An ascending HNN extension of a free group inside $\operatorname {SL}_{2} \mathbb {C}$

Abstract:

We give an example of a subgroup of $\operatorname {SL}_{2}\mathbb {C}$ 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 $\operatorname {SL}_{2}\mathbb {C}$ 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 $\pi _1(M)$ in $\operatorname {SL}_{2}\mathbb {C}$ 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.