9.2 Følner sequences and their properties
5 min read•july 30, 2024
Følner sequences are a key tool in understanding . These finite subsets of a group become increasingly invariant under group actions, providing a concrete way to visualize and work with the abstract concept of amenability.
By studying Følner sequences, we can bridge group theory and functional analysis. They allow us to construct invariant means, prove ergodic theorems, and analyze group structure. This connection opens up applications in diverse areas of mathematics and beyond.
Følner Sequences for Amenable Groups
Definition and Characterization
Top images from around the web for Definition and Characterization
Dual graph - Wikipedia View original
Is this image relevant?
Category:Geometric group theory - Wikimedia Commons View original
Is this image relevant?
Dual graph - Wikipedia View original
Is this image relevant?
Category:Geometric group theory - Wikimedia Commons View original
Is this image relevant?
1 of 2
Top images from around the web for Definition and Characterization
Dual graph - Wikipedia View original
Is this image relevant?
Category:Geometric group theory - Wikimedia Commons View original
Is this image relevant?
Dual graph - Wikipedia View original
Is this image relevant?
Category:Geometric group theory - Wikimedia Commons View original
Is this image relevant?
1 of 2
- Følner sequences consist of finite subsets of a group becoming increasingly invariant under group actions
- Sequence (Fn) of finite subsets of G satisfies for any g in G, symmetric difference between Fn and gFn, normalized by size of Fn, approaches zero as n approaches infinity
- Provide geometric characterization of amenable groups connecting abstract amenability to concrete subsets
- Existence of equates to group amenability
- Allow construction of invariant means on amenable groups bridging group theory and functional analysis
- Generalize to metric spaces and graphs extending applicability beyond group theory
Mathematical Formulation
- Formal definition uses symmetric difference and limit notation:
- states for finite K ⊂ G and ε > 0, finite F ⊂ G exists where |KF \ F| < ε|F|
- Concept of almost invariant sets crucial finite subsets close to invariant under action of given finite subset
- Følner ratio quantifies measuring proportion of elements remaining after group action
Examples and Applications
- Finite groups always have Følner sequences (single set containing all elements)
- Infinite cyclic group Z has Følner sequence Fn = {-n, -n+1, ..., n-1, n}
- Free groups lack Følner sequences demonstrating non-amenability
- Used in to prove mean ergodic theorem for amenable group actions
- Applied in operator algebras to study group von Neumann algebras and crossed products
Existence of Følner Sequences
Proof Strategy
- Relies on Følner condition for every finite K ⊂ G and ε > 0, finite F ⊂ G exists with |KF \ F| < ε|F|
- Involves iterative process starting with arbitrary finite subset
- Progressively refines subset to satisfy Følner condition for increasing sequence of finite subsets and decreasing ε values
- Utilizes concept of almost invariant sets finite subsets close to invariant under action of given finite subset
- Employs compactness arguments (Tychonoff's theorem) to extract Følner sequence from net of almost invariant sets
- Demonstrates equivalence between Følner sequences and other amenability characterizations (invariant means, Reiter condition)
Key Steps and Concepts
- Begin with countable dense subset {g1, g2, ...} of G and decreasing sequence εn approaching 0
- Construct sequence of finite sets Fn satisfying |giFn Δ Fn| < εn|Fn| for i ≤ n
- Use diagonal argument to extract subsequence satisfying Følner condition for all gi
- Prove resulting sequence satisfies Følner condition for all g ∈ G using density of {gi}
- Utilize measure-theoretic concepts (symmetric difference, normalized measures)
- Apply Banach-Alaoglu theorem to construct invariant mean from Følner sequence
Examples and Counterexamples
- Amenable groups (abelian, solvable) always possess Følner sequences
- Construct explicit Følner sequence for Z2: Fn = {(x,y) ∈ Z2 : |x|, |y| ≤ n}
- (free groups, groups containing free subgroups) lack Følner sequences
- Lamplighter group Z2 ≀ Z has Følner sequence despite exponential growth
Properties of Følner Sequences
Asymptotic Invariance
- Key property sequences become increasingly invariant under group actions
- Quantified using Følner ratio measuring proportion of elements remaining after group action
- Asymptotic invariance expressed mathematically:
- Implies convergence of normalized characteristic functions in L1(G)
- Allows approximation of group elements by finite subsets with controlled error
Growth and Structure
- Følner sequences exhibit subadditivity Følner ratio of union bounded by sum of individual ratios
- Growth rate related to group's growth providing insights into structure and properties
- For groups of polynomial growth, Følner sequences can be chosen with polynomial growth rate
- Følner sequences in amenable groups can be chosen to be nested (Fn ⊂ Fn+1)
- Quasi-invariance property for any finite K ⊂ G, Følner sequence eventually almost invariant under its action
- Følner sequences relate to isoperimetric inequalities and amenability at infinity
Examples and Applications
- In Z, Følner sequence [-n,n] has linear growth matching group's growth rate
- Heisenberg group has Følner sequence with cubic growth reflecting its polynomial growth rate
- Følner sequences in lamplighter group Z2 ≀ Z exhibit exponential growth
- Used to compute L2-Betti numbers and von Neumann dimension of group representations
- Apply in analysis of random walks on groups studying return probabilities and entropy
Applications of Følner Sequences
Invariant Means and Ergodic Theory
- Construct invariant means on amenable groups providing concrete realization of amenability
- Develop ergodic theorems for amenable group actions generalizing classical results
- Prove pointwise ergodic theorem for amenable groups using Følner sequences
- Establish Ornstein-Weiss lemma fundamental result in ergodic theory of amenable group actions
- Apply to study unique ergodicity and mixing properties of group actions
Group Invariants and Structure
- Compute group invariants (L2-Betti numbers, von Neumann dimension of representations)
- Analyze asymptotic behavior of random walks (return probabilities, entropy)
- Prove results about amenable group structure (existence of nilpotent subgroups of bounded index)
- Study growth functions and isoperimetric profiles of amenable groups
- Investigate quasi-isometry invariants of groups using Følner sequences
Extended Mathematical Applications
- Operator algebras use Følner sequences to study group von Neumann algebras and crossed products
- Dynamical systems apply Følner sequences in topological dynamics and symbolic dynamics
- Geometric measure theory utilizes Følner sequences for differentiation theorems on groups
- Number theory employs Følner sequences in study of equidistribution and Diophantine approximation
- Theoretical computer science uses Følner sequences in analysis of group-based algorithms
Key Terms to Review (18)
Amenable groups: 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.
