action on a tree

A tree is a graph with no loops — branch as you may, you never come back to where you started except by retracing your steps. When a group acts on a tree by symmetries (isometries), the rigid, loop-free shape forces the group's structure into the open. Where the elements move points and which points they fix becomes a blueprint you can read the whole group off of. Trees are the ideal stage because their geometry is so constrained that group actions on them cannot hide.

Concretely, a group G acts on a simplicial tree T by automorphisms — permuting vertices and edges, preserving incidence. One classifies each nontrivial element by its behavior: an elliptic element fixes a vertex, while a hyperbolic element translates along a unique invariant geodesic line (its axis) by a positive amount, its translation length. The action is without inversions if no element swaps the two ends of an edge; one can always pass to the barycentric subdivision to ensure this.

The structure of the action is captured by the quotient graph T/G together with the stabilizers of vertices and edges, assembled into a graph of groups. The two simplest patterns are emblematic: a group acting on a tree with a single edge orbit and two vertex orbits builds an amalgamated free product, while a single edge orbit and one vertex orbit builds an HNN extension. This dictionary is exactly Bass-Serre theory.