How to construct Fundamental Domains

Question

I understand what a Fundamental Domain is, however, I have a difficulty in understanding the algorithm to construct Fundamental Domain of $SL(2,\mathbb{Z})$. Here are the lecture notes which are quite helpful:

Keith Conrad Summer School

In these notes, he says that $``$For some $n$, $T^ng_0 \tau$ has real part between $-1/2$ and $1/2$.$"$,(page 7, before (3.4)). Why/how $1/2$?

Our instructor says that there is an algorithm for constructing Fundamental Domain of $SL(2,\mathbb{Z})$. He takes a point $iy$ in upper half plane which is not stabilized by $S$ and $T$ that generate $SL(2,\mathbb{Z})$.(Say $2i$). ($S$ and $T$ are same in the lecture notes of Prof. Conrad). Then he looks at the image under $S$ of $2i$, then takes a perpendicular bisector(under hyperbolic distance) between $2i$ and $S(2i)$, takes the area of the part where $2i$ is. Then he does the same for $T$, then he does the same for $T^{-1}$, takes the intersecting area between them, this is the Fundamental Domain of $SL(2,\mathbb{Z})$.

What is the reasoning behind the algorithm? Where do I stop?

How do I know that I should use $T^{-1}$, too?

How do I generalize this algorithm to construct Fundamental Domain of $\\$ $\Gamma(2)=\langle z+2, z/(2z+1)\rangle$? Is there any "practical" way to show that this area has one orbit element from each orbit?

Thanks in advance.

Answer 1

Your post has maybe too many questions in it; it's probably better to look at them in one post at a time. So I'm going to stick to your first question.

To start with, if $g_0\tau = x+iy$ then $Tg_0\tau=(x+1)+iy$. It follows that $$\{T^ng_0\tau \mid n \in \mathbb Z\} = \{(x+n)+iy \mid n \in \mathbb Z\} $$ and therefore the set of real parts of the numbers $\{T^ng_0\tau \mid n \in \mathbb Z\}$ is equal to $$\{x+n \mid n \in \mathbb Z\} $$ Since adding an integer does not change the fractional part of the number, there exists an integer $m$ such that $x+m \in [0,1)$. If $x+m \in [0,1/2)$ then set $n=m$, whereas if $x+m \in [1/2,1)$ then set $n=m-1$, and now you've found the integer $n$ such that $x+n \in [-1/2,1/2)$. Therefore, the real part of $T^n g_0 \tau$ is in $[-1/2,1/2]$.