, Volume 3, Issue 1, pp 51-89

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

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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. uniformly exponential growth,

  2. growth tightness,

  3. regularity of growth series,

  4. classical growth series (with respect to word lengths),

  5. growth series with respect to weights,

  6. complete growth series,

  7. spectral radius of simple random walks on Cayley graphs.

The first author acknowledges support from the Foundation National Suisse de la Recherche Scientifique, from the Russian Foundation for Fundamental Research, Grant 96-01-00974, and from INTAS, Grant 94-3420.