  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. The
main ingredient in our construction is a specific finite volume (noncompact)
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 its fundamental group 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.
