Fixed Price of Groups and Percolation

TL;DR

This paper establishes a dichotomy for finitely generated groups, showing they either have fixed price or exhibit infinitely many infinite clusters in some Cayley graph under Bernoulli percolation.

Contribution

It proves a new fundamental dichotomy linking fixed price property and percolation behavior in finitely generated groups.

Findings

Either the group has fixed price or some Cayley graph has infinitely many infinite clusters.

The result applies to all finitely generated groups, unifying group theory and percolation theory.

Provides a new perspective on the relationship between group properties and percolation phenomena.

Abstract

We prove that for every finitely generated group $\Gamma$, at least one of the following holds: (1) $\Gamma$ has fixed price; (2) each of its Cayley graphs $G$ has infinitely many infinite clusters for some Bernoulli percolation on $G$.