Let $\Sigma$ be a compact closed connected oriented surface of genus $g>1$. Klarreich proved that the space of ending laminations $\mathcal{EL}(\Sigma)$ is the ideal boundary of the curve complex $C(\Sigma)$, and it is known that this boundary is connected by Leininger-Schleimer and Gabai.
For a complete hyperbolic metric on $\Sigma \times \mathbb{R}$ without cusps, Brock-Canary-Minsky proved that the metric is determined up to isometry by invariants of the ends $( \nu_-, \nu_+ )$, which are conformal structures on the geometrically finite pieces and ending laminations of the geometrically infinite ends.
Before the general case of the ending lamination theorem, Minsky proved it for “thick” hyperbolic manifolds which have a lower bound on the injectivity radius. He characterized thick manifolds in terms of their end invariants $(\nu_-,\nu_+)$. One can see that if one leaves one end invariant $\nu_+$ fixed and varies the other end invariant $\nu_=$, the $+$ end remains thick independent of the other invariant $\nu_-$, and hence one has a collection of “thick” ending laminations $\mathcal{EL}_{thick}(\Sigma)$ associated to the geometrically infinite thick ends of hyperbolic manifolds.
Is the space of thick ending laminations $\mathcal{EL}_{thick}(\Sigma)$ path connected?
A variation on this question: If we fix one end invariant $\nu_-$ (let’s say it is a conformal structure), and vary the other $\nu_+ \in \mathcal{EL}(\Sigma)$, keeping the $+$ end uniformly thick with some lower bound $\epsilon >0 $ on the injectivity radius, is this space of ending laminations path-connected?
Note that $ \mathcal{EL}_{thick}(\Sigma) $ will be the union of these ending laminations for fixed $\nu_-$ and letting $\epsilon \to 0$, so a positive answer to the second question implies a positive answer to the first.
