A Liouville hyperbolic souvlaki
Published version
Peer-reviewed
Repository URI
Repository DOI
Change log
Authors
Carmesin, J
Federici, B
Georgakopoulos, A
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. In an appendix by Gábor Pete and Gourab Ray, our construction is extended to yield a unimodular graph with the above properties.
Description
Keywords
Liouville property, hyperbolic graph, infinite graph, amenability, transience, flow, harmonic function
Journal Title
Electronic Journal of Probability
Conference Name
Journal ISSN
1083-6489
1083-6489
1083-6489
Volume Title
22
Publisher
Institute of Mathematical Statistics
Publisher DOI
Sponsorship
Supported by an EPSRC grant EP/L505110/1, and by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 639046).