Let $G$ be the set of all homeomorphisms $f$ of $\mathbb{R}^2$ such that

both $f$ and $f^{-1}$ are polynomial maps.

We equip $G$ with the compact open topology and the obvious group structure.

Is $G$ with the compact open topology a topological group? Is it an infinite dimensional Lie Group?What can be said about finite dimensional Lie groups which are contained in $G$?(Are there some examples other than (subgroups of ) Affine$(\mathbb{R}^n)$?

Algebraic Automorphisms of Affine Spaceby H. Kraft, Progress in Math. 80 (Birkhäuser), p. 81-105. $\endgroup$ – abx Jul 20 '17 at 6:09