is hyperbolic, then for all $x\in T$, the projection of $x$ on $A_g$ is the projection of $x$ on $[g{^{-1}}.x,g.x]$. In particular, if the midpoint of $[x,g.x]$ exists, then it lies in $A_g$. It also follows that if $g$ is hyperbolic and if $g{^{-1}}.x,x,g.x$ are aligned (in any order) then they lie on the axis of $g$.

If an abelian group $A$ acts by isometries on $\Lambda$–tree $T$ and contains a hyperbolic element $g$, then all the hyperbolic elements of $A$ have the same axis $l$, $A$ contains no inversion, and all elliptic elements fix $l$. We say that $l$ is the *axis* of the abelian group $A$. The axis of $A$ can be characterized as the only *closed* $A$–invariant linear subtree of $T$, or as the only *maximal* $A$–invariant linear subtree of $T$.

Finally if a group $\Gamma$ has a non-abelian action without inversion on a $\Lambda$–tree $T$, then there is a unique minimal non-empty $\Gamma$–invariant subtree ${T_{\mathrm{min}}}(\Gamma)$ [@Chi_introduction Theorem 3.4.1]. This applies if $\Gamma$ is non-abelian and acts freely on $T$. Note however that ${T_{\mathrm{min}}}(\Gamma)$ is not a closed subtree of $T$ in general (even if $\Lambda={{\mathbb{R}}}$); if $\Lambda$ is not archimedean, ${T_{\mathrm{min}}}(\Gamma)$ may also fail to coincide 