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Hyperbolic Orbifolds of Minimal Volume

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Abstract

We provide a survey of hyperbolic orbifolds of minimal volume, starting with the results of Siegel in two dimensions and with the contributions of Gehring, Martin and others in three dimensions. For higher dimensions, we summarise some of the most important results, due to Belolipetsky, Emery and Hild, by discussing related features such as hyperbolic Coxeter groups, arithmeticity and consequences of Prasad’s volume, as well as canonical cusps, crystallography and packing densities. We also present some new results about volume minimisers in dimensions six and eight related to Bugaenko’s cocompact arithmetic Coxeter groups.

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Acknowledgments

The author would like to thank Vincent Emery for helpful discussions concerning Sec. 4.2.2. and Matthieu Jacquemet for his technical support. She was partially supported by Schweizerischer Nationalfonds 200020-144438.

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Authors and Affiliations

  1. Department of Mathematics, University of Fribourg, 1700 , Fribourg, Switzerland

    Ruth Kellerhals

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  1. Ruth Kellerhals
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Correspondence to Ruth Kellerhals.

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Communicated by Matti Vuorinen.

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Kellerhals, R. Hyperbolic Orbifolds of Minimal Volume. Comput. Methods Funct. Theory 14, 465–481 (2014). https://doi.org/10.1007/s40315-014-0074-y

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