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?
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2$\begingroup$ For this strategy to work, you also need to assume that $\phi$ preserves each boundary component. But yes, there are many, many examples of hyperbolic manifolds with $n$ boundary components, for any $n$, and many other ways to construct them. The question of computing the moduli is more difficult/interesting. $\endgroup$– HJRWCommented Apr 14, 2021 at 15:30
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1$\begingroup$ Cross-posted: math.stackexchange.com/questions/4101645/… $\endgroup$– Moishe KohanCommented Apr 14, 2021 at 23:40
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2 Answers
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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.
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3$\begingroup$ The alternating pretzel link $P(2,2,\ldots,2)$ is hyperbolic provided it has at least 3 components. The number of components is given by the number of 2's. $\endgroup$ Commented Apr 14, 2021 at 19:40
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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 $({\mathbb H}^2)^n$.
answered Apr 14, 2021 at 23:52