## Abstract

We give an example of a subgroup of SL _{2} 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 SL _{2} 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 π _{1} (M) in SL _{2} 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.

Original language | English (US) |
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Pages (from-to) | 3131-3136 |

Number of pages | 6 |

Journal | Proceedings of the American Mathematical Society |

Volume | 134 |

Issue number | 11 |

DOIs | |

State | Published - Nov 2006 |

Externally published | Yes |

## Keywords

- Ascending HNN extension
- Hyperbolic 3-manifold
- SL ℂ

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics

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