## Tagged Questions

2answers
241 views

### finite index subgroup of a fuchsian group

Given G, a fuchsian group and a finite sub set A of G. Does there exist a finite index subgroup H in G such that inter section of A with H is empty?
1answer
77 views

### Fundamental domain for subgroup of fuchsian Schottky group.

Let G be a Fuchsian Schottky group defined by a possibly infinite set of disjoint halfplanes {C_i}_i. Let F be the fundamental domain obtained by intersecting the complements of th …
2answers
154 views

### Fuchsian groups and automorphisms of Riemann surfaces

Let $\Gamma \subseteq PSL_2(\mathbb{R})$ be a Fuchsian group, possibly containing elliptic elements. Is it true that $N(\Gamma) / \Gamma$, where $N(\Gamma)$ the normalizer of \$\Gam …
1answer
95 views

### Fuchsian groups and their normalizers

Let $\Gamma \leq PSL_2(\mathbb{R})$ be a Fuchsian group. What is the relation between $N(\Gamma) = \{ \alpha \in PSL_2(\mathbb{R}) \mid \alpha \Gamma \alpha^{-1} = \Gamma \}$ and \$ …
0answers
83 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 …
1answer
197 views

### How do you find the genus of a Fuchsian group derived from a quaternion algebra?

Let $G$ be a Fuchsian group with normalizer $N(G)$ inside $PSL(2,13)$ Due to the Hurwitz formula, it suffices to find a presentation of $G$ of the form: \langle x_1,\ldots,x_r,a …
1answer
262 views

### Are any two Dirichlet domains for a Fuchsian group “comparable”?

Let $\Gamma$ be a [EDIT: finitely generated] Fuchsian group of the first kind (i.e. a discrete subgroup of $PSL_2(\mathbf{R})$ acting on the upper half-plane admitting a fundamenta …
4answers
350 views

### Growth of smallest closed geodesic in congruence subgroups?

Let $\Gamma$ be one of the classical congruence subgroups $\Gamma_0(N)$, $\Gamma_1(N)$ and $\Gamma(N)$ of $SL(2, \mathbb{Z})$. How does the lower bound for the length of primitive …
1answer
310 views

### Non congruence subgroups containing congruence subgroups.

Does there exist Fuchsian groups, which is not conjugated in $SL(2, \mathbb{R})$ to a subgroup of $SL(2, \mathbb{Z})$, but still contains a congruence subgroup?
1answer
391 views

### How nice are representation varieties of Fuchsian groups?

Background Let $S_{g,n}$ be an oriented surface of genus $g$, with $n$ punctures. We explicitly prohibit the non-hyperbolic cases: $g=0$, $n=0,1,2$. $g=1$, $n=0$. Let \$\Gamma …
1answer
322 views

### The smallest positive eigenvalue and the length of the shortest geodesic

I'm confused about some things concerning lengths of geodesics on Riemann surfaces and positive eigenvalues of the Laplacian. Moreover, I'm also interested in the relation between …
1answer
196 views

### Genus of arithmetic surface groups

It is known that for each genus, only finitely many points in the moduli space of hyperbolic genus g surfaces are arithmetic. I'm wondering if an existence result is known: for wh …
1answer
253 views

### Arithmetic Fuchsian group

Dear all, I have the following questions: Are all Fuchsian groups of signature $(0;2,2,2,\infty)$ arithmetic? What is known about the trace fields of these groups? Best, K.
2answers
414 views

### Cusp width for an arbitraty Fuchsian group

In Shimura's Intro to Arithmetic Theory of Automorphic Forms, he defines a cusp of a Fuchsian group $\Gamma$ as a point $s \in \mathbb{R} \cup \{ \infty \}$ that is fixed by a para …
0answers
198 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 …

15 30 50 per page
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 35, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8794677257537842, "perplexity_flag": "middle"}