Invariant spanning double rays in amenable groups

TL;DR

This paper establishes an equivalence between amenability of finitely generated groups and the existence of invariant random spanning double rays in their Cayley graphs, extending known results about invariant spanning trees.

Contribution

It introduces the concept of invariant random spanning double rays in powers of Cayley graphs, generalizing previous results on invariant spanning trees and amenability.

Findings

Equivalence between amenability and invariant double rays in Cayley graphs.

Extension of invariant spanning tree results to double rays in graph powers.

Provides new characterizations of amenability through invariant spanning structures.

Abstract

A well-known result of Benjamini, Lyons, Peres, and Schramm states that if $G$ is a finitely generated Cayley graph of a group $\Gamma$, then $\Gamma$ is amenable if and only if $G$ admits a $\Gamma$-invariant random spanning tree with at most two ends. We show that this is equivalent to the existence of a $\Gamma$-invariant random spanning double ray in a power of $G$.