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Quasi-regular graphs, cogrowth, and amenability

Pages: 678 - 687, Issue Special, July 2003

 Abstract        Full Text (183.6K)              

Sam Northshield - Dept. of Mathematics, Plattsburgh State University, Plattsburgh, NY 12901, United States (email)

Abstract: We extend Grigrochuk’s cogrowth criterion for amenability of groups to the case of non-regular graphs for which a certain regularity condition is satisfied. The proof involves generalized Laplacians which are inverses of growth series and whose determinants are closely related to zeta functions of graphs.

Keywords:  Amenability, growth, cogrowth, Martin boundary, generalized Laplacian, harmonic functions.
Mathematics Subject Classification:  Primary 31C20; Secondary 31C35, 43A07, 31C05, 39A12.

Received: September 2002;      Revised: March 2003;      Published: April 2003.