Thanks for your answer @HJRW. It would be very interesting to see whether one could prove that the property of a homomorphism to $\mathbb{Z}$ being dominated by a splitting over finitely presented subgroups is “open” in the same sense as the BNS invariants.

Thanks all for your answers. Indeed I am in fact interested in studying kernes of homomorphisms $\pi_1(M) \to \mathbb{Z}$ and whether they can be realised as direct limits of “nice” subgroups of $\pi_1(M)$

Apologies, I forgot to add that I also need the submanifold to be compact.

@HJRW ah of course, thank you. Sorry, I fundamentally misunderstood a part of the definition. It is clear to me now why there should only ever be one non-trivial deformation space.

Apologies for making myself unclear. In my question, $\mathcal{A}$ is the set of subgroups containing edge stabilisers of the splitting. So for instance, if $\Sigma$ is a large enough closed surface, $G = \pi_1(\Sigma)$ and $\mathcal{A}$ is the set of cyclic subgroups of $G$, then any two non-isotopic pair of pants decompositions of $\Sigma$ induce two cyclic splittings which live in two distinct deformation spaces (which are singletons).