Abstract
A conjecture of Benjamini & Schramm from 1996 states that any finitely generated group that is not a finite extension of $\mathbb{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.
Highlights
Bernoulli percolation on a graph is the process where each edge of the graph is deleted or kept independently. This model has its origin in statistical physics [10], but gives rise to interesting and beautiful mathematics even in “non-realistic” geometries, such as Cayley graphs of abstract groups
Interesting in these cases is the relation between the algebraic properties of the group and the behavior of the percolation process
In this paper we are concerned with the property of the existence of a non-trivial percolation phase, usually known as “pc < 1”
Summary
Bernoulli percolation on a graph is the process where each edge of the graph is deleted or kept independently This model has its origin in statistical physics [10], but gives rise to interesting and beautiful mathematics even in “non-realistic” geometries, such as Cayley graphs of abstract groups. Interesting in these cases is the relation between the algebraic properties of the group and the behavior of the percolation process. 1.1 Percolation on groups Let G be a finitely generated group. By using Gromov’s theorem regarding groups of polynomial growth [17], and the structure of nilpotent groups (see e.g. Chapter 7.9, proof of Theorem 7.18, in [20]). See Theorem 6 below for the definitions and precise statement
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