An ascending HNN extension of a free group inside SL_2 ℂ
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 π1(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.
Additional Information© 2006 American Mathematical Society. The copyright for this article reverts to public domain after 28 years from publication. Received by the editors February 18, 2005 and, in revised form, June 7, 2005. Article electronically published on May 18, 2006. Communicated by Ronald A. Fintushel. Both authors were partially supported by the U.S. National Science Foundation (grant #DMS-0405491) and the Sloan Foundation.
Published - CALpams06.pdf