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The fixed point set of the involution in the universal cover is a totally geodesic copy of hyperbolic space of codimension 1. So that gives you the first totally geodesic hypersurface in some finite …

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

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 …

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 …

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 …

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

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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …