Abstract
This paper presents a curvature-free version of the Log ( 2 k β 1 ) \text {Log}(2k-1) Theorem of Anderson, Canary, Culler, and Shalen [J. Differential Geometry 44 (1996), pp. 738β782]. It generalizes a result by Hou [J. Differential Geometry 57 (2001), no. 1, pp. 173β193] and its proof is rather straightforward once we know the work by Lim [Trans. Amer. Math. Soc. 360 (2008), no. 10, pp. 5089β5100] on volume entropy for graphs. As a byproduct we obtain a curvature-free version of the Collar Lemma in all dimensions.
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