Amenable group

Definition

An amenable group is a type of group that has a specific property related to its actions on sets, particularly in terms of measures and averages. This property allows for the existence of an invariant mean on the space of bounded functions defined on the group. Amenable groups are significant because they connect group theory with analysis and topology, influencing various areas such as harmonic analysis and ergodic theory.

5 Must Know Facts For Your Next Test

1. Amenable groups include all finite groups, abelian groups, and certain types of infinite groups, like free groups on two or more generators.
2. A crucial characterization of amenable groups is that they admit an invariant mean for bounded functions, which implies that averages converge in a certain sense.
3. Amenability can be tested using Følner sequences, which provide a method for verifying if a group can be averaged over its finite subsets.
4. Certain properties of amenable groups make them particularly useful in various branches of mathematics, including ergodic theory, where they often appear in the study of dynamical systems.
5. Every amenable group is also a member of the class of groups that have the fixed point property, meaning they exhibit certain geometric behaviors related to topological spaces.

Review Questions

• What are some key properties that define an amenable group, and how do they connect to its role in measure theory?
  • Key properties that define an amenable group include the existence of an invariant mean and the ability to construct Følner sequences. These properties connect deeply to measure theory since they allow for averaging processes over the group's elements, leading to conclusions about their actions on various sets. In this way, amenable groups serve as a bridge between abstract algebra and analysis.
• Discuss how the concept of Følner sequences is used to determine whether a group is amenable.
  • Følner sequences provide a practical approach to determining if a group is amenable by examining finite subsets within the group. If there exists such a sequence where the ratio of the size of intersections with translates remains bounded, it suggests that averages taken over these subsets will converge to a limit. This behavior reflects the underlying structure of the group, leading to insights into its amenability.
• Evaluate the implications of amenability in various mathematical contexts, including its influence on dynamics and topology.
  • The implications of amenability extend across multiple mathematical fields, influencing areas such as dynamical systems and topology. In dynamics, amenable groups often facilitate the existence of invariant measures and ergodic properties, allowing for predictable behavior over time. In topology, amenable groups relate to spaces exhibiting fixed point properties, showing how algebraic structures can interact with topological aspects, thereby enriching our understanding of both disciplines.

Related terms

Invariant Mean: An invariant mean is a functional that assigns an average value to a bounded function over a group in such a way that it remains unchanged under the action of the group.

Fixed Point Property:

The fixed point property refers to the characteristic of a space where every continuous map from the space into itself has at least one fixed point.

Følner Sequence: A Følner sequence is a sequence of finite subsets of a group that allows for the averaging of functions over these subsets, facilitating the study of amenability.