Can a hyperbolic three-manifold have  toric boundary components? 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?
 A: The simplest construction I can think of is as follows.  Let $X$ be the complement of the Borromean rings in the three-sphere.  Then an $n$-fold cover of $X$ (unwrapping only one boundary torus) will again be hyperbolic and will have $2n + 1$ torus boundary components.
A: As for conformal moduli of the tori (more precisely, Teichmuller parameters) that appear: It is hard to tell, afaik, there is no explicit description. We know that these will be elements of $\bar{{\mathbb Q}}$ (this is essentially due to Selberg, 1960) and that they form a dense subset in ${\mathbb H}^2$ in the case of one boundary torus. See
Nimershiem, Barbara E., Isometry classes of flat 2-tori appearing as cusps of hyperbolic 3-manifolds are dense in the moduli space of the torus, Johannson, Klaus (ed.), Low-dimensional topology. Proceedings of a conference, held May 18-26, 1992 at the University of Tennessee, Knoxville, TN, USA. Cambridge, MA: International Press. Conf. Proc. Lect. Notes Geom. Topol. 3, 133-142 (1994). ZBL0852.57009.
I did not check, but, most likely, the result also holds in the case of $n$ boundary tori, where density is understood in  $({\mathbb H}^2)^n$.
