Subgroups of Profinite Groups Acting on Trees

Abstract

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.

The geometry of profinite graphs with applications to free groups and finite monoids

“…We continue to use the notation [v, w] to denote this connection, which we call the geodesic between v and w. A tree-like graph must be connected since any two vertices have a connection. We begin with some basic properties of tree-like profinite graphs analogous to the case of R-trees [6], [59]. We note that a finite graph Γ is tree-like if and only if it has no circuits of length greater than one (that is, a finite graph Γ is tree-like if and only if it is a tree, perhaps with loops adjoined to some vertices).…”

“…A profinite graph is a topological graph Γ which is a projective limit of finite, discrete graphs. It is well known (see [59], [41]) that Γ is profinite if and only if V (Γ) and E(Γ) are both compact, totally disconnected Hausdorff spaces (sometimes called Boolean spaces). We shall usually refer to such spaces as profinite sets.…”

“…Following Gildenhuys and Ribes [17], and Zalesskiȋ and Mel'nikov [59], a profinite graph Γ is connected (that is, connected as a profinite graph; the terminology profinitely connected is used in [5], [6], [7], [38]) if each finite continuous morphic image is connected (as an abstract graph). For instance, Γ A ( F H (A)) is connected, as it is the projective limit of the Cayley graphs of the finite quotients of F H (A).…”

“…For a profinite ring R with unity, there is a homological notion of what it means for a profinite graph to be an R-tree [17], [59], [6], [5] (see section 9 for more details). For a pseudovariety of groups H, a graph is said to be an H-tree if it is an R-tree, where R = lim ← − {Z/nZ | Z/nZ ∈ H} is the free pro-cyclic pro-H group, considered as a ring.…”

“…A key property of R-trees, R = 0, is the following (see, for instance, [17], [59], [6]): in an R-tree Γ any pair of vertices v, w has a unique minimal connection (minimality is in terms of containment); this connection is usually denoted by [v, w]. This geometric condition is a key ingredient in the papers [5], [6], [52] and in the proof of the Ribes and Zalesskiȋ product theorem [39], [40], [52].…”