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Phase Transitions in Infinitely Generated Groups, and Related Problems in Additive Number Theory

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Published/Copyright: August 4, 2011
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Integers
From the journal Volume 11 Issue 4

Abstract

Let A be an infinite set of generators for a group G, and let LA(r) denote the number of elements of G whose word length with respect to A is exactly r. The growth function LA is a function from the nonnegative integers ℕ0 to the set ℕ0 ∪ {∞}. The purpose of this note is to determine all growth functions associated to infinite generating sets for groups.

Keywords.: Additive Bases; Infinitely Generated Groups; Growth in Groups; Geometric Group Theory
Received: 2009-09-30
Accepted: 2009-12-27
Published Online: 2011-08-04
Published in Print: 2011-August

© de Gruyter 2011

Articles in the same Issue

  1. The Number of Solutions of λ(x) = n
  2. Monochromatic Sums Equal to Products in ℕ
  3. On the Sum of Reciprocal Generalized Fibonacci Numbers
  4. Recursively Self-Conjugate Partitions
  5. Modular Hyperbolas and the Coefficients of (x–1 + 6 + x)k
  6. Algebraic Proof for the Geometric Structure of Sumsets
  7. An Erdős–Fuchs Type Theorem for Finite Groups
  8. Coincidences of Catalan and q-Catalan Numbers
  9. Phase Transitions in Infinitely Generated Groups, and Related Problems in Additive Number Theory
  10. Perfect Numbers with Identical Digits
  11. Remarks on the Pólya–Vinogradov Inequality
  12. Heights of Divisors of xn – 1
  13. Bernoulli Numbers and Generalized Factorial Sums
  14. Witten Volume Formulas for Semi-Simple Lie Algebras
https://doi.org/10.1515/integ.2011.037
Keywords for this article
Additive Bases; Infinitely Generated Groups; Growth in Groups; Geometric Group Theory

Articles in the same Issue

  1. The Number of Solutions of λ(x) = n
  2. Monochromatic Sums Equal to Products in ℕ
  3. On the Sum of Reciprocal Generalized Fibonacci Numbers
  4. Recursively Self-Conjugate Partitions
  5. Modular Hyperbolas and the Coefficients of (x–1 + 6 + x)k
  6. Algebraic Proof for the Geometric Structure of Sumsets
  7. An Erdős–Fuchs Type Theorem for Finite Groups
  8. Coincidences of Catalan and q-Catalan Numbers
  9. Phase Transitions in Infinitely Generated Groups, and Related Problems in Additive Number Theory
  10. Perfect Numbers with Identical Digits
  11. Remarks on the Pólya–Vinogradov Inequality
  12. Heights of Divisors of xn – 1
  13. Bernoulli Numbers and Generalized Factorial Sums
  14. Witten Volume Formulas for Semi-Simple Lie Algebras
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