3,4,5,6$ and showed the growth rates are always Perron numbers in the paper [20]. In this paper, we consider the growth rates of cofinite 3-dimensional hyperbolic Coxeter groups some of whose dihedral angles are $\frac{\pi}{m}$ for $m\geq{7}$ and prove that their growth rates are always Perron numbers. Moreover, by combining with the result in [20], we conclude that the growth rates of cofinite 3-dimensional hyperbolic Coxeter groups are always Perron numbers in general.

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We studied the growth rates of cofinite 3-dimensional hyperbolic Coxeter groups whose dihedral angles are of the form $\frac{\pi}{m}$ for $m=2,3,4,5,6$ and showed the growth rates are always Perron numbers in the paper [20]. In this paper, we consider the growth rates of cofinite 3-dimensional hyperbolic Coxeter groups some of whose dihedral angles are $\frac{\pi}{m}$ for $m\geq{7}$ and prove that their growth rates are always Perron numbers. Moreover, by combining with the result in [20], we conclude that the growth rates of cofinite 3-dimensional hyperbolic Coxeter groups are always Perron numbers in general.

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Available abstract

We studied the growth rates of cofinite 3-dimensional hyperbolic Coxeter groups whose dihedral angles are of the form $\frac{\pi}{m}$ for $m=2,3,4,5,6$ and showed the growth rates are always Perron numbers in the paper [20]. In this paper, we consider the growth rates of cofinite 3-dimensional hyperbolic Coxeter groups some of whose dihedral angles are $\frac{\pi}{m}$ for $m\geq{7}$ and prove that their growth rates are always Perron numbers. Moreover, by combining with the result in [20], we conclude that the growth rates of cofinite 3-dimensional hyperbolic Coxeter groups are always Perron numbers in general.

Key concepts: Coxeter group, Mathematics, Dihedral group, Dihedral angle, Combinatorics, Pure mathematics, Group (periodic table), Physics

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