  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.
