Generating the Möbius group with involution conjugacy classes
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- by Ara Basmajian and Karan Puri
- Proc. Amer. Math. Soc. 140 (2012), 4011-4016
- DOI: https://doi.org/10.1090/S0002-9939-2012-11253-5
- Published electronically: February 29, 2012
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Abstract:
A $k$-involution is an involution with a fixed point set of codimension $k$. The conjugacy class of such an involution, denoted $S_k$, generates $\text {M\"ob}(n)$(the group of isometries of hyperbolic $n$-space) if $k$ is odd and its orientation-preserving subgroup if $k$ is even. In this paper, we supply effective lower and upper bounds for the $S_k$ word length of $\text {M\"ob}(n)$ if $k$ is odd and the $S_k$ word length of $\text {M\"ob}^+(n)$ if $k$ is even. As a consequence, for a fixed codimension $k$, the length of $\text {M\"ob}^{+}(n)$ with respect to $S_k$, $k$ even, grows linearly with $n$, with the same statement holding for $\text {M\"ob}(n)$ in the odd case. Moreover, the percentage of involution conjugacy classes for which $\text {M\"ob}^{+}(n)$ has length two approaches zero as $n$ approaches infinity.References
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Bibliographic Information
- Ara Basmajian
- Affiliation: Department of Mathematics, Graduate Center and Hunter College, CUNY, New York, New York 10016
- Email: abasmajian@gc.cuny.edu
- Karan Puri
- Affiliation: Department of Mathematics, Queensborough Community College, CUNY, Bayside, New York 11364
- Email: kpuri@qcc.cuny.edu
- Received by editor(s): August 15, 2010
- Received by editor(s) in revised form: April 19, 2011
- Published electronically: February 29, 2012
- Additional Notes: The first author was supported in part by PSC-CUNY Grant 627 14-00 40
- Communicated by: Michael Wolf
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 4011-4016
- MSC (2010): Primary 51M10; Secondary 30F40
- DOI: https://doi.org/10.1090/S0002-9939-2012-11253-5
- MathSciNet review: 2944740