Asymptotic Group Theory: Asymptotic group theory studies the behavior of groups at infinity, often through the lens of large-scale geometry and combinatorial properties. This area focuses on how group properties change when looking at sequences of finite groups or examining the limits of group actions on spaces, like metric spaces. It connects deeply with concepts such as Følner sequences, which help capture the essence of how groups can behave in asymptotic contexts.
Asymptotic Invariance: Asymptotic invariance refers to the property of a sequence or a function where certain invariants remain unchanged as one approaches infinity. This concept is crucial in understanding Følner sequences, as it helps establish how the behavior of groups can be approximated by their actions on large sets. It highlights the relationship between group growth and properties that do not change under scaling, providing a framework for analyzing asymptotic behavior in group theory.
Boundedness: Boundedness refers to a property of a Følner sequence where the size of the sets in the sequence remains uniformly bounded by some fixed value, regardless of their position in the sequence. This characteristic ensures that the sets do not grow indefinitely and allows for meaningful averaging processes within the context of group actions. It serves as a crucial condition in establishing various properties related to amenability and the asymptotic behavior of groups.
Cayley Graph: A Cayley graph is a graphical representation of a group that illustrates the structure of the group by connecting its elements with edges based on group operations. This graph helps visualize relationships between group elements and can reveal geometric properties, which makes it a vital tool in understanding group theory from a geometric perspective.
E. a. van den berg: E. A. van den Berg is a mathematician known for his work on Følner sequences, which are important in the study of the asymptotic behavior of groups in geometric group theory. His contributions help clarify the relationships between group actions and their invariant measures, emphasizing how these sequences can be used to analyze the growth of groups and their properties in a measurable way.
Erdős–Szekeres Theorem: The Erdős–Szekeres theorem states that any sequence of at least $n^2$ distinct real numbers contains a monotonic subsequence of length at least $n$. This theorem is significant in combinatorial mathematics, particularly in the study of order types and the properties of sequences, which relates closely to Følner sequences by examining structural properties of sets within groups.
Ergodic theory: Ergodic theory is a branch of mathematics that studies the long-term average behavior of dynamical systems and their statistical properties. It connects concepts from measure theory and topology, focusing on how a system evolves over time and whether its time averages converge to space averages. This theory plays a significant role in understanding group actions on spaces, particularly in relation to Følner sequences, amenable groups, and various key theorems.
Følner condition: The følner condition is a property of groups that describes the existence of certain subsets, known as Følner sets, which allow for a form of averaging over the group that approximates the behavior of the group as a whole. A group satisfies the følner condition if, for every finite subset of the group and for every positive number, there exists a finite subset such that the boundary of this subset is small relative to its size. This concept is closely linked to the study of amenable groups, their sequences, and various examples that illustrate the condition's relevance.
Følner sequence: 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.
Gromov's Theorem: Gromov's Theorem states that a finitely generated group has polynomial growth if and only if it is virtually nilpotent, meaning it contains a nilpotent subgroup of finite index. This theorem connects geometric properties of groups to algebraic structures, providing a crucial bridge between group theory and geometric group theory.
Hyperbolic Groups: Hyperbolic groups are a class of finitely generated groups that exhibit a certain geometric property, characterized by the existence of a hyperbolic metric on their Cayley graphs. This property leads to unique and powerful features, such as exponential growth rates and specific algebraic structures, making them crucial in understanding geometric group theory.
Invariant measure: An invariant measure is a measure that remains unchanged under the action of a group on a measurable space. This concept is crucial for understanding how dynamics interact with geometrical structures, particularