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

Jérémie Brieussel Institut/Laboratoire Montpelliérain Alexander Grothendieck (IMAG) (UMR 5149), Université de Montpellier, 34090 Montpellier, France Tianyi Zheng Department of Mathematics, Stanford University, Stanford (Palo Alto) CA 94305

Group Theory and Lie Theory Metric Geometry Probability mathscidoc:2203.17002

Annals of Mathematics, 193, (1), 1-105, 2021.1
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 Lp-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 Lp-compression exponent of a group and its wreath product with the cyclic group forp in [1,2].
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@inproceedings{jérémie2021speed,
  title={Speed of random walks, isoperimetry and compression of finitely generated groups},
  author={Jérémie Brieussel, and Tianyi Zheng},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220318130511383436007},
  booktitle={Annals of Mathematics},
  volume={193},
  number={1},
  pages={1-105},
  year={2021},
}
Jérémie Brieussel, and Tianyi Zheng. Speed of random walks, isoperimetry and compression of finitely generated groups. 2021. Vol. 193. In Annals of Mathematics. pp.1-105. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220318130511383436007.
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