# The group of polynomial homeomorphism of the plane

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$(\mathbb{R}^n)$?

• This group has been heavily studied in algebraic geometry. For a nice survey, you can have a look at Algebraic Automorphisms of Affine Space by H. Kraft, Progress in Math. 80 (Birkhäuser), p. 81-105. – abx Jul 20 '17 at 6:09
• @abx Very great comment. Thank you! – Ali Taghavi Jul 20 '17 at 6:13
• When a group $G$ is given "is $G$ a topological group" makes no sense. One should rather ask whether $G$ admits interesting/natural group topologies. – YCor Jul 20 '17 at 8:10
• @YCor I think In the first version of the question, I had also pointed out to compact open topology. May be I did not understand your comment, correctly? – Ali Taghavi Jul 20 '17 at 9:04
• @YCor What about if we replace the compact open topology by another one: For example We identify $G$ with certain subspace of $\mathbb{R}^{\infty}$. with the product topology?(The identification of the space of polynomials with the space of their coefficient). – Ali Taghavi Jul 20 '17 at 9:55