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 $[1,2]$.