A Liouville hyperbolic souvlaki

TL;DR

This paper constructs specific bounded-degree graphs with unique hyperbolic and Liouville properties, providing counterexamples to existing conjectures and questions in graph theory and geometric group theory.

Contribution

It introduces new examples of graphs with hyperbolic and Liouville properties that challenge previous assumptions and conjectures.

Findings

Constructed a transient bounded-degree graph with no transient subgraph embeddable in finite genus surfaces.

Created a Liouville, Gromov-hyperbolic graph with trivial boundary and no transient subtree.

Provided counterexamples to conjectures by Benjamini and Schramm.

Abstract

We construct a transient bounded-degree graph no transient subgraph of which embeds in any surface of finite genus. Moreover, we construct a transient, Liouville, bounded-degree, Gromov--hyperbolic graph with trivial hyperbolic boundary that has no transient subtree. This answers a question of Benjamini. This graph also yields a (further) counterexample to a conjecture of Benjamini and Schramm.