The growth rates of ideal coxeter polyhedra in hyperbolic 3-space

Jun Nonaka; Ruth Kellerhals

doi:10.3836/tjm/1502179234

The growth rates of ideal coxeter polyhedra in hyperbolic 3-space

Jun Nonaka, Ruth Kellerhals

Waseda University Senior High School

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

Abstract

In [7], Kellerhals and Perren conjectured that the growth rates of the reflection groups given by compact hyperbolic Coxeter polyhedra are always Perron numbers. We prove that this conjecture holds in the context of ideal Coxeter polyhedra in H³. Our methods allow us to bound from below the growth rates of composite ideal Coxeter polyhedra by the growth rates of its ideal Coxeter polyhedral constituents.

Original language	English
Pages (from-to)	379-391
Number of pages	13
Journal	Tokyo Journal of Mathematics
Volume	40
Issue number	2
DOIs	https://doi.org/10.3836/tjm/1502179234
Publication status	Published - 2017 Dec

Keywords

Cusp
Growth rate
Hyperbolic Coxeter polyhedron
Perron number

ASJC Scopus subject areas

Mathematics(all)

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10.3836/tjm/1502179234

Cite this

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title = "The growth rates of ideal coxeter polyhedra in hyperbolic 3-space",

abstract = "In [7], Kellerhals and Perren conjectured that the growth rates of the reflection groups given by compact hyperbolic Coxeter polyhedra are always Perron numbers. We prove that this conjecture holds in the context of ideal Coxeter polyhedra in H3. Our methods allow us to bound from below the growth rates of composite ideal Coxeter polyhedra by the growth rates of its ideal Coxeter polyhedral constituents.",

keywords = "Cusp, Growth rate, Hyperbolic Coxeter polyhedron, Perron number",

author = "Jun Nonaka and Ruth Kellerhals",

note = "Funding Information: ACKNOWLEDGMENT. The first author would like to show his greatest appreciation to Professor Hiroyasu Izeki who provided valuable comments and suggestions. He would also like to thank Dr. Alexander Kolpakov for valuable comments. The second author was partially supported by Schweizerischer Nationalfonds 200020–156104. The authors would also like to thank the referee for helpful comments.",

year = "2017",

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doi = "10.3836/tjm/1502179234",

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T1 - The growth rates of ideal coxeter polyhedra in hyperbolic 3-space

AU - Nonaka, Jun

AU - Kellerhals, Ruth

N1 - Funding Information: ACKNOWLEDGMENT. The first author would like to show his greatest appreciation to Professor Hiroyasu Izeki who provided valuable comments and suggestions. He would also like to thank Dr. Alexander Kolpakov for valuable comments. The second author was partially supported by Schweizerischer Nationalfonds 200020–156104. The authors would also like to thank the referee for helpful comments.

PY - 2017/12

Y1 - 2017/12

N2 - In [7], Kellerhals and Perren conjectured that the growth rates of the reflection groups given by compact hyperbolic Coxeter polyhedra are always Perron numbers. We prove that this conjecture holds in the context of ideal Coxeter polyhedra in H3. Our methods allow us to bound from below the growth rates of composite ideal Coxeter polyhedra by the growth rates of its ideal Coxeter polyhedral constituents.

AB - In [7], Kellerhals and Perren conjectured that the growth rates of the reflection groups given by compact hyperbolic Coxeter polyhedra are always Perron numbers. We prove that this conjecture holds in the context of ideal Coxeter polyhedra in H3. Our methods allow us to bound from below the growth rates of composite ideal Coxeter polyhedra by the growth rates of its ideal Coxeter polyhedral constituents.

KW - Cusp

KW - Growth rate

KW - Hyperbolic Coxeter polyhedron

KW - Perron number

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JO - Tokyo Journal of Mathematics

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The growth rates of ideal coxeter polyhedra in hyperbolic 3-space

Abstract

Keywords

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