Orbit Closures of the Witt Group Actions

We prove that for any prime $p$ there exists an algebraic action of the two-dimensional Witt group $W_2(p)$ on an algebraic variety $X$ such that the closure in $X$ of the $W_2(p)$-orbit of some point $x\in X$ contains infinitely many $W_2(p)$-orbits. This is related to the problem of extending, from the case of characteristic zero to the case of characteristic $p$, the classification of connected affine algebraic groups $G$ such that every algebraic $G$-variety with a dense open $G$-orbit contains only finitely many $G$-orbits.