Which elements of the fundamental group can be realized as transversals of a taut foliation?

Question

Specifically in a closed, orientable 3-manifold. I'm not necessarily looking for a complete answer, as I don't expect one. Is there prior literature on this question? Also interested in this question but for elements of $H^1(M, \mathbb{Z})$. A necessary condition is that if $\Sigma$ is a compact leaf of your taut foliation, it must have nonzero algebraic intersection with the homology class of the curve. I can't figure out if this is sufficient, or what other necessary conditions there may be.

Answer 1

One perspective on this question is to consider the leaf space of the pullback $\tilde{\mathcal{F}}$ of the taut foliation $\mathcal{F}$ of $M$ to the universal cover ($\tilde{M} \cong \mathbb{R}^3$ foliated by planes in the interesting case) together with the action of the fundamental group on the space of leaves $\Lambda=\tilde{M}/\tilde{\mathcal{F}}$, which is a simply-connected but possibly non-Hausdorff 1-manifold by a result of Palmeira.
I think that a loop will be homotopic to be transverse to the foliation if and only if the corresponding group element acts by a translation on part of this manifold $\Lambda$. This seems quite complicated though since a non-Hausdorff 1-manifold can have branching like a tree, so let’s consider the case in which the leaf space $\Lambda$ is Hausdorff.

In this case, the leaf space $\Lambda$ is homeomorphic to $\mathbb{R}$. An example to keep in mind is a 3-manifold fibering over $S^1$. Assume that the foliation $\mathcal{F}$ is cooriented. Then an element $g\in \pi_1(M)$ acts on $\mathbb{R}$ by a homeomorphism $f: \mathbb{R}\to \mathbb{R}$ which preserves orientation. Suppose that $f(x)\neq x$ for some $x\in \mathbb{R}$. Then consider a point in a leaf $L \subset \tilde{\mathcal{F}}$ of the foliation in the preimage of $x$, and its image under the covering translation $g(x)\subset g(L)$. One may connect $x$ to $g(x)$ by an interval transverse to $\tilde{\mathcal{F}}$, and hence projecting to a closed loop transverse to $\mathcal{F}$ in $M$. Thus the only elements of the fundamental group that cannot be homotoped to be transverse to $\mathcal{F}$ are the elements that fix $\Lambda$ pointwise. In the case of a fibration, these are the elements in the fiber surface subgroup. In general, they will correspond to a normal subgroup of $\pi_1(M)$ which is the intersection of the fundamental groups of the leaves of the foliation.