Speed of random walks, isoperimetry and compression of finitely generated groups

Abstract

We give a solution to the inverse problem (given a prescribed function, find a corresponding group) for large classes of speed, entropy, isoperimetric profile, return probability and $L_{p}$-compression functions of finitely generated groups. For smaller classes, we give solutions among solvable groups of exponential volume growth. As corollaries, we prove a recent conjecture of Amir on joint evaluation of speed and entropy exponents and we obtain a new proof of the existence of uncountably many pairwise non-quasi-isometric solvable groups, originally due to Cornulier and Tessera. We also obtain a formula relating the $L_{p}$-compression exponent of a group and its wreath product with the cyclic group for $p$ in $[1,2]$.

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      doi = {10.4007/annals.2013.177.3.7},
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      publisher = {Amer. Math. Soc., Providence, RI},
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      isbn = {978-0-8218-4739-8},
      mrclass = {60J10 (60-01 60J05 60K35 60K37 68U20 68W20)},
      mrnumber = {2466937},
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      volume = {42},
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      year = {2016},
      pages = {xv+699},
      isbn = {978-1-107-16015-6},
      mrclass = {60C05 (05C05 05C81 28A80 60J10 60J80 60K35 82B41)},
      mrnumber = {3616205},
      mrreviewer = {Laurent Miclo},
      doi = {10.1017/9781316672815},
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      year = {2008},
      pages = {Art. ID rnn 076, 34},
      issn = {1073-7928},
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      mrnumber = {2439557},
      mrreviewer = {Tatiana Smirnova-Nagnibeda},
      doi = {10.1093/imrn/rnn076},
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      journal = {Duke Math. J.},
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      mrclass = {20F65 (51F99)},
      mrnumber = {2783928},
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      author = {Saloff-Coste, Laurent},
      title = {Lectures on finite {M}arkov chains},
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      venue = {{S}aint-{F}lour, 1996},
      series = {Lecture Notes in Math.},
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      mrclass = {60J27 (15A60 47D07 60J10)},
      mrnumber = {1490046},
      mrreviewer = {James Allen Fill},
      doi = {10.1007/BFb0092621},
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Authors

Jérémie Brieussel

Université de Montpellier, Institut Montpelliérain Alexander Grothendieck, Montpellier, France

Tianyi Zheng

Department of Mathematics, Stanford, University, Stanford, CA, USA

Current address:

Department of Mathematics, UC San Diego, CA, USA