Translation length functions of non-simplicial trees Let $G$ be a finitely generated group. By a theorem of Culler and Morgan, the set of non-abelian (not necessarily simplicial) minimal $\mathbb{R}$-trees with isometric $G$-action injects into the infinite-dimensional vector space $\mathbb{R}^G$ by assigning to each non-abelian tree $T$ its translation length function $l_T\colon G\to\mathbb{R},\ g\mapsto l_T(g)=\inf_{x\in T}d(x,gx)$.
Is there a way to read off of $l_T\in\mathbb{R}^G$ if $T$ is simplicial? For example, if the entries of $l_T$ are bounded below by some positive constant $c>0$, can $T$ still be non-simplicial?
Here, by a simplicial tree I mean an $\mathbb{R}$-tree whose set of vertices (i.e. points that when removed disconnect the tree into more than two components) is discrete and closed.
 A: A partial answer: suppose in addition that there are finitely many
orbits of branch points. (Branch point = vertex in the OQ.) This condition has been
considered by Jiang and Guirardel. For example Jiang
showed that this condition is satisfied if the given minimal action (of f.g.
$G$ on an $\mathbb{R}$-tree) is free.
Note $\ell(g)\leq L_x(g)=d(x,gx)$ for all $x\in X$, so if the
non-zero values of $\ell$ are bounded away from zero by $c$ then
$d(x,gx)>c$ for all $x\in X$ and $g\in G$, that is, orbits of each
point are discrete. Therefore the set of all branch points, as a
finite union of closed and discrete subsets is closed and discrete.
In the opposite direction, if $x$ is a branch point and $c_x>0$ such
that every other branch point is further from $x$ than $c_x$, then
$L_x(g)> c_x$ for all $g$ that doesn't fix $x$. Since $L_{\gamma
x}(g)=L_x(\gamma^{-1}g\gamma)$, the number $c_x$ depends only on the orbit, not
on $x$ itself. Choosing $c_x$ for each orbit and taking the minimum
$c$ we get $L_x(g)=d(x,gx)>c$ for all branch points $x$.
However, making the step from $L_x(g)$ bounded away from zero to
$\ell(g)$ bounded away from zero requires more input.
A: This question has been answered by Misha, in the comments, as follows.

Given a discrete length function (I think, it suffices to assume it us bounded below) one defines a discrete pretree on which the group still acts. This discrete pretree extends canonically to a discrete tree, so the group still acts. This was worked out by Bowditch and Guirardel long ago

