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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Compact hyperbolic 4-manifolds of small volume
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by Marston Conder and Colin Maclachlan PDF
Proc. Amer. Math. Soc. 133 (2005), 2469-2476 Request permission


We prove the existence of a compact non-orientable hyperbolic 4-manifold of volume $32\pi ^{2}/3$ and a compact orientable hyperbolic 4-manifold of volume $64\pi ^{2}/3$, obtainable from torsion-free subgroups of small index in the Coxeter group $[5,3,3,3]$. At the time of writing these are the smallest volumes of any known compact hyperbolic 4-manifolds.
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Additional Information
  • Marston Conder
  • Affiliation: Department of Mathematics, University of Auckland, Private Bag 92019 Auckland, New Zealand
  • MR Author ID: 50940
  • ORCID: 0000-0002-0256-6978
  • Email:
  • Colin Maclachlan
  • Affiliation: Department of Mathematical Sciences, University of Aberdeen, Aberdeen AB24 3UE, Scotland, United Kingdom
  • Email:
  • Received by editor(s): August 29, 2003
  • Published electronically: March 14, 2005
  • Additional Notes: This research was supported by grants from the N.Z. Marsden Fund (grant no. UOA 124) and the N.Z. Centres of Research Excellence Fund (grant no. UOA 201)
  • Communicated by: Linda Keen
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 133 (2005), 2469-2476
  • MSC (2000): Primary 57M50, 20F55; Secondary 51M20, 20B40
  • DOI:
  • MathSciNet review: 2138890