Non-linear ascending HNN extensions of free groups

is called an ascending HNN extension of G (or the mapping torus of the endomorphism φ). In particular, the ascending HNN extensions of free groups of finite rank are simply the groups given by presentations 〈x1, ..., xn, t | txit−1 = wi, i = 1, ..., n〉, where w1, ..., wn are words generating a free subgroup of rank n. In [BS], Borisov and Sapir proved that all ascending HNN extensions of linear groups are residually finite. In particular the ascending HNN extensions of free groups are residually finite. After [BS], the question of linearity of these groups became very interesting. This question is especially interesting for ascending HNN extensions of free groups because most of these groups are hyperbolic [I.Kap], and because this class of groups contains most 1-related groups [BS]. Let H = HNNφ(Fn) be an ascending HNN extension of a free group. If n = 1 then H is a Baumslag-Solitar group BS(n, 1), so it is inside SL2(Q). If n = 2 and φ is an automorphism then the linearity of H follows from the linearity of Aut(F2). The linearity of Aut(F2) follows from two facts: Dyer, Formanek and Grossman [DFG] reduced the linearity of Aut(F2) to the linearity of the braid group B4; the linearity of B4 was proved by Krammer [Kra]. In this note, we prove that in the case when φ is not an automorphism, the situation is different.