21
votes
Why are Fuchsian groups interesting?
About the relation to fractals: for Fuchsian groups of the first kind, the limit set has Hausdorff dimension 1, i.e., it is not fractal.
However, for all other quasifuchsian groups of the first kind,...
- 10.4k
18
votes
17
votes
Equations defining hyperbolic geodesics in $\mathbb C \setminus\{0,1\}$
I wondered this myself, I made some similar pictures to approximate the stable lamination of a pseudo-Anosov map (the blue curve is a geodesic, the other colors horocycles).
Every geodesic on the ...
- 68.8k
13
votes
Equations defining hyperbolic geodesics in $\mathbb C \setminus\{0,1\}$
$\def\CC{\mathbb{C}}\def\HH{\mathbb{H}}\def\ZZ{\mathbb{Z}}\def\RR{\mathbb{R}}\def\Id{\mathrm{Id}}$These geodesics are always algebraic. We can understand their equations using the classical modular ...
- 156k
13
votes
Why are Fuchsian groups interesting?
Fuchsian groups, particularly those which are cocompact, form the tips of several big mathematical icebergs. To put this less metaphorically, several discoveries about Fuchsian groups, obtained using ...
- 15.4k
13
votes
Why are Fuchsian groups interesting?
Fuchsian groups occur naturally in JSJ-theory. In 3-manifolds they occur as the base Groups of the Seifert pieces and in geometric group theory they occur as (quotients of) enclosing groups (also ...
- 566
11
votes
Why are Fuchsian groups interesting?
Fuchsian groups are important for the theory of dessin d'enfants in arithmetic geometry. Wolfart wrote a survey here: http://www.math.uni-frankfurt.de/~wolfart/Artikel/abc.pdf .
One typical ...
9
votes
Accepted
Hyperbolic Metric on a Riemann Surface
Choose an horocycle around the puncture. Then the end delimited by the horocycle is isometric to a cusp, which is obtained by quotienting the following domain of the Poincaré half-plane
$$
C_R = \{ z \...
- 18.7k
7
votes
Why are Fuchsian groups interesting?
Are the Fuchsian groups with fixed points interesting from a geometric perspective?
Yes, notice $PSL(2,\mathbb{Z})$ fixes points. The fundamental domain in this ...
- 5,259
6
votes
Canonical immersion of the double torus
This is not really an answer, but rather a longish comment and a suggestion about how one might focus the question a bit better.
First, when one asks for a 'canonical' isometric embedding into some ...
- 108k
6
votes
Accepted
Reference request: geometric finiteness of Fuchsian groups
Yes, this is a theorem of Siegel, and a reference with simple proof is
Theorem XI.12 in
M. Tsuji, Potential theory in modern function theory, Maruzen, Tokyo 1959 (there is an AMS Chelsea reprint, 1975)...
- 91.8k
5
votes
Accepted
What is the homeomorphism from $\Gamma \backslash T_1 \mathbb{H}$ to $T_1(\Gamma \backslash \mathbb{H})$
I'll define the map and leave the proof that it is a diffeo to you. Assume $M$ is a Riemannian manifold and $\Gamma$ is a torsion free group that acts properly discontinuously by isometries on $M$. ...
- 11.6k
5
votes
Equations defining hyperbolic geodesics in $\mathbb C \setminus\{0,1\}$
Sorry for a non-answer. Pictures are so beautiful I could not resist and made another illustration. Shown in red is the image under $\lambda$ of the geodesic from $\mathbb H$ that is the semicircle ...
5
votes
Equations defining hyperbolic geodesics in $\mathbb C \setminus\{0,1\}$
Let me rephrase the proof of @Ian Agol as I understand it.
Let $\Gamma(2)$ be the congruence subgroup
of level $2$ (it consists of all $2\times 2$ integer matrices $A$ with
${\mathrm{det}\, }A=1$ and $...
- 91.8k
5
votes
Accepted
Is every Cantor set $C\subseteq\mathbb R^{\infty}$ the limit set of a Fuchsian group?
