Journal
ANNALS OF MATHEMATICS
Volume 193, Issue 1, Pages 1-105Publisher
Princeton Univ, Dept Mathematics
DOI: 10.4007/annals.2021.193.1.1
Keywords
random walks; speed; entropy; uniform embedding of groups; wreath products; isoperimetry
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The study addresses the inverse problem of finite generated groups involving speed, entropy, isoperimetric profile, return probability, and L-p-compression functions. Additionally, it proves a recent conjecture related to joint evaluation of speed and entropy exponents and provides a new proof of the existence of uncountably many pairwise non-quasi-isometric solvable groups. Furthermore, a formula linking the L-p-compression exponent of a group and its wreath product with the cyclic group for p in [1, 2] is obtained.
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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