Controlling LEF growth in some group extensions

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.