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Electron capture by slow multiply charged ions colliding on rare-gas targets is known to populate highly excited states of the projecile with very large cross sections. For double electron capture, (n, n') series with n = n' are, in general, preferentially populated. Though theseslates arcalmost camplctelyautoioniilng, important radiative decay following double capture has been reported. Recently the fluorescence from n<< n' Rydberg states has been observed. These findings are important since they strongly modify the charge and radiation balance of the multiple charge exchange reactions. We propose here a mechanism based on post-collisional effects which feeds Rydberg states from the quasi-symmetrical doubly excited states initially populated. A simple quantitative estimate of this effect is proposed on the basis of the well known 'Barker and Berry' formula.

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We prove that the profinite completion of the fundamental group of a compact 3-manifold M satisfies a Tits alternative: if a closed subgroup H does not contain a free pro-p subgroup for any p, then H is virtually soluble, and furthermore of a very particular form. In particular, the profinite completion of the fundamental group of a closed, hyperbolic 3-manifold does not contain a subgroup isomorphic to Z 2 . This gives a profinite characterization of hyperbolicity among irreducible 3-manifolds. We also characterize Seifert fibred 3-manifolds as precisely those for which the profinite completion of the fundamental group has a non-trivial procyclic normal subgroup. Our techniques also apply to hyperbolic, virtually special groups, in the sense of Haglund and Wise. Finally, we prove that every finitely generated pro-p subgroup of the profinite completion of a torsion-free, hyperbolic, virtually special group is free pro-p.In a heuristic commonly used to distinguish 3-manifolds M and N, one computes all covers M 1 , . . . , M m of M and N 1 , . . . , N n of N up to some small degree, and then compares the resulting finite lists (M i ) and (N j ). If they can be distinguished, then one has a proof that M and N were not homeomorphic.It is natural to ask whether this method always works. A more precise question was formulated by Calegari-Freedman-Walker in [CFW10], who asked whether the fundamental group of a 3-manifold is determined by its finite quotients, or equivalently, by its profinite completion. (A standard argument shows that two finitely generated groups have the same set of finite quotients if and only if their profinite completions are isomorphic. In fact, the Geometrization theorem is equivalent to the Hyperbolization theorem and the Seifert conjecture, together with the Elliptization theorem, which asserts that M is spherical if and only if π 1 M is finite [Sco83]. Since 3-manifold groups are residually finite [Hem87], π 1 M is finite if and only if its profinite completion is, so there is no distinct profinite analogue of the Elliptization theorem. Theorems A and B can therefore be thought of as providing 2 a complete profinite analogue of the Geometrization theorem (although one should note, of course, that our proofs rely essentially on Geometrization). In Theorem 8.4, we proceed to show that the profinite completion of the fundamental group detects the geometry of a closed, orientable, irreducible 3-manifold.Alternatively, Theorem A can be thought of as a classification result for the abelian subgroups of profinite completions of fundamental groups of hyperbolic manifolds. In fact, we prove a much more general 'Tits alternative' for profinite completions of 3-manifold groups. We use the notation Z π to denote p∈π Z p .Theorem C. If M is any compact 3-manifold and H is a closed subgroup of π 1 M that does not contain a free non-abelian pro-p subgroup for any prime p then H is on the following list:1. H is conjugate to the completion of a virtually soluble subgroup of π 1 M; or, where π and σ a...