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Entropy and drift in word hyperbolic groups

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Abstract

The fundamental inequality of Guivarc’h relates the entropy and the drift of random walks on groups. It is strict if and only if the random walk does not behave like the uniform measure on balls. We prove that, in any nonelementary hyperbolic group which is not virtually free, endowed with a word distance, the fundamental 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 of Ancona and Blachère–Haïssinsky–Mathieu. For non-admissible measures, this follows from a counting result, interesting in its own right: we show that, in any infinite index subgroup, the number of non-distorted points is exponentially small compared to the growth of balls in the whole group. The uniformity is obtained by studying the behavior of measures that degenerate towards a measure supported on an elementary subgroup.

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Acknowledgements

Frédéric Mathéus would like to thank Pr. De-Jun Feng for suggesting Propositions 6.2 and 6.3.

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Authors and Affiliations

  1. Laboratoire Jean Leray, CNRS UMR 6629, Université de Nantes, 2 rue de la Houssinière, 44322, Nantes, France

    Sébastien Gouëzel

  2. L.M.B.A., UMR 6205, Univ. Bret. Sud, BP 573, 56017, Vannes, France

    Frédéric Mathéus

  3. CNRS, IRMAR - UMR 6625, Univ. Rennes, 35000, Rennes, France

    François Maucourant

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  1. Sébastien Gouëzel
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  3. François Maucourant
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Correspondence to Sébastien Gouëzel.

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Gouëzel, S., Mathéus, F. & Maucourant, F. Entropy and drift in word hyperbolic groups. Invent. math. 211, 1201–1255 (2018). https://doi.org/10.1007/s00222-018-0788-y

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