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Controlling LEF growth in some group extensions.

Henry Bradford1

  • 1Christ's College, University of Cambridge, St Andrew's Street, Cambridge, CB2 3BU UK.

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Summary
This summary is machine-generated.

Researchers show that smooth growth functions between n! and exp(exp(n)) can approximate the LEF growth function of finitely generated groups. This finding advances understanding of group growth and structure.

Keywords:
Finitely presented groupsLEFSchreier graphs

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Area of Science:

  • Group Theory
  • Geometric Group Theory
  • Metric Geometry

Background:

  • Finitely generated groups are fundamental objects in mathematics.
  • The LEF (locally embeddable finite) growth function quantifies the complexity of these groups.
  • Understanding group growth is crucial for classifying and analyzing group structures.

Purpose of the Study:

  • To investigate the possible values of LEF growth functions for finitely generated groups.
  • To determine if a wide range of growth functions can be realized by such groups.
  • To explore the relationship between group structure and its growth properties.

Main Methods:

  • Studying orders of finite groups admitting local embeddings in word metrics.
  • Estimating the LEF growth of semidirect products of finitely supported permutations and groups.
  • Utilizing sequences of finitely presented subgroups with short relative presentations.

Main Results:

  • Proved that any sufficiently smooth increasing function between n! and exp(exp(n)) is close to an LEF growth function.
  • Established estimates for the LEF growth of specific group constructions (FSym(Ω) ⋊ Γ and EΩ(R) ⋊ Γ).
  • Demonstrated the flexibility of LEF growth functions in approximating various growth rates.

Conclusions:

  • The LEF growth function is highly versatile, capable of approximating a broad spectrum of growth rates.
  • The study provides new insights into the structure of finitely generated groups via their growth functions.
  • This work contributes to the ongoing effort to connect algebraic properties of groups with their geometric behavior.