$\forall M \in \mathrm{SL}(2, \mathbb{R})$ has a neighborhood diffeomorphic to $\mathbb{R}^3$?

Question

$\forall M \in \mathrm{SL}(2, \mathbb{R})$ has a neighborhood diffeomorphic to $\mathbb{R}^3$?

Known that $\mathrm{SL}(2, \mathbb{R})$ is a three-dimensional manifold.

Answer 1

(Just so this question has an answer.)

The answer to your question is yes. Any point in a smooth $n$-dimensional manifold has a neighbourhood diffeomorphic to $\mathbb{R}^n$ (this is part of the definition). In particular, any $M \in \operatorname{SL}(2, \mathbb{R})$ has a neighbourhood diffeomorphic to $\mathbb{R}^3$ as $\operatorname{SL}(2, \mathbb{R})$ is a smooth three-dimensional manifold.