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

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    $\begingroup$ 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. $\endgroup$
    – abx
    Commented Jul 20, 2017 at 6:09
  • $\begingroup$ @abx Very great comment. Thank you! $\endgroup$
    – Ali Taghavi
    Commented Jul 20, 2017 at 6:13
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    $\begingroup$ 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. $\endgroup$
    – YCor
    Commented Jul 20, 2017 at 8:10
  • $\begingroup$ @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? $\endgroup$
    – Ali Taghavi
    Commented Jul 20, 2017 at 9:04
  • $\begingroup$ @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). $\endgroup$
    – Ali Taghavi
    Commented Jul 20, 2017 at 9:55

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