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Generating the Möbius group with involution conjugacy classes

Authors: Ara Basmajian and Karan Puri
Journal: Proc. Amer. Math. Soc. 140 (2012), 4011-4016
MSC (2010): Primary 51M10; Secondary 30F40
Published electronically: February 29, 2012
MathSciNet review: 2944740
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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\uml 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\uml ob}(n)$ if $ k$ is odd and the $ S_k$ word length of $ \text {M\uml ob}^+(n)$ if $ k$ is even. As a consequence, for a fixed codimension $ k$, the length of $ \text {M\uml ob}^{+}(n)$ with respect to $ S_k$, $ k$ even, grows linearly with $ n$, with the same statement holding for $ \text {M\uml ob}(n)$ in the odd case. Moreover, the percentage of involution conjugacy classes for which $ \text {M\uml ob}^{+}(n)$ has length two approaches zero as $ n$ approaches infinity.

References [Enhancements On Off] (What's this?)

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Additional Information

Ara Basmajian
Affiliation: Department of Mathematics, Graduate Center and Hunter College, CUNY, New York, New York 10016

Karan Puri
Affiliation: Department of Mathematics, Queensborough Community College, CUNY, Bayside, New York 11364

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
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.