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2016 MATRIX Annals pp 323–347Cite as

Unravelling the Dodecahedral Spaces

Part of the MATRIX Book Series book series (MXBS,volume 1)

Abstract

The hyperbolic dodecahedral space of Weber and Seifert has a natural non-positively curved cubulation obtained by subdividing the dodecahedron into cubes. We show that the hyperbolic dodecahedral space has a 6-sheeted irregular cover with the property that the canonical hypersurfaces made up of the mid-cubes give a very short hierarchy. Moreover, we describe a 60-sheeted cover in which the associated cubulation is special. We also describe the natural cubulation and covers of the spherical dodecahedral space (aka Poincaré homology sphere).

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Acknowledgements

Research of the first author was supported by the Einstein Foundation (project Einstein Visiting Fellow Santos). Research of the second author was supported in part under the Australian Research Council’s Discovery funding scheme (project number DP160104502). The authors thank Schloss Dagstuhl Leibniz-Zentrum für Informatik and the organisers of Seminar 17072, where this work was completed.

The authors thank Daniel Groves and Alan Reid for their encouragement to write up these results, and the anonymous referee for some insightful questions and comments which triggered us to find a special cover.

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Authors and Affiliations

  1. Discrete Geometry Group, Mathematical Institute, Freie Universität Berlin, Arnimallee 2, 14195, Berlin, Germany

    Jonathan Spreer

  2. School of Mathematics and Statistics F07, The University of Sydney, Camperdown, NSW, 2006, Australia

    Stephan Tillmann

Authors
  1. Jonathan Spreer
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  2. Stephan Tillmann
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Correspondence to Jonathan Spreer .

Editor information

Editors and Affiliations

  1. University of Melbourne, Melbourne, Australia

    Prof. Jan de Gier

  2. University of Western Australia, Crawley, Australia

    Cheryl E. Praeger

  3. UCLA, Los Angeles, USA

    Prof. Terence Tao

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Spreer, J., Tillmann, S. (2018). Unravelling the Dodecahedral Spaces. In: de Gier, J., Praeger, C., Tao, T. (eds) 2016 MATRIX Annals. MATRIX Book Series, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-319-72299-3_17

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