Let $\Gamma$ be a lattice in $G = \text{SL}(2,\mathbb{R})$ and consider the subgroups $$ N^- := \Bigl\{ \begin{pmatrix}1 & 0 \\ x & 1\end{pmatrix} : x \in \mathbb{R}\Bigr\} $$ and $$ A^+ := \Bigl\{ \begin{pmatrix}e^\tau & 0 \\ 0 & e^{-\tau}\end{pmatrix} : \tau \geq 0\Bigr\} $$ and $$ P := \Bigl\{ \begin{pmatrix}* & * \\ 0 & *\end{pmatrix} \in G\Bigr\} $$ Then one knows that $U := N^- P$ is dense and open in $G$ and $\Gamma u (A^+)^{-1}$ is dense in $G$ for all $u \in U$. My question is:
Given $g \in G$, can we construct sequences $(a_n)_n$ in $A^+$, $(\gamma_n)_n$ in $\Gamma$ and $(u_n)_n$ in $U$ such that the following holds?
- $\gamma_nu_n^{-1}a_n^{-1} \to g$
- $u_n \to e$
A statement like this is used in Mostow's book Strong Rigidity of Locally Symmetric Spaces in the proof of Lemma 8.5 on page 65 without explanation. However, starting with a sequence $(u_n)_n \to e$, I can not see how to get the sequences $(\gamma_n)_n$ and $(a_n)_n$ in a uniform way.
Edit: I have now written down the question with specific groups.