Reduction of orbits to the standard domain

Fix $N>0$ and suppose $\mathrm{\mid }c\tau +d\mathrm{\mid }\le N$. Since $\mathrm{Im}(c\tau +d)=cy$ and $\mathrm{Re}(c\tau +d)=cx+d$, [F1] gives $\mathrm{\mid }c\mathrm{\mid }y\le N$, that is $\mathrm{\mid }c\mathrm{\mid }\le N\mathrm{/}y$; by [F2] only finitely many integers $c$ satisfy this. For each such $c$, [F1] gives $\mathrm{\mid }cx+d\mathrm{\mid }\le N$, so $-N-cx\le d\le N-cx$, and [F2] leaves only finitely many integers $d$. Hence only finitely many pairs $(c,d)$ satisfy $\mathrm{\mid }c\tau +d\mathrm{\mid }\le N$.

The set $V:=\{\mathrm{\mid }c\tau +d\mathrm{\mid }:\gamma \in {\mathrm{\Gamma }}^{\mathrm{\prime }}\text{ with bottom row }(c,d)\}$ is nonempty (the identity has value $\mathrm{\mid }0\cdot \tau +1\mathrm{\mid }=1$) and every element is positive. Applying 1.1 with $N=1$ shows that the elements of $V$ are among the finitely many numbers $\mathrm{\mid }c\tau +d\mathrm{\mid }$ attached to pairs with $\mathrm{\mid }c\tau +d\mathrm{\mid }\le 1$, together with values $>1$; hence $V$ has a least element $m>0$, realized by some ${\gamma }_0\in {\mathrm{\Gamma }}^{\mathrm{\prime }}$. Since $\mathrm{\Im }(\gamma \cdot \tau )=y\mathrm{/}\mathrm{\mid }c\tau +d{\mathrm{\mid }}^2$, the point ${\tau }_0:={\gamma }_0\cdot \tau$ of the orbit has maximal imaginary part $y\mathrm{/}{m}^2$: for every $\gamma \in {\mathrm{\Gamma }}^{\mathrm{\prime }}$ with bottom row $(c,d)$ one has $\mathrm{\mid }c\tau +d\mathrm{\mid }\ge m$, so $\mathrm{\Im }(\gamma \cdot \tau )=y\mathrm{/}\mathrm{\mid }c\tau +d{\mathrm{\mid }}^2\le y\mathrm{/}{m}^2=\mathrm{\Im }{\tau }_0$. Choose $n\in \mathbb{Z}$ with $\mathrm{\mid }\mathrm{\Re }{\tau }_0-n\mathrm{\mid }\le 1\mathrm{/}2$, possible by taking $n=\lfloor \mathrm{\Re }{\tau }_0+1\mathrm{/}2\rfloor$ [F2]; then ${\tau }_1:=T^{-n}\cdot {\tau }_0$ satisfies $\mathrm{\Im }{\tau }_1=\mathrm{\Im }{\tau }_0$ (translation does not change the imaginary part) and $\mathrm{\mid }\mathrm{\Re }{\tau }_1\mathrm{\mid }=\mathrm{\mid }\mathrm{\Re }{\tau }_0-n\mathrm{\mid }\le 1\mathrm{/}2$. If $\mathrm{\mid }{\tau }_1\mathrm{\mid }<1$, then $S{\tau }_1\in {\mathrm{\Gamma }}^{\mathrm{\prime }}\cdot \tau$ and $\mathrm{\Im }(S{\tau }_1)=\mathrm{\Im }{\tau }_1\mathrm{/}\mathrm{\mid }{\tau }_1{\mathrm{\mid }}^2>\mathrm{\Im }{\tau }_1$ by [F1] and [F3], contradicting the maximality of $\mathrm{\Im }{\tau }_0=\mathrm{\Im }{\tau }_1$. Hence $\mathrm{\mid }{\tau }_1\mathrm{\mid }\ge 1$, and ${\tau }_1\in \bar{D}$ is the required ${\mathrm{\Gamma }}^{\mathrm{\prime }}$-equivalent point.