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

By Jérémie Brieussel, Tianyi Zheng. Annals of Mathematics, 2015

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We give a solution to the inverse problem (given a function, find a corresponding group) for large classes of speed, entropy, isoperimetric profile, return probability and $L_p$-compression functions of finitely generated groups of exponential volume growth. For smaller classes, we give solutions among solvable groups. 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 $[1,2]$.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2152fc4d0a55dedf1cd4fea4666a6397dab1749c