Amenable groups

Definition

Amenable groups are a class of groups that exhibit a certain kind of 'smallness' or 'finiteness' in their behavior, specifically relating to the existence of Følner sequences, which allow for averages over the group to be well-behaved. This concept connects closely with growth functions, as amenable groups typically have polynomial growth rather than exponential growth, indicating that they don't expand too rapidly. Additionally, examples of amenable groups include abelian groups and finite groups, while non-examples can include free groups and non-abelian simple groups.

5 Must Know Facts For Your Next Test

1. An amenable group can be characterized by the existence of a left-invariant mean on bounded functions defined on the group.
2. All finite groups are amenable because they can trivially have Følner sequences, where each sequence consists of the entire group.
3. Abelian groups are always amenable due to their structure allowing for the existence of suitable Følner sequences.
4. Free groups are classic examples of non-amenable groups because they exhibit exponential growth and lack Følner sequences.
5. The property of being amenable is preserved under taking extensions; if a group has an amenable normal subgroup, the quotient group will also be amenable.

Review Questions

• How does the concept of growth functions relate to amenable groups and what can it tell us about their structure?
  • Growth functions are essential in understanding the behavior of amenable groups because they typically exhibit polynomial growth, indicating that as you look at larger and larger portions of the group, their size doesn't expand too quickly. This contrasts with non-amenable groups which often show exponential growth. Thus, examining the growth function provides insights into whether a group is amenable and helps categorize its complexity based on how it expands.
• Discuss the significance of Følner sequences in determining whether a group is amenable and provide an example.
  • Følner sequences are crucial for identifying amenable groups as they provide a method for averaging functions over the group. If a group has a Følner sequence, it implies that one can find a form of stability in the average values taken across finite subsets. For example, consider the integers under addition; their Følner sequences might consist of intervals like {0,...,n}, showing how averages can stabilize and leading us to classify them as an amenable group.
• Evaluate how understanding amenable groups influences our perspective on more complex structures within group theory.
  • Understanding amenable groups helps highlight fundamental properties that differentiate simple and complex structures within group theory. By identifying amenability, we can better comprehend the limitations and behaviors of various group types. For instance, knowing that free groups are non-amenable guides us in recognizing their more intricate dynamics compared to finite or abelian groups. This contrast aids in shaping theories around stability, averaging, and even applications in areas like functional analysis and probability theory on groups.

Related terms

Følner Sequence: A sequence of finite subsets of a group that captures the idea of averaging over the group in a way that the average becomes increasingly stable as the size of the subsets grows.

Growth Function: A function that describes how the number of elements within a ball of radius n in a group grows as n increases, which can be polynomial, exponential, or other types.

Non-amenable Groups: Groups that do not possess Følner sequences or have exponential growth, typically leading to complicated structures and behaviors.