Journal of Dynamical and Control Systems

, Volume 3, Issue 1, pp 51-89

First online:

On problems related to growth, entropy, and spectrum in group theory

  • R. GrigorchukAffiliated withSteklov Mathematical Institute
  • , P. De La HarpeAffiliated withSection de Mathématiques, Université de Genève, C.P. 240

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We review some known results and open problems related to the growth of groups. For a finitely generated group Γ, given whenever necessarytogether with a finite generating set, we discuss the notions of
  1. (1)

    uniformly exponential growth,

  2. (2)

    growth tightness,

  3. (3)

    regularity of growth series,

  4. (4)

    classical growth series (with respect to word lengths),

  5. (5)

    growth series with respect to weights,

  6. (6)

    complete growth series,

  7. (7)

    spectral radius of simple random walks on Cayley graphs.


1991 Mathematics Subject Classification


Key words and phrases

Growth growth series entropy spectral radius Laplacian group Cayley graph quasi-isometry