The modular-group tag has no wiki summary.

**4**

votes

**3**answers

184 views

### Enumerating cosets of the modular group

Suppose we chose the generators $f(z) = z+1, g(z) = z-1, h(z) = -1/z$ for the modular group $\Gamma$ (i.e. the group of fractional linear tranformations $z \mapsto (az +b)/(cz+d)$ with $a,b,c,d \in ...

**1**

vote

**1**answer

694 views

### A question about Ahlfors's proof of modular function being a covering space of the twice punctured plane

I have a question about Ahlfors's proof of modular function being a covering space of the twice punctured plane .See Ahlfors' complex analysis, second edition, page 272. You can either explain or ...

**11**

votes

**1**answer

349 views

### Is there a faithful transitive locally finite action of the modular group?

Is there a faithful transitive action of $G = \mathrm{PSL}_2(\mathbb{Z})$ on $\mathbb{Z}$ such that orbits under each $g \in G$ are finite?

**4**

votes

**2**answers

171 views

### Images of the fundamental domain of $\text{SL}_2(\mathbb{Z})\backslash \mathbb{H}$ whose Euclidean area is large

Let $S$ be a compact subset of the closure of the upper half plane. (Assume that $S$ is a (Euclidean) rectangular box, if you wish.) Let $D$ be the standard fundamental domain of ...

**3**

votes

**2**answers

189 views

### the Action of $SL_2(\mathbb{R})$ on a fundamental domain of $\Gamma$

Let $F=\{z\in H: |z|\geq1, |Re(z)|\leq 1/2\}$. It is a fundamental domain of the modular group $\Gamma$ acting on the upper half plane $H$. (Strictly speaking, one should take part of the boundary ...

**10**

votes

**3**answers

789 views

### Congruence subgroups as abstract groups

This is probably well know, and maybe even trivial, but not to me. Consider for concreteness the subgroup
$$
\pm\Gamma_0(3)=\left\{\begin{pmatrix}a & b \\ c & ...

**10**

votes

**4**answers

908 views

### Analytic function avoiding elements of the modular group

A friend recently told me the following two facts, for which he cannot recall a proof or a reference
(but he remembers seeing them in the literature):
Let $f$ be a holomorphic function mapping the ...

**4**

votes

**4**answers

585 views

### Are there noncongruence subgroups (of finite index) of the modular group generated only by 2 or 3 elements?

By the modular group I either mean $SL(2,\mathbb{Z})$, or $PSL(2,\mathbb{Z})$.
Where can I find examples of these?
Another question - is there a good (ideally analytical, but possibly ...

**6**

votes

**4**answers

988 views

### Does every polynomial diophantine equation have solutions modulo p?

Obviously, this is not exactly true; what I am really asking is whether any diophantine polynomial equation with integer coefficients (let's call them DPEICs) who's solution does not admit ...

**6**

votes

**3**answers

675 views

### Are congruence subgroups of the modular group finitely presented?

Are the congruence subgroups of the modular group $\Gamma\equiv\mathrm{PSL}\left(2,\mathbb{Z}\right)$ (e.g. $\Gamma\left(n\right)$, $\Gamma_{0}\left(n\right)$, $\Gamma_{1}\left(n\right)$ etc.) ...

**3**

votes

**1**answer

198 views

### Simplifying presentations of modular subgroups

I've been using the Reidemeister-Schreier process (detailed in e.g. Holt et al. - Handbook of Computational Group Theory) to find the presentations of various modular subgroups. For example, this ...

**1**

vote

**2**answers

236 views

### Identifying Subgroups of the Modular Group via Permutation Representations on Cosets

Suppose we have a known group G and an unknown subgroup H. The permutation representation of G on the cosets of H gives a permutation group C, which is known. Is it possible to identify the generators ...

**8**

votes

**3**answers

517 views

### Congruence Subgroups as Open Subgroups of the Modular Group Under the Right Topology

It occurred to me that a subgroup of the modular group $\Gamma$ is a congruence subgroup iff it contains a subgroup of the form $\Gamma(N)$, while a subgroup of a general topological group is open iff ...

**8**

votes

**1**answer

562 views

### Automorphisms of a matrix in Smith normal form?

Added: Amritanshu Prasad's answer makes it clear that I am really asking for a description of the group of integer unimodular matrices $P$ such that $D^{-1}PD$ is also integer. These matrices are ...

**15**

votes

**2**answers

1k views

### Distinguishing congruence subgroups of the modular group

This question is something of a follow-up to
Transformation formulae for classical theta functions .
How does one recognise whether a subgroup of the modular group
$\Gamma=\mathrm{SL}_2(\mathbb{Z})$ ...