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

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- by Danny Calegari and Nathan M. Dunfield PDF
- Proc. Amer. Math. Soc.
**134**(2006), 3131-3136 Request permission

## 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.## References

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## Additional Information

**Danny Calegari**- Affiliation: Department of Mathematics 253-37, California Institute of Technology, Pasadena, California 91125
- MR Author ID: 605373
- Email: dannyc@caltech.edu
**Nathan M. Dunfield**- Affiliation: Department of Mathematics 253-37, California Institute of Technology, Pasadena, California 91125
- MR Author ID: 341957
- ORCID: 0000-0002-9152-6598
- Email: dunfield@caltech.edu
- 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
- © Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**134**(2006), 3131-3136 - MSC (2000): Primary 20E06; Secondary 57Mxx
- DOI: https://doi.org/10.1090/S0002-9939-06-08398-5
- MathSciNet review: 2231894