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An ascending HNN extension of a free group inside $ \operatorname{SL}_{2} \mathbb{C}$

Authors: Danny Calegari and Nathan M. Dunfield
Journal: Proc. Amer. Math. Soc. 134 (2006), 3131-3136
MSC (2000): Primary 20E06; Secondary 57Mxx
Published electronically: May 18, 2006
MathSciNet review: 2231894
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Abstract: We give an example of a subgroup of $ \operatorname{SL}_{2}\mathbb{C}$ which is a strictly ascending HNN extension of a non-abelian finitely generated free group $ F$. In particular, we exhibit a free group $ F$ in $ \operatorname{SL}_{2}\mathbb{C}$ of rank $ 6$ which is conjugate to a proper subgroup of itself. This answers positively a question of Drutu and Sapir (2005). The main ingredient in our construction is a specific finite volume (non-compact) hyperbolic 3-manifold $ M$ which is a surface bundle over the circle. In particular, most of $ F$ comes from the fundamental group of a surface fiber. A key feature of $ M$ is that there is an element of $ \pi_1(M)$ in $ \operatorname{SL}_{2}\mathbb{C}$ with an eigenvalue which is the square root of a rational integer. We also use the Bass-Serre tree of a field with a discrete valuation to show that the group $ F$ we construct is actually free.

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Additional Information

Danny Calegari
Affiliation: Department of Mathematics 253-37, California Institute of Technology, Pasadena, California 91125

Nathan M. Dunfield
Affiliation: Department of Mathematics 253-37, California Institute of Technology, Pasadena, California 91125

Keywords: Ascending HNN extension, $\operatorname{SL}_{2}{\mathbb C}$, hyperbolic 3-manifold
Received by editor(s): February 18, 2005
Received by editor(s) in revised form: June 7, 2005
Published electronically: May 18, 2006
Additional Notes: Both authors were partially supported by the U.S. N.S.F. and the Sloan Foundation.
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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