In Henry Wilton's excellent paper "Hall's Theorem for limit groups" (Geom. Funct. Anal. 18, pp. 271–303, 2008 ) he proves the following result (see his paper or here and here for the relevant definitions).

Wilton's Theorem: Limit groups are subgroup separable.

This theorem has me wondering whether, in some cases, it is possible to see this result by passing through free group quotients. To make this precise, say a group $G$ is freely subgroup separable if any finitely generated and infinite-index (see Ian Agol's answer, below) $H \leq G$ is the intersection of subgroups in $\{ H \Delta : G/\Delta \text{ is a free group} \}$.

Question 1: Which limit groups are freely subgroup separable? In particular, are surface groups freely subgroup separable?

If the answer to the above question is "all of them", then this, along with the fact that free groups are subgroup separable, would imply Wilton's Theorem, so I suspect the answer is more complicated than that. In light of this, here is a refinement of Question 1.

Question 2: Given a limit group, $G$, for which finitely generated subgroups $ H \leq G$, do we have $H = \cap \{ H \Delta : G/\Delta \text{ is a free group} \}$?

In Henry Wilton's excellent paper "Hall's Theorem for limit groups" (Geom. Funct. Anal. 18, pp. 271–303, 2008 ) he proves the following result (see his paper or here and here for the relevant definitions).

Wilton's Theorem: Limit groups are subgroup separable.

This theorem has me wondering whether, in some cases, it is possible to see this result by passing through free group quotients. To make this precise, say a group $G$ is freely subgroup separable if any finitely generated and infinite-index (see Ian Agol's answer, below) $H \leq G$ is the intersection of subgroups in $\{ H \Delta : G/\Delta \text{ is a free group} \}$.

Question 1: Which limit groups are freely subgroup separable? In particular, are surface groups freely subgroup separable?

If the answer to the above question is "all of them", then this, along with the fact that free groups are subgroup separable, would imply Wilton's Theorem, so I suspect the answer is more complicated than that. In light of this, here is a refinement of Question 1.

Question 2: Given a limit group, $G$, for which finitely generated subgroups $ H \leq G$, do we have $H = \cap \{ H \Delta : G/\Delta \text{ is a free group} \}$?

In Henry Wilton's excellent paper "Hall's Theorem for limit groups" (Geom. Funct. Anal. 18, pp. 271–303, 2008 ) he proves the following result (see his paper or here and here for the relevant definitions).

Wilton's Theorem: Limit groups are subgroup separable.

This theorem has me wondering whether, in some cases, it is possible to see this result by passing through free group quotients. To make this precise, say a group $G$ is freely subgroup separable if any finitely generated (and infinite-index?) $H \leq G$ is the intersection of subgroups in $\{ H \Delta : G/\Delta \text{ is a free group} \}$.

Question 1: Which limit groups are freely subgroup separable? In particular, are surface groups freely subgroup separable?

If the answer to the above question is "all of them", then this, along with the fact that free groups are subgroup separable, would imply Wilton's Theorem, so I suspect the answer is more complicated than that. In light of this, here is a refinement of Question 1.

Question 2: Given a limit group, $G$, for which finitely generated subgroups $ H \leq G$, do we have $H = \cap \{ H \Delta : G/\Delta \text{ is a free group} \}$?

In Henry Wilton's excellent paper "Hall's Theorem for limit groups" (Geom. Funct. Anal. 18, pp. 271–303, 2008 ) he proves the following result (see his paper or here and here for the relevant definitions).

Wilton's Theorem: Limit groups are subgroup separable.

This theorem has me wondering whether, in some cases, it is possible to see this result by passing through free group quotients. To make this precise, say a group $G$ is freely subgroup separable if any finitely generated and infinite-index (see Ian Agol's answer, below) $H \leq G$ is the intersection of subgroups in $\{ H \Delta : G/\Delta \text{ is a free group} \}$.

Question 1: Which limit groups are freely subgroup separable? In particular, are surface groups freely subgroup separable?

If the answer to the above question is "all of them", then this, along with the fact that free groups are subgroup separable, would imply Wilton's Theorem, so I suspect the answer is more complicated than that. In light of this, here is a refinement of Question 1.

Question 2: Given a limit group, $G$, for which finitely generated subgroups $ H \leq G$, do we have $H = \cap \{ H \Delta : G/\Delta \text{ is a free group} \}$?