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

 

Arithmeticity, discreteness and volume


Authors: F. W. Gehring, C. Maclachlan, G. J. Martin and A. W. Reid
Journal: Trans. Amer. Math. Soc. 349 (1997), 3611-3643
MSC (1991): Primary 30F40, 20H10, 57N10
MathSciNet review: 1433117
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Abstract: We give an arithmetic criterion which is sufficient to imply the discreteness of various two-generator subgroups of $\textup {PSL}(2,\mathbf {c})$. We then examine certain two-generator groups which arise as extremals in various geometric problems in the theory of Kleinian groups, in particular those encountered in efforts to determine the smallest co-volume, the Margulis constant and the minimal distance between elliptic axes. We establish the discreteness and arithmeticity of a number of these extremal groups, the associated minimal volume arithmetic group in the commensurability class and we study whether or not the axis of a generator is simple. We then list all ``small'' discrete groups generated by elliptics of order $2$ and $n$, $n=3,4,5,6,7$.


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Additional Information

F. W. Gehring
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1003
Email: fgehring@math.lsa.umich.edu

C. Maclachlan
Affiliation: Department of Mathematics, University of Aberdeen, Aberdeen, Scotland
Email: cmac@math.aberdeen.ac.uk

G. J. Martin
Affiliation: Department of Mathematics, University of Auckland, Auckland, New Zealand and Australian National University, Canberra, Australia
Email: martin@math.auckland.ac.nz

A. W. Reid
Affiliation: Department of Mathematics, University of Texas, Austin, Texas 78712
Email: areid@math.utexas.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-97-01989-2
PII: S 0002-9947(97)01989-2
Received by editor(s): January 4, 1996
Additional Notes: Research supported in part by grants from the U. S. National Science Foundation, the N.Z. Foundation of Research, Science and Technology, the Australian Research Council, the U.K. Royal Society and the U.K. Scientific and Engineering Research Council. We wish also to thank the University of Texas at Austin and the University of Auckland for their hospitality during part of this work. We are grateful to D. J. Lewis who gave us a proof for Theorem 5.12 and to K. N. Jones who computed the co-volumes that appear in Table 11.
Article copyright: © Copyright 1997 American Mathematical Society