Let $G$ be an infinite group generated by all its order two elements.
Is there something interesting that can be said about such $G$?
The group $G$ I had in mind is the group of automorphisms of $k[x,y]$, $k$ is a field of characteristic zero; if I am not wrong, $G$ equals its subgroup generated by all involutions (the group generated by all involutions is a normal subgroup of $G$, and this fact somehow implies my claim, hopefully).
Please see this very nice paper which implies, in case the group of automorphisms of $k[x,y]$ indeed equals its subgroup generated by all involutions (of $k$-algebras), that every $k$-algebra automorphism of $k[x,y]$ is a product of at most four involutions (since a $k$-algebra involution is in particular a $k$-vector space involution; just forget the multiplication in $k[x,y]$).
Thank you very much!
One thing you can say of a group $G$ generated by its elements of order 2 is that its abelianization $G_{\mathrm{ab}}=G/[G,G]$ is a 2-elementary abelian group. Indeed, $G_{\mathrm{ab}}$, being a quotient of $G$ is also generated by elements of order 2, and for an abelian group, this means being 2-elementary abelian (which in turn is equivalent to being isomorphic to a vector space over $\mathbf{Z}/2\mathbf{Z}$).