Open Access
2017 A Liouville hyperbolic souvlaki
Johannes Carmesin, Bruno Federici, Agelos Georgakopoulos
Electron. J. Probab. 22: 1-19 (2017). DOI: 10.1214/17-EJP44


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. In an appendix by Gábor Pete and Gourab Ray, our construction is extended to yield a unimodular graph with the above properties.


Download Citation

Johannes Carmesin. Bruno Federici. Agelos Georgakopoulos. "A Liouville hyperbolic souvlaki." Electron. J. Probab. 22 1 - 19, 2017.


Received: 14 April 2016; Accepted: 5 March 2017; Published: 2017
First available in Project Euclid: 25 April 2017

zbMATH: 1361.05028
MathSciNet: MR3646062
Digital Object Identifier: 10.1214/17-EJP44

Primary: 05C63 , 05C81 , 31C20 , 57M15

Keywords: amenability , flow , Harmonic function , hyperbolic graph , infinite graph , Liouville property , transience

Vol.22 • 2017
Back to Top