A Liouville hyperbolic souvlaki
  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. 
Johannes Carmesin, Bruno Federici and Agelos Georgakopoulos
math.CO math.PR