Weighted cogrowth formula for free groups

Johannes Jaerisch, Katsuhiko Matsuzaki

Research output: Contribution to journalArticlepeer-review


We investigate the relationship between geometric, analytic and probabilistic indices for quotients of the Cayley graph of the free group Cay.Fn/ by an arbitrary subgroup G of Fn. Our main result, which generalizes Grigorchuk’s cogrowth formula to variable edge lengths, provides a formula relating the bottom of the spectrum of weighted Laplacian on Gn Cay.Fn/ to the Poincaré exponent of G. Our main tool is the Patterson–Sullivan theory for metric trees.

Original languageEnglish
Pages (from-to)349-368
Number of pages20
JournalGroups, Geometry, and Dynamics
Issue number2
Publication statusPublished - 2020


  • Bottom of spectrum
  • Cayley graph of free group
  • Cogrowth
  • Discrete Laplacian
  • Green function
  • Patterson–Sullivan theory
  • Spectral radius

ASJC Scopus subject areas

  • Geometry and Topology
  • Discrete Mathematics and Combinatorics


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