Finite generation of intersections of free subgroups

Question

Let $G$ be a finitely generated group, and let $F_1, F_2$ be two subgroups of $G$ which are free of finite rank at least 2. I am wondering what conditions can be placed on $G$ so that $F_1\cap F_2$ is finitely generated. For example:

• It is a classical result of Howson that if $G$ is itself free then $F_1\cap F_2$ will be finitely generated.
• I. Kapovich (Subgroup properties of fully residually free groups. Trans. Amer. Math. Soc. 354 (2002), no. 1, 335–362.) extended Howson's result to limit groups and certain hyperbolic groups. However, in these cases all finitely generated subgroups have pairwise intersections which are finitely generated. So this is stronger than the situation I am asking about.

On the other hand, if we set $G$ to be a free product of two free groups amalgamated across a non-finitely generated subgroup, so $G=F_1\ast_CF_2$ where $C$ is not finitely generated, then $F_1\cap F_2=C$ is not finitely generated. ($G=F(a, b)\times\mathbb{Z}$ gives a finitely presented example of this phenomenon - see wikipedia.)

I'm interested in any class of groups where for all $F_1, F_2$ free, $F_1\cap F_2$ is finitely generated, but I was specifically wondering:

1. Let $G$ be hyperbolic. Is $F_1\cap F_2$ necessarily finitely generated? (Rips' construction gives finitely generated $H, K<G$ with $H\cap K$ not finitely generated, but the subgroups given by Rips are not in free.)
2. Let $G$ be a one-relator group. Is $F_1\cap F_2$ necessarily finitely generated? (The above example $F_1\ast_C F_2$ has cohomological dimension 2, so possibly can embed into a one-relator group.)

Answer 1

The answer to both 1. and 2. is no.

Burns and Brunner showed that free-by-cyclic groups do not have the "finitely generated intersection property" (a.k.a. Howson property), which was recently generalized by Bamberger and Wise in Failure of the finitely generated intersection property for ascending HNN extensions of free groups. In fact, it is proved that there are finitely generated free subgroups with infinitely generated intersection. A random 2-generator 1-relator group will be hyperbolic. That it is free-by-cyclic with positive probability is Theorem 6.1 in:

Dunfield, Nathan M.; Thurston, Dylan P., A random tunnel number one 3-manifold does not fiber over the circle, Geom. Topol. 10, 2431-2499 (2006). ZBL1139.57018.

Burns, R. G.; Brunner, A. M., Two remarks on the group property of Howson, Algebra Logic 18, 319-325 (1980); translation from Algebra Logika 18, 513-522 (1979). ZBL0448.20030.