A full characterization of invariant embeddability of unimodular planar graphs

TL;DR

This paper characterizes when unimodular random planar graphs can be invariantly embedded into Euclidean or hyperbolic planes, extending previous results to graphs with multiple ends based on Halin's characterization.

Contribution

It provides a full characterization of invariant embeddability for unimodular planar graphs, including those with multiple ends, using Halin's graph-theoretic criteria.

Findings

Unimodular graphs with multiple ends can be invariantly embedded into Euclidean or hyperbolic planes.

Embedding depends on the graph's amenability.

Extension of previous results to a broader class of graphs.

Abstract

When can a unimodular random planar graph be drawn in the Euclidean or the hyperbolic plane in a way that the distribution of the random drawing is isometry-invariant? This question was answered for one-ended unimodular graphs in \cite{benjamini2019invariant}, using the fact that such graphs automatically have locally finite (simply connected) drawings into the plane. For the case of graphs with multiple ends the question was left open. We revisit Halin's graph theoretic characterization of graphs that have a locally finite embedding into the plane. Then we prove that such unimodular random graphs do have a locally finite invariant embedding into the Euclidean or the hyperbolic plane, depending on whether the graph is amenable or not.