Maximal holonomy of infra-nilmanifolds with -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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be the quaternionic Heisenberg group of real dimension and let denote the maximal order of the holonomy groups of all infra-nilmanifolds with -geometry. We prove that . As an application, by applying Kim and Parker's result, we obtain that the minimum volume of a -dimensional quaternionic hyperbolic manifold with cusps is at least

**1.**Harold Brown, Rolf Bülow, Joachim Neubüser, Hans Wondratschek, and Hans Zassenhaus,*Crystallographic groups of four-dimensional space*, Wiley-Interscience [John Wiley & Sons], New York-Chichester-Brisbane, 1978. Wiley Monographs in Crystallography. MR**0484179****2.**Jacek Cygan,*A tangential convergence for bounded harmonic functions on a rank one symmetric space*, Trans. Amer. Math. Soc.**265**(1981), no. 2, 405–418. MR**610957**, 10.1090/S0002-9947-1981-0610957-4**3.**Ku Yong Ha, Jang Hyun Jo, Seung Won Kim, and Jong Bum Lee,*Classification of free actions of finite groups on the 3-torus*, Topology Appl.**121**(2002), no. 3, 469–507. MR**1909004**, 10.1016/S0166-8641(01)00090-6**4.**Sa’ar Hersonsky and Frédéric Paulin,*On the volumes of complex hyperbolic manifolds*, Duke Math. J.**84**(1996), no. 3, 719–737. MR**1408542**, 10.1215/S0012-7094-96-08422-7**5.**I. Kim and J. R. Parker, Geometry of quaternionic hyperbolic manifolds,*Math. Proc. Cambridge Philos. Soc.***135**(2003), 291-310.**6.**K. B. Lee and Frank Raymond,*Topological, affine and isometric actions on flat Riemannian manifolds*, J. Differential Geom.**16**(1981), no. 2, 255–269. MR**638791****7.**Kyung Bai Lee, Joon Kook Shin, and Shoji Yokura,*Free actions of finite abelian groups on the 3-torus*, Topology Appl.**53**(1993), no. 2, 153–175. MR**1247674**, 10.1016/0166-8641(93)90134-Y**8.**Kyung Bai Lee and Andrzej Szczepański,*Maximal holonomy of almost Bieberbach groups for 𝐻𝑒𝑖𝑠₅*, Geom. Dedicata**87**(2001), no. 1-3, 167–180. MR**1866847**, 10.1023/A:1012032913680**9.**J. Wolf, Spaces of Constant Curvature, 5th ed., Publish or Perish, Wilmington, 1984.**10.**S. Wolfram, Mathematica, Wolfram Research, 1993.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
20H15,
20F18,
20E99,
53C55

Retrieve articles in all journals with MSC (2000): 20H15, 20F18, 20E99, 53C55

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:
https://doi.org/10.1090/S0002-9947-04-03511-1

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