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We give a solution to the inverse problem (given a function, find acorresponding group) for large classes of speed, entropy, isoperimetricprofile, return probability and $L_p$-compression functions of finitelygenerated groups of exponential volume growth. For smaller classes, we givesolutions among solvable groups. As corollaries, we prove a recent conjectureof Amir on joint evaluation of speed and entropy exponents and we obtain a newproof of the existence of uncountably many pairwise non-quasi-isometricsolvable groups, originally due to Cornulier and Tessera. We also obtain aformula relating the $L_p$-compression exponent of a group and its wreathproduct with the cyclic group for $p$ in $[1,2]$.

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