An ascending HNN extension of a free group inside

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 which is a strictly ascending HNN extension of a non-abelian finitely generated free group . In particular, we exhibit a free group in of rank 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 which is a surface bundle over the circle. In particular, most of comes from the fundamental group of a surface fiber. A key feature of is that there is an element of in 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 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

Email:
dannyc@caltech.edu

**Nathan M. Dunfield**

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

Email:
dunfield@caltech.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-06-08398-5

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.