## New answers tagged hyperbolic-geometry

2
votes

### For which quadratic number field, the algebraic integers are cusps for some Coxeter group?

This is a partial answer to say that this won't happen very often.
Let's work with the triangle group where $s_1 s_2$ has order $p$, $s_2 s_3$ has order $q$ and $s_1 s_3$ has order $\infty$. Let $\...

1
vote

### Simple curves on hyperbolic tori

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 ...

3
votes

Accepted

### Guts of 3-manifolds for sutured manifolds and pared manifolds

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 ...

4
votes

Accepted

### A formula for the cross-ratio in terms of hyperbolic data

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, ...

2
votes

### Who first used the cross-ratio to describe shapes in hyperbolic geometry?

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 ...

4
votes

Accepted

### Geodesic laminations on the 4-punctured sphere

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 ...

1
vote

### Is there a general dilogarithm formula for the Cheegerâ€“Chernâ€“Simons class?

As noted in the comments, a paper [1] of Garoufalidis, D. Thurston, and Zickert answers this question for $\operatorname{SL}_n(\mathbb{C})$.
[1]
Garoufalidis, Stavros; Thurston, Dylan P.; Zickert, ...

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