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All two-bridge knots are hyperbolic except for the $(2, k)$ torus links. For an “as simple as possible” (but still fairly difficult) proof, see Theorem 10.17 of Purcell’s book Hyperbolic Knot Theory.

Yes, all hyperbolic three-manifold fundamental groups can be generated by loxodromic elements. For, suppose that $\Gamma = \{ \gamma_i \}$ is a generating set. Take $\gamma$, a loxodromic element wh …

This three-manifold can also be constructed by taking a genus two surface $S$, crossing with the interval $I$ to get $S \times I$, and attaching a pair of one-handles both of which connect $S \times \ …

The fixed points of $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, acting on $\mathbb{CP}^1$, are found by solving, for $z$ and $w$, the equation $azw + bw^2 = cz^2 + dzw$. Much of the time you can …

The answer to the first question is "yes" and the answer to the second is "no", assuming that you are looking for a covering which is a locally isometric. If you do not require a locally isometric co …

Question: Is the length of homotopically non-trivial loops in $M(f)$ bounded below in terms of the genus $g$ of the fiber? Answer: No. Here is one family of examples. Suppose that $S$ is the closed o …

They do not give the definition of conjugate geodesic, but we can make an educated guess. Suppose that $F$ is a free group of rank two. (So, isomorphic to the fundamental group of the once-punctured …

Edited: to reflect the correct definitions. Question 1: Why are the guts well-defined? Answer 1: By the JSJ theory there is a unique collection of $I$-bundles (and Seifert fibered spaces) that contai …

Your formula can be found on page 355 (near the end of Section 7.4) of Marden's book Outer circles: an introduction to hyperbolic 3-manifolds. In the second edition of the book, with the title, Hyperb …

Thurston (even before his lecture series in 1979) was probably the first to use cross-ratios to describe the shapes of hyperbolic tetrahedra. He was certainly the first to use these shapes to compute …

The answer to your question is "no". There are simple geodesics in $S = S_{0, 4}$ that lie in a compact subsurface of $S$, but whose closures contain no simple closed geodesic. Here is the usual "fir …

One explanation is to look at the "spun triangulation" SnapPy gives to the Weeks manifold. This is composed of two ideal tetrahedra with volumes ~0.83828404504 and ~0.10442331774 adding to the famili …

There is a great deal of overlap between this question and the following questions asked by the same person: About isotopy and homotopy About isotopy of simple close curve However, the exact questio …

I think that the answer is “yes, such a constant $K$ exists.” Suppose that $g$ is the given mapping class in the stabiliser of $\gamma$. Replace $g$ by a large power (bounded by the topology of $S$) …

For question one: one possible qualitative approach is to consider the "thick-thin" decomposition of the given metric on $S$. For question two: the hyperbolic metric on a surface and the "moduli of a …