Amenability

Definition

Amenability refers to a property of groups that allows for the existence of a finitely additive invariant mean on bounded functions defined on the group. This concept is closely tied to the idea of how groups can be 'nicely' approximated by their finite subgroups, influencing the growth and classification of groups. Amenable groups exhibit certain behaviors in their structure that can provide insight into their growth types and their classification in terms of polynomial growth.

5 Must Know Facts For Your Next Test

1. An amenable group can be approximated by its finite subgroups, which suggests that they have a form of controllable structure and behavior.
2. All abelian groups are amenable, which indicates that simple algebraic properties can lead to this significant topological characteristic.
3. Groups with polynomial growth are always amenable, showing a direct relationship between growth types and amenability.
4. The existence of an invariant mean on an amenable group allows for averaging processes that can be used in various mathematical analyses.
5. Non-amenable groups often exhibit 'large' or 'complicated' behavior, making them less tractable in certain mathematical contexts.

Review Questions

• How does amenability influence the classification of groups and their growth types?
  • Amenability directly impacts the classification of groups as it provides a criterion for distinguishing between different types of groups based on their structural properties. For example, amenable groups can be characterized by their ability to be approximated by finite subgroups, leading to connections with their growth rates. Specifically, groups exhibiting polynomial growth are always amenable, revealing how these concepts intertwine in group theory.
• Discuss the relationship between Gromov's Theorem and the concept of amenability in groups.
  • Gromov's Theorem establishes a significant connection between growth rates and the structural nature of groups, stating that a finitely generated group has polynomial growth if and only if it is virtually nilpotent. This directly ties into the concept of amenability because all groups with polynomial growth are also amenable. Thus, Gromov's work highlights how these two properties can help categorize and understand groups within geometric group theory.
• Evaluate how the property of amenability might affect the study of infinite groups compared to finite ones.
  • The study of infinite groups with the property of amenability presents unique challenges and insights compared to finite groups. Infinite amenable groups can be analyzed through their approximation by finite subgroups, allowing for certain averaging processes that are not applicable to non-amenable infinite groups. In contrast, non-amenable infinite groups often showcase more complex behavior that resists simplification, thereby necessitating different analytical approaches. Understanding amenability thus becomes crucial in delineating between manageable infinite structures and those that are more resistant to traditional methods.

Related terms

Finitely Additive Mean: A finitely additive mean is a function that assigns an average value to bounded functions over a group in such a way that it respects finite disjoint unions of sets.

Growth Rate: The growth rate of a group describes how the number of elements in its balls of finite radius increases with respect to the radius, influencing the classification of groups.

Gromov's Theorem:

Gromov's Theorem states that a finitely generated group has polynomial growth if and only if it is virtually nilpotent, connecting amenability with growth characteristics.