An ascending HNN extension of a free group inside SL ₂ℂ

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.