Electronic Journal of Probability

A Liouville hyperbolic souvlaki

Johannes Carmesin, Bruno Federici, and Agelos Georgakopoulos

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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.

Article information

Electron. J. Probab., Volume 22 (2017), paper no. 36, 19 pp.

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

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Zentralblatt MATH identifier

Primary: 57M15: Relations with graph theory [See also 05Cxx] 05C63: Infinite graphs 05C81: Random walks on graphs 31C20: Discrete potential theory and numerical methods

Liouville property hyperbolic graph infinite graph amenability transience flow harmonic function

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Carmesin, Johannes; Federici, Bruno; Georgakopoulos, Agelos. A Liouville hyperbolic souvlaki. Electron. J. Probab. 22 (2017), paper no. 36, 19 pp. doi:10.1214/17-EJP44. https://projecteuclid.org/euclid.ejp/1493085635

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