Subgroups of Fundamental Groups of Graphs of Groups

Abstract

This chapter contains structural results about subgroups of fundamental groups $\varPi(\mathcal{G}, \varGamma)$ of graphs of profinite groups $(\mathcal{G}, \varGamma)$; these include free and amalgamated products of profinite groups, HNN extensions, etc. Unlike fundamental groups of graphs of abstract groups, where often one can use a combinatorial description of the elements of the fundamental group, to obtain structural results in the profinite case one relies exclusively on geometric methods, primarily analysis of actions of profinite groups on profinite trees. In particular there are results on finite subgroups, normalizers, normal subgroups, etc.

As an application, this chapter contains an analogue of the classical Kurosh subgroup theorem for the free product of abstract groups. It describes the structure of an open subgroup $H$ of a free pro-$\mathcal{C}$ product of pro-$\mathcal{C}$ groups as a free pro-$\mathcal{C}$ product of a free pro-$\mathcal{C}$ group and intersections of $H$ with conjugates of the free factors. The original free product may have a finite or infinite number of free factors.