Profinite Graphs and Groups pp 111-136 | Cite as

# Profinite Groups Acting on \(\mathcal{C}\)-Trees

## Abstract

The first section of this chapter is concerned with fixed points under the action of a pro-\(\pi\) group acting on \(\pi\)-tree. In particular, it is proved that if a pro-\(\pi\) group acts on a \(\pi\)-tree, the subset of fixed points is a \(\pi\)-subtree (if it is not empty) and that a finite \(\pi\)-group acting on a \(\pi\)-tree fixes a vertex. As a consequence it is shown that the smallest \(\pi\)-subtree \([v,w]\) containing two distinct vertices \(v,w\) of a \(\pi\)-tree must contain edges. One also deduces that under some mild conditions, if a profinite group \(G\) acts on a \(\pi\)-tree, then this tree contains a unique minimal \(G\)-invariant \(\pi\)-subtree. This is a very useful tool in many applications.

The second section contains a description of the structure of a pro-\(\pi\) group that acts faithfully and irreducibly on a \(\pi\)-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-\(\pi\) group that acts on a \(\pi\)-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.

## References

- Hall, M. Jr.: The Theory of Groups. Macmillan Co., New York (1959) Google Scholar
- Huppert, B.: Endliche Gruppen I. Die Grundlehren der Mathematischen Wissenschaften, vol. 134. Springer, Berlin (1967) Google Scholar