Profinite groups acting on C-trees
The first section of this chapter is concerned with fixed points under the action of a pro- π group acting on π -tree. In particular, it is proved that if a pro- π group acts on a π -tree, the subset of fixed points is a π -subtree (if it is not empty) and that a finite π -group acting on a π -tree fixes a vertex. As a consequence it is shown that the smallest π -subtree [ v, w] containing two distinct vertices v, w of a π -tree must contain edges. One also deduces that under some mild conditions, if a profinite group G acts on a π -tree, then this tree contains a unique minimal G -invariant π -subtree. This is a very useful tool in many applications. The second section contains a description of the structure of a pro- π group that acts faithfully and irreducibly on a π -tree: it must have a nonabelian free pro- p subgroup with an induced free action, or solvable of a very specific form. More generally, one has a description of the possible alternative structures of a pro- π group that acts on a π -tree without fixed points: it contains a nonabelian pro- p subgroup that acts freely or the quotient modulo the stabilizer of some edge is solvable of a special type.
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Ribes, L. (2017). Profinite groups acting on C-trees. In Ergebnisse der Mathematik und ihrer Grenzgebiete. doi:10.1007/978-3-319-61199-0_4