Følner sequence

Definition

A følner sequence is a sequence of finite subsets of a group that grows in size while satisfying a specific condition related to the group’s action on itself. It provides a way to study the asymptotic behavior of groups, particularly in terms of their amenability, by examining how the group elements can be approximated by these finite sets. The concept is essential for analyzing properties like the existence of a left-invariant mean on a group.

5 Must Know Facts For Your Next Test

1. Følner sequences are used to establish whether a group is amenable; if such a sequence exists, the group is amenable.
2. A Følner sequence satisfies the condition that for every finite subset of the group, the number of elements from that subset that intersect with the Følner sets grows proportionally to the size of the Følner sets themselves.
3. The existence of a Følner sequence can help demonstrate properties like the convergence of averages over the group.
4. Følner sequences provide insight into the large-scale geometry of groups, influencing areas such as geometric group theory and topology.
5. In many cases, particularly for finitely generated groups, constructing explicit Følner sequences can be achieved through careful selection of growing finite subsets.

Review Questions

• How does a Følner sequence help in determining whether a group is amenable?
  • A Følner sequence provides a framework to analyze whether a group is amenable by showing that for any finite subset of the group, there exists an associated Følner set such that the intersection with that finite set behaves proportionally as both grow. If such sequences can be constructed for a group, it indicates that averages taken over these sets converge and thus confirms the group's amenability. The relationship between Følner sequences and invariant means is crucial here.
• Discuss how constructing explicit Følner sequences for finitely generated groups can influence their analysis in geometric group theory.
  • Constructing explicit Følner sequences for finitely generated groups aids in understanding their asymptotic behavior and large-scale geometry. This construction often involves selecting finite subsets that grow in a controlled manner, allowing researchers to explore properties such as growth rates and convergence behaviors in relation to left-invariant means. This analysis is critical in geometric group theory because it connects algebraic properties with geometric structures, enhancing our comprehension of group actions on spaces.
• Evaluate the implications of Følner sequences in relation to left-invariant means and their role in studying group dynamics.
  • Følner sequences have significant implications for the existence and properties of left-invariant means within groups. The connection between these sequences and invariant means reveals insights into how functions can be averaged over groups while maintaining structural integrity under translations. Evaluating this relationship deepens our understanding of not just individual groups but also broader categories of groups with shared characteristics. By analyzing these connections, one can uncover patterns in group dynamics, which are essential in various mathematical applications, including probability theory and harmonic analysis.

Related terms

Amenable Group: A type of group that has an invariant mean on bounded functions, which often relates to the existence of a Følner sequence.

Left-Invariant Mean: A functional that assigns an average value to functions defined on a group while being invariant under left translations by group elements.

Group Action: An operation by which a group acts on a set, allowing for the study of symmetries and structures within that set.