Nonamenable Liouville Graphs

  • Itai Benjamini
Part of the Lecture Notes in Mathematics book series (LNM, volume 2100)


In this section we present an example of a bounded degree graph with a positive Cheeger constant (i.e. nonamenable graph) which is Liouville, that is, it admits no non constant bounded harmonic functions. This example shows that the theorem proved in Sect. 12 cannot be extended to general graphs.


Harmonic Function Binary Tree Cayley Graph General Graph Harmonic Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. [BC96]
    I. Benjamini, J. Cao, Examples of simply-connected Liouville manifolds with positive spectrum. Differ. Geom. Appl. 6(1), 31–50 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [BR11]
    I. Benjamini, D. Revelle, Instability of set recurrence and Green’s function on groups with the Liouville property. Potential Anal. 34(2), 199–206 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [BS96a]
    I. Benjamini, O. Schramm, Harmonic functions on planar and almost planar graphs and manifolds, via circle packings. Invent. Math. 126(3), 565–587 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [GKN12]
    R. Grigorchuk, V.A. Kaimanovich, T. Nagnibeda, Ergodic properties of boundary actions and the Nielsen-Schreier theory. Adv. Math. 230(3), 1340–1380 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [Gri99]
    G. Grimmett, Percolation. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 321, 2nd edn. (Springer, Berlin, 1999)Google Scholar
  6. [HLW06]
    S. Hoory, N. Linial, A. Wigderson, Expander graphs and their applications. Bull. Am. Math. Soc. (N.S.) 43(4), 439–561 (2006) (electronic)Google Scholar
  7. [Kle10]
    B. Kleiner, A new proof of Gromov’s theorem on groups of polynomial growth. J. Am. Math. Soc. 23(3), 815–829 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [KV83]
    V.A. Kaĭmanovich, A.M. Vershik, Random walks on discrete groups: boundary and entropy. Ann. Probab. 11(3), 457–490 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [Lyo87]
    T. Lyons, Instability of the Liouville property for quasi-isometric Riemannian manifolds and reversible Markov chains. J. Differ. Geom. 26(1), 33–66 (1987)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Itai Benjamini
    • 1
  1. 1.Department of MathematicsThe Weizmann Institute of ScienceRehovotIsrael

Personalised recommendations