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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 …

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, …

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 …

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 …

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 …

Cederberg's A Course in Modern Geometries does some of this.

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 …

I remember finding examples like this in graduate school. Draw the figure eight knot and then draw a parallel pushoff of a meridian. Snappea gives you a volume for this link. (It should tell you th …

I would guess that the rank could go down in general, but that something like the following should be true (and may well be, but my memory is a little foggy). Let $N$ be your $3$-manifold obtained by …

Regarding Question 2, you get lots of examples that are rigid for the lifting map $M(\Gamma) \to M(\Gamma_B)$. Let $B$ be finitely generated subgroup of $\Gamma$ (considered a fuchsian group) such t …

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 …

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

Have you tried something like this (you probably have): Take a diagram of a complicated hyperbolic knot and draw a diagram for the Whitehead double. Then change a crossing in the clasp. I'd like to t …

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 …

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 …