On the topology of relative orbits for actions of algebraic groups over complete fields (Q617805)

Suppose \(k\) is a field which is complete with respect to a non-trivial valuation of real rank \(1\) (for example, \(k\) could be a \(\mathfrak{p}\)-adic field). Let \(G\) be a linear algebraic \(k\)-group acting on an affine \(k\)-variety \(V\). Suppose \(x \in V(k)\) is a closed \(k\)-point. The main aim of this paper is to investigate connections between the Zariski closedness of the \(G\)-orbit \(G\cdot x\) in \(V\) and the Hausdorff closedness of the so-called relative orbit \(G(k)\cdot x\) in \(V(k)\). The methods involve finding ways of equipping certain cohomology groups with a topology. The paper contains a review of many of the existing results in the literature, in particular results by \textit{D. Birkes} [Ann. Math. (2) 93, 459--475 (1971; Zbl 0198.35001)] and \textit{R. J. Bremigan} [J. Reine Angew. Math. 453, 21--47 (1994; Zbl 0808.14040)], and also presents some new results. The authors state that proofs of their results will appear elsewhere.