Metabelian groups with large return probability

TL;DR

This paper characterizes metabelian groups based on their return probability decay rate, linking it to algebraic properties like Krull dimension, and provides bounds and embeddings respecting this dimension.

Contribution

It offers a purely algebraic characterization of return probability decay in finitely generated metabelian groups and extends embedding theorems to respect Krull dimension.

Findings

Characterization of groups with return probability ~exp(-n^{1/3})

Lower bounds on return probability for groups with torsion derived subgroup

A variation of Kaloujinine and Krasner's embedding theorem respecting Krull dimension

Abstract

We study the return probability of finitely generated metabelian groups. We give a characterization of metabelian groups whose return probability is equivalent to $\exp(-n^\frac1{3} )$ in purely algebraic terms, namely the Krull dimension of the group. Along the way, we give lower bounds on the return probability for metabelian groups with torsion derived subgroup, according to the dimension. We also establish a variation of the famous embedding theorem of Kaloujinine and Krasner for metabelian groups that respects the Krull dimension.