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

Abstract

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 $[1,2]$.

References in corpus (2)

• Random walks and isoperimetric profiles under moment conditions
• About the speed of random walks on solvable groups

Cited by in corpus (8)

• Følner functions and the generic Word Problem for finitely generated amenable groups
• Separation profiles, isoperimetry, growth and compression
• Shalom's property $H_{\mathrm{FD}}$ and extensions by $\mathbb{Z}$ of locally finite groups
• Amenability versus non-exactness of dense subgroups of a compact group
• Densely related groups
• Soluble groups with no $\mathbb{Z}\wr \mathbb{Z}$ sections
• Poincaré profiles of lamplighter diagonal products
• Isoperimetric profiles and random walks on some groups defined by piecewise actions