I'm wondering if I can construct a class of hyperbolic three-manifolds with $n$ toric boundaries. My idea is to take a bordered hyperbolic Riemann surface $\Sigma_{g,n}$, of genus $g$ with geodesic boundary lengths $\ell_1,\dots,\ell_n$, and then fibre it over the circle as $$ M = \Sigma_{g,n} \times [0,1] \,/\sim \qquad (x,t)\sim (\varphi(x),t+1), $$ where $\varphi(x)$ is a Pseudo-Anosov element of the mapping class group of $\Sigma_{g,n}$. I would imagine that each boundary component locally looks like a product of two circles, so that $\partial M$ is the disconnected sum of $n$ tori. How do I prove/disprove that this is true? If true, what are the complex structure moduli $\tau_1,\dots, \tau_n$ of the boundary tori?