subdivision for simplicial trees to get rid of inversions: consider $\Tilde\Lambda=\frac12 \Lambda$, and let $\Tilde T$ be the $\Tilde\Lambda$–tree obtained by the extension of scalars $\Lambda\subset\Tilde\Lambda$. Then the natural extension of $g$ to $\Tilde T$ fixes a unique point in $\Tilde T$ (in particular, $g$ is elliptic in $\Tilde T$). If $g$ is elliptic or is an inversion, its translation length $l_T(g)$ is defined to be $0$.

If $g$ is hyperbolic, then the set $A_g$ is non-empty, and is a maximal linear subtree of $T$, and is thus closed in $T$. It is called *the axis* of $g$. Moreover, the restriction of $g$ to $A_g$ is conjugate to the action of a translation $\tau\co a\mapsto a+l_T(g)$ on a $\tau$–invariant convex subset of $\Lambda$ for some positive $l_T(g)\in\Lambda$. The *translation length* of $g$ is the element $l_T(g)\in\Lambda_{>0}$. If $p$ is the projection of $x$ on $A_g$, then for $k\neq 0$, $d(x,g^k.x)=2d(x,p)+|k|l_T(g)$.

Note that it may happen that $A_g$ is not isometric to $\Lambda$. For instance, if $\Lambda={{\mathbb{R}}}^2$, the axis of an element $g$ with infinitesimal translation length can be of the form $I\times {{\mathbb{R}}}$ where $I$ is any non-empty interval in ${{\mathbb{R}}}$ which can be open, semi-open or closed.

If $g$ 