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The fundamental inequality of Guivarc'h relates the entropy and the drift ofrandom walks on groups. It is strict if and only if the random walk does notbehave like the uniform measure on balls. We prove that, in any nonelementaryhyperbolic group which is not virtually free, endowed with a word distance, thefundamental inequality is strict for symmetric measures with finite support,uniformly for measures with a given support. This answers a conjecture of S.Lalley. For admissible measures, this is proved using previous results ofAncona and Blach{\`e}re-Ha{\"i}ssinsky-Mathieu. For non-admissible measures,this follows from a counting result, interesting in its own right: we showthat, in any infinite index subgroup, the number of non-distorted points isexponentially small. The uniformity is obtained by studying the behavior ofmeasures that degenerate towards a measure supported on an elementary subgroup.

Entropy and drift in word hyperbolic groups