Group action on a tree

A group action on a tree is a homomorphism to the tree's combinatorial isometry group. Barycentric subdivision removes edge inversions without changing the essential action.

Every inversion-free tree isometry is either elliptic, fixing a vertex, or hyperbolic, translating along a unique bi-infinite geodesic.

An elliptic tree isometry fixes a vertex. Its fixed-point set is a nonempty convex subtree.

A hyperbolic tree isometry has positive translation length and preserves a unique bi-infinite geodesic.

The axis of a hyperbolic tree isometry $g$ is its unique invariant line. The isometry acts on this line by translation through its translation length.

The translation length of an isometry $g$ of a metric space is $\ell (g)={\inf }_{x}d(x,gx)$. For a hyperbolic tree isometry, it is attained exactly on the axis of a tree isometry.

If two elliptic isometries $g,h$ of a tree have elliptic product $gh$, then their fixed subtrees intersect.

Finite families of convex subtrees of a tree have the Helly property: if every pair intersects, then their total intersection is nonempty.