  This paper presents a curvature-free version of the Log(2k-1) Theorem of
Anderson, Canary, Culler & Shalen [ACCS96]. It generalizes a result by Hou
[Hou01] and its proof is rather straightforward once we know the work by Lim
[Lim08] on volume entropy for graphs. As a byproduct we obtain a curvature-free
version of the Collar Lemma in all dimensions.
