Questions tagged [fuchsian-groups]
In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R)
15
questions with no upvoted or accepted answers
7
votes
0
answers
156
views
The relation between Wolf's and Teichmüller's parametrization of the Teichmüller space
Let $\mathcal{T}_g$ be the Teichmüller space of Riemannian surface structures on an oriented 2-dimensional manifold of genus $g$. Fix a point $S \in \mathcal{T}_g$. There are two different ways to ...
7
votes
0
answers
272
views
Hyperbolic group with boundary $S^1$ implies virtually Fuchsian via bounded cohomology?
Question: Is there an approach to $\partial G \cong S^1$ implies virtually Fuchsian using bounded cohomology of $\mathrm{Homeo^+} (S^1)$? If not is there a reason to believe it wouldn't work, or maybe ...
4
votes
0
answers
172
views
Asymptotics of arithmetic Fuchsian groups and Shimura curves.
I'm interested in what is known/expected about some families of arithmetic Fuchsian groups. Here is the simplest family that I'm interested in: Let $E = Z[\omega]$, where $\omega = e^{2 \pi i / 3}$. ...
3
votes
0
answers
75
views
Spectral theory for Fuchsian groups of the first kind
There are tons of material on the spectral theory of $L^2(\Gamma\backslash G)$ for a lattice $\Gamma$ in $G=PSL_2({\mathbb R})$. There are also many papers on the case of $\Gamma$ being convex-...
3
votes
0
answers
178
views
One question about iteration on groups
Let $G$ be a finitely generated group, $H$ a subgroup of $G$ of index $n$, with $a_i$ a set of coset representatives and $$G=\displaystyle\bigcup_{i=1}^nH{a_i}.$$
Let $\phi:H\rightarrow G$ be a ...
3
votes
0
answers
339
views
Discussion of specific arithmetic triangle groups?
Arithmetic triangle groups were classified in Takeuchi, Arithmetic triangle groups, J. Math. Soc. Japan Volume 29, Number 1 (1977), 91-106. The (2,3,7) case was discussed in detail in a number of ...
2
votes
0
answers
67
views
Maximal orders and surface subgroups of even genus
Let $A$ be a quaternion algebra over a totally real number field $k$. Suppose that $A$ splits at exactly one real place of $A$. Let $\mathcal{O}$ be a maximal order in $A$. Then $\mathcal{O}$ contains ...
2
votes
0
answers
182
views
Regarding fundamental domain of 2 genus surface
Let $\mathbb{H}^2$ be the hyperbolic plane with $(2,3,7)$ tiling. Let $\Gamma$ be a subgroup of $(2,3,7)$ triangle group such that $\mathbb{H}^2/\Gamma$ is the compact orientable surface of genus 2 ...
2
votes
0
answers
142
views
example of fuchsian groups acting on 2-sphere by G. Martin
Currently I am reading a paper "Infinite group actions on spheres" by Gaven Martin. I am a first year graduate students and I got lots of questions, so one of them is about the following example: (...
1
vote
0
answers
87
views
Almost modularity of Belyi curves and etale fundamental group of non-Belyi curves
Belyi's theorem states that every curve defined over $\mathbb{\bar Q}$ is almost modular (obtained from $\mathbb{H}^2/\Gamma,\ \Gamma$ a finite index subgroup of $PSL(2,\mathbb{Z})$), after ...
1
vote
0
answers
58
views
Variation of the geometry of a Dirichlet region as the defining point varies
Let $\Gamma$ a Fuchsian group acting on the hyperbolic plane $\mathfrak{H}$. For me, I am most interested in the case where $\Gamma$ has a fundamental domain that is a finite-polygon with all ...
1
vote
0
answers
130
views
Ideal Ford domain (for finite index subgroup)
Let $G$ be a lattice Fuchsian group with parabolic elements, seen as a discrete subgroup of matrices
$
g=
\begin{pmatrix}
\alpha & \overline{\beta} \\
\beta & \overline{\alpha}
\end{pmatrix}
$...
1
vote
0
answers
87
views
Arithmetic product and sum of limit sets of non-elementary Fuchsian group of second kind
Let $L \subset \mathbb{R}$ be a limit set of a Fuchsian group $\Gamma$. If $\Gamma$ is a non-elementary Fuchsian group of second kind, then $L$ is a Cantor set. For example: $\Gamma= \bigg\langle \...
0
votes
0
answers
148
views
Classification of fundamental domains of a fuchsian group
Let $G$ be the (2,3,7) triangle group. We can see it as symmetry group of (2,3,7) tiling of the hyperbolic plane or symmetry group of $[3^7]$ tiling of the hyperbolic plane. This contains translations,...
0
votes
0
answers
263
views
Is the absolute value of the j-invariant bounded from below on an annulus
Let $j:\mathbf{H}\to \mathbf{C}$ be the $j$-invariant. It's a modular function for $\Gamma(1) = \textrm{PSL}_2(\mathbf{Z})$.
For $\epsilon>0$ small, let $B(\epsilon)$ be the image of the strip $$\{...