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

 

On 2-Generator Subgroups of $SO(3)$


Authors: Charles Radin and Lorenzo Sadun
Journal: Trans. Amer. Math. Soc. 351 (1999), 4469-4480
MSC (1991): Primary 51F25, 52C22
Published electronically: June 10, 1999
MathSciNet review: 1624202
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Abstract: We classify all subgroups of $SO(3)$ that are generated by two elements, each a rotation of finite order, about axes separated by an angle that is a rational multiple of $\pi$. In all cases we give a presentation of the subgroup. In most cases the subgroup is the free product, or the amalgamated free product, of cyclic groups or dihedral groups. The relations between the generators are all simple consequences of standard facts about rotations by $\pi$ and $\pi/2$. Embedded in the subgroups are explicit free groups on 2 generators, as used in the Banach-Tarski paradox.


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Additional Information

Charles Radin
Affiliation: Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712
Email: radin@math.utexas.edu

Lorenzo Sadun
Affiliation: Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712
Email: sadun@math.utexas.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02397-1
PII: S 0002-9947(99)02397-1
Received by editor(s): October 13, 1997
Published electronically: June 10, 1999
Additional Notes: Research of the first author was supported in part by NSF Grant No. DMS-9531584.
Research of the second author was supported in part by NSF Grant No. DMS-9626698.
Article copyright: © Copyright 1999 American Mathematical Society