Indicable groups and $p_c<1$

TL;DR

This paper proves a conjecture relating group properties to percolation phases, using novel algebraic and probabilistic methods to establish non-trivial percolation in certain finitely generated groups.

Contribution

It confirms the Benjamini & Schramm conjecture for groups with non-trivial real-valued homomorphisms, introducing new methods combining algebraic and probabilistic techniques.

Findings

Proved the conjecture for groups with non-trivial homomorphisms into real numbers.

Reduced the problem to hereditary just-infinite groups.

Developed new methods combining EIT and evolving sets techniques.

Abstract

A conjecture of Benjamini & Schramm from 1996 states that any finitely generated group that is not a finite extension of Z has a non-trivial percolation phase. Our main results prove this conjecture for certain groups, and in particular prove that any group with a non-trivial homomorphism into the additive group of real numbers satisfies the conjecture. We use this to reduce the conjecture to the case of hereditary just-infinite groups. The novelty here is mainly in the methods used, combining the methods of EIT and evolving sets, and using the algebraic properties of the group to apply these methods.