## On the topology of relative orbits for actions of algebraic groups over complete fields.(English)Zbl 1217.14031

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 D. Birkes [Ann. Math. (2) 93, 459–475 (1971; Zbl 0198.35001)] and 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.

### MSC:

 14L24 Geometric invariant theory 14L30 Group actions on varieties or schemes (quotients) 20G15 Linear algebraic groups over arbitrary fields

### Keywords:

closed orbits; local fields; algebraic group actions

### Citations:

Zbl 0198.35001; Zbl 0808.14040
Full Text:

### References:

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