All Questions
13 questions
3
votes
1
answer
158
views
Geometry and topology of Fuchsian character varieties
Consider the hyperbolic space, $\mathbb H^2$. A Fuchsian group is a discrete subgroup of $\text{PSL}(2,\mathbb R)$. We can generate tessellations, especially $\{p,q\} \;\text{tesellations}$ of $\...
- 357
3
votes
1
answer
375
views
Reference for triangle groups
Can anyone suggest to me some references for studying triangle groups? Especially the existence of finite index subgroups, subgroups isomorphic to fundamental groups of compact surfaces etc.
- 613
1
vote
1
answer
239
views
Area of fundamental domain of Fuchsian group and index of a Fuchsian group in the triangle group
Let $\mathbb{H}$ be the upper half plane model of hyperbolic geometry. Let $\Gamma$ be the Fuchsian group such that $\mathbb{H}/\Gamma$ is the compact orientable surface of genus $2$.
Suppose $\Gamma =...
- 613
0
votes
0
answers
164
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,...
- 613
1
vote
0
answers
68
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 ...
- 5,349
1
vote
0
answers
142
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}
$...
- 33
2
votes
1
answer
219
views
Dirichlet region of a free group
Let $G$ be a non-uniform lattice Fuchsian group and let $P$ be a Dirichlet region for $G$. In particular $G$ has parabolic elements, $P$ is not compact and has finite area. We are in the unit disc. Is ...
- 33
3
votes
2
answers
159
views
What is the homeomorphism from $\Gamma \backslash T_1 \mathbb{H}$ to $T_1(\Gamma \backslash \mathbb{H})$
Let $\mathbb{H}$ be hyperbolic plane, $\Gamma$ is a discrete subgroup of $PSL_2(\mathbb{R}$) so that $\Gamma \backslash \mathbb{H}$ is a compact hyperbolic surface. Maybe it will be very simple to you ...
- 53
30
votes
7
answers
5k
views
Why are Fuchsian groups interesting?
I keep hearing that fuchsian groups are interesting for other reasons than the Fuchsian model for hyperbolic Riemann surfaces.
What are those reasons?
Are the Fuchsian groups with fixed points ...
- 893
0
votes
2
answers
352
views
Reference for 'Normal Subgroups of Fuchsian Groups'
I am looking for a reference on how to explicitly construct normal subgroups of a given Fuchsian group. I appreciate any help.
- 989
3
votes
1
answer
139
views
Classification of maximal nonuniform Fuchsian lattices existent?
I am interested in the set of all non-cocompact Fuchsian lattices which all have a distinguished point as cusp, say $\infty$ in the upper half plane model of the hyperbolic plane. Of course, the ...
- 245
3
votes
1
answer
232
views
Arithmetic Fuchsian lattices that are not finite index subgroups of Eichler orders?
Lindenstrauss' proof of AQUE (arithmetic quantum unique ergodicity) assumes that the Fuchsian lattice is an Eichler order or, if I understand it correctly, a finite index subgroup of an Eichler order. ...
- 245
2
votes
1
answer
360
views
Can the finiteness of a Burnside group with two generators be checked algorithmically by using Fuchsian von Dyck groups?
BIG EDIT of the previous question "Coverings of the free Burnside groups", never answered.
In the paper http://link.springer.com/article/10.1007/BF00046586 (last section) there is an interesting ...
- 2,527