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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Maximal holonomy of infra-nilmanifolds with $2$-dimensional quaternionic Heisenberg geometry


Authors: Ku Yong Ha, Jong Bum Lee and Kyung Bai Lee
Journal: Trans. Amer. Math. Soc. 357 (2005), 355-383
MSC (2000): Primary 20H15, 20F18, 20E99, 53C55
Published electronically: May 10, 2004
MathSciNet review: 2098099
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Abstract: Let $\mathbf{H}_{4n-1}(\mathbb{H} )$ be the quaternionic Heisenberg group of real dimension $4n-1$ and let $I_{n}$ denote the maximal order of the holonomy groups of all infra-nilmanifolds with $\mathbf{H}_{4n-1}(\mathbb{H} )$-geometry. We prove that $I_2=48$. As an application, by applying Kim and Parker's result, we obtain that the minimum volume of a $2$-dimensional quaternionic hyperbolic manifold with $k$ cusps is at least $\frac{\sqrt{2}k}{720}.$


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Additional Information

Ku Yong Ha
Affiliation: Department of Mathematics, Sogang University, Seoul 121-742, Korea
Email: kyha@sogang.ac.kr

Jong Bum Lee
Affiliation: Department of Mathematics, Sogang University, Seoul 121-742, Korea
Email: jlee@sogang.ac.kr

Kyung Bai Lee
Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019
Email: kb_lee@math.ou.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03511-1
PII: S 0002-9947(04)03511-1
Keywords: Almost Bieberbach group, holonomy group, quaternionic Heisenberg group, quaternionic hyperbolic manifold
Received by editor(s): March 23, 2003
Received by editor(s) in revised form: August 25, 2003
Published electronically: May 10, 2004
Additional Notes: This research was supported in part by grant No. R01-1999-000-00002-0(2002) from the interdisciplinary Research program, and by grant No. R14-2002-044-01002-0(2002) from ABRL of KOSEF
This work was done while the second-named author was visiting the Department of Mathematics at the University of Oklahoma. He expresses his sincere thanks for their hospitality.
Article copyright: © Copyright 2004 American Mathematical Society