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Space of thick ending laminations

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 $...
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Are the unit tangent spaces over hyperbolic surfaces always Seifert?

I saw on Wikipedia (in the geometrization conjecture article) that the unitary tangent space over a surface S with finite volume and genus > 1 is a Seifert manifold. What if we do not assume that S ...