Expanding on my comments, here are some obstructions coming from Hausdorff dimension and self-similarity.
One observation is that every nonelementary Kleinian group $\Gamma$ has positive critical ...
- 12.2k
4
votes
Accepted
Positive genus Fuchsian groups
Yes, this is true, but proving this is easier than finding a reference.
Every finitely-generated matrix group (e.g. a lattice in $PSL(2, {\mathbb R})$ contains a torsion-free subgroup. The general ...
- 12.2k
4
votes
Accepted
Fuchsian group which is derived from a division quaternion algebra, Mixing flows on the quotient space
1 - There is an explicit reference given in the book: Borel, Harish-chandra, arithmetic subgroups of algebraic groups, 1962. This is the general result for matrix groups. A simpler proof has been ...
- 18.7k
4
votes
Reference for openness of subspace of PSL(2,R) representation variety corresponding to Teichmüller space
$\DeclareMathOperator\PSL{PSL}$Here is a complement to Jean Raimbault's first comment (I would have posted it as a comment, but I have not yet unlocked that privilege). Let $S$ be a thrice-punctured ...
- 143
4
votes
Accepted
Analytic continuation of the Eisenstein series defined over Hecke and Fricke subgroups
One natural generalization of $E_s$ to $\Gamma_0(N) \backslash \mathbb{H}$ is the family of Eisenstein series, $$E_{s; \chi_1, \chi_2}(\tau) = \sum_{\gamma \in \Gamma_{\infty} \backslash \mathrm{SL}...
- 56
4
votes
Accepted
Area of fundamental domain of Fuchsian group and index of a Fuchsian group in the triangle group
EDIT: My answer (below) is for the original question. The current question has been modified to include my answer.
The area of a fundamental domain for $\Gamma$ is $4\pi = -2\pi \chi(\mathbb{H} / \...
- 28.1k
4
votes
Accepted
Reference for triangle groups
It seems that you are interested in “Fenchel’s conjecture” stating that (essentially) all two-orbifolds are finitely (orbifold) covered by surfaces. The case of triangle orbifolds is the hardest, and ...
- 28.1k
3
votes
Reference for 'Normal Subgroups of Fuchsian Groups'
Fuchsian groups are residually finite, so they have many finite index normal subgroups. (Even better, surface groups are residually free!)
Torsion free fuchsian groups (aka surface groups) are ...
- 28.1k
3
votes
Reference for openness of subspace of PSL(2,R) representation variety corresponding to Teichmüller space
You know, I also have been looking for a reference for openness in the type preserving setting and haven't been able to find it. (Actually, I'd like to have a theorem like this that works for ...
- 532
3
votes
$PSL_2(\mathbb{R})$ representations of free groups
I am essentially just repeating Will's answer, but giving a slightly different point-of-view (and a relevant reference).
Let $F_r$ be a free group of rank $r>1$. Given a discrete faithful ...
- 8,529
3
votes
Accepted
$PSL_2(\mathbb{R})$ representations of free groups
It suffices to count the punctures and boundary components. For each puncture or boundary component, the loop around it gives a conjugacy class in the fundamental group, well-defined up to inversion. ...
- 148k
3
votes
How to construct a group of Möbius transformations corresponding to a given fundamental triangle?
My question [is] how to construct the generators $T$, $S$ of the group of Möbius transformations such that this triangle is its fundamental domain?
Via large amounts of hyperbolic trigonometry. See ...
- 28.1k
3
votes
Accepted
Examples of group families with solvable uniform word problem
Derek Holt is the expert here - I hope he will correct me where I err.
There are many such families. Here are a few ones that spring to mind (or were mentioned in the comments above), in no ...
- 28.1k
2
votes
Accepted
Dirichlet region of a free group
The statement is false. Here is a counter-example. Let $T$ be an ideal triangle (say in the unit disk model). Let $S$ be the surface obtained by doubling $T$ across it’s boundary: that is, take two ...
- 28.1k
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