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Shin and Strenner have shown that the conjecture is false when 3g + n > 4. See http://arxiv.org/abs/1410.6974

As Ryan points out, the interesting case is when the fiber is 2-dimensional. As Igor points out, this is a difficult open problem when the fiber has dimension 2. When the fiber is a surface $F$, th …

Sixty is an upper bound. Hodgson and Kerckhoff's Universal Hyperbolic Dehn Filling theorem ("Universal bounds for hyperbolic Dehn surgery." Annals of Mathematics. 162(1), 367-421) says that, in a one …

I think this should just follow from the exponential mixing of the geodesic flow (due to Pollicott). Exponential mixing says that there is a constant $q$ such that if you have two smooth functions $f …

This phenomenon is called "cosmetic surgery." If $K$ is an amphichiral knot in the $3$--sphere with exterior $M_K$, then $M_K(p/q) \cong - M_K(-p/q)$. So if $p/q$ is a hyperbolic filling slope, the …

It follows from Thurston's Covering Theorem that there are no such examples. The covering theorem says that if a degenerate end is infinite-to-one under a covering map, then you are (virtually) in th …

The conformal structure on the cuspidal torus is usually called the "cusp shape." See Adams, Hildebrand, Weeks Hyperbolic invariants of knots and links and McReynolds, Arithmetic cusp shapes are dense …

For the arithmetic point of view you mention, Maclachlan and Reid's book "The arithmetic of hyperbolic 3-manifolds" is a great reference. In case you're interested, there are also many geometric ways …

An explicit interpolation is pretty easy to find. (I'll use the upper half plane model.) $f$ is given by $\left( \begin{array}{cc} a & b \\\ c & d \end{array} \right)$. $f$ is elliptic, parabolic, …

If you don't care about explicitly constructing the metric, you can do this: Let $N$ be a closed orientable $3$-manifold. Let $\Gamma$ be any finite graph such that each component has negative Euler …

Let $Y$ be a compact hyperbolic surface. There are only finitely many closed geodesics in $Y$ whose lengths are less that $\ell$. Since $\pi_1(Y)$ is residually finite, there is a normal subgroup of …

Infinitely many Dehn fillings on the figure eight knot complement $M_8$ have this property: All but finitely many fillings on $M_8$ are hyperbolic, by Thurston's hyperbolic Dehn filling theorem. The …

Here are some: Stillwell, "Geometry of Surfaces." Bonahon, "Low-dimensional geometry. From Euclidean surfaces to hyperbolic knots." Katok, "Fuchsian groups."

An end of an orientable finite volume hyperbolic $3$--manifold always has a neighborhood homeomorphic to $S^1 \times S^1 \times \mathbb{R}$, so no. Introductory texts on hyperbolic manifolds will con …

It is worth pointing out that $T$ may lie in the image of $i_*$: There is an $M$ such that $H_1(M) \cong \mathbb{Z}^{n} \oplus T$ and $H_{\mathrm{cusp}}(M) \cong \mathbb{Z}^{n-\mathrm{number\ of\ cus …