Reading Ratner's paper "Ragunathan's conjectures for SL(2,R)"

Question

Hello everyone (interested),

I am trying to read Marina Ratner's paper "Ragunathan's conjectures for $SL_{2}(R)$" (Israel Journal of Mathematics 80 (1992), 1-31). There is a claim right at the end of page 7 which puzzles me (I reproduce it below).

Firstly, so far I could not come up with a proof of this claim, but presumably even more seriously, I have no idea what the motivation/underlying idea is. Probably I am missing something obvious. Perhaps someone out there knows this? I'll copy the claim in question for you, but it perhaps makes more sense to look it up in the actual paper:

Let $\Gamma$ be a non-uniform (= not cocompact) lattice in $SL_{2}(R)$. Then there are $r_{0}>1,g_{1},\ldots,g_{n}\in SL_{2}(R)$ and $\gamma_{1},\ldots,\gamma _{n}\in\Gamma$ so that for $E_{i}:=\{g_{i}% \begin{pmatrix} e^{t} & \\ & e^{-t}% \end{pmatrix}% \begin{pmatrix} 1 & \ast\\ & 1 \end{pmatrix}% \begin{pmatrix} \cos\theta & \sin\theta\\ -\sin\theta & \cos\theta \end{pmatrix} \mid r_{0}<t<\infty\}$ we have

1. $g_{i}^{-1}\gamma_{i}g_{i}=\{% \begin{pmatrix} 1 & s\\ & 1 \end{pmatrix} \mid s<0\}$ for $i=1,\ldots,n$;
2. $\Gamma\setminus SL_{2}(R)-\cup_{i}\{{E_{i}}\}$ is compact,
3. $\gamma_{i}E_{i}=E_{i}$ for all $i=1,\ldots,n$;
4. $\gamma E_{i}\cap E_{i}=\varnothing$ for $\gamma\in\Gamma-<g_{i}>$
5. $\gamma E_{i}\cap E_{j}=\varnothing$ for $i\neq j,$ $\gamma\in\Gamma$ arbitrary,
6. $d(x,(\Gamma \setminus \operatorname*{id})x)=d(x,\gamma_{i}x)$ for $x\in E_{i}$.

Why is this true? Is there some intuition or geometric meaning in the $E_{i}$? Thank you very much for your help!

Answer 1

The $g_i$'s represent the cusps of $\Gamma\backslash\mathrm{PSL}_2(\mathbb{R})/\mathrm{PSO}_2(\mathbb{R})=\Gamma\backslash\mathcal{H}$, the $E_i$'s are corresponding cuspidal zones, and $\gamma_i$'s are the parabolic motions fixing the corresponding cusps. See Section 2.2 in Iwaniec: Spectral methods of automorphic forms.