A Notion of Dimension based on Probability on Groups

TL;DR

This paper introduces a new notion of dimension for infinite groups based on percolation theory, linking probabilistic decay rates to geometric and algebraic properties of groups.

Contribution

It defines the percolation dimension of groups, explores its properties, and relates it to growth rates, providing a novel perspective connecting probability and group theory.

Findings

Percolation dimension is monotone decreasing with respect to subgroups and quotients.

For certain groups, percolation dimension coincides with the growth rate exponent.

The notion provides a new tool to analyze the structure of infinite groups.

Abstract

We introduce notions of dimension of an infinite group, or more generally, a metric space, defined using percolation. Roughly speaking, the percolation dimension $pdim(G)$ of a group $G$ is the fastest rate of decay of a symmetric probability measure $\mu$ on $G$, such that Bernoulli percolation on $G$ with connection probabilities proportional to $\mu$ behaves like a Poisson branching process with parameter 1 in a sense made precise below. We show that $pdim(G)$ has several natural properties: it is monotone decreasing with respect to subgroups and quotients, and coincides with the growth rate exponent for several classes of groups.