Let G be a group and let T be a tree on which G acts. This chapter deals with minimal G -invariant subtrees of T. This is done both when G is an abstract group and T an abstract tree, and when G is a pro- C group and T a C -tree; in fact the interest lies in the study of both cases together and the relationship with each other. For example, attached to a finitely generated free-by-finite group R, there is a graph of finite groups (G, Δ) over a finite graph Δ, so that R is its fundamental group; moreover R acts naturally on the universal covering graph Sabs of this graph of groups. Let b∈ R act freely on Sabs (a hyperbolic element); then there is a unique minimal 〈 b〉 -invariant subtree Lb of Sabs (called the ‘Tits line’ or the ‘axis’ of b). In parallel, the profinite completion R of R is the profinite fundamental group of (G, Δ), and Sabs is naturally densely embedded in the standard profinite tree S of (G, Δ); then it is proved that the closure L¯b of Lb in S is precisely the unique minimal 〈 b〉 ‾ -invariant subtree of S. Analogous results can be obtained for different types of groups R, such as free or amalgamated products of abstract groups. Knowledge about these sort of minimal subtrees can be used to obtain information about properties of R, such as conjugacy of elements or subgroups.
|Series||Ergebnisse der Mathematik und ihrer Grenzgebiete|
Ribes, L. (2017). Minimal subtrees. In Ergebnisse der Mathematik und ihrer Grenzgebiete. doi:10.1007/978-3-319-61199-0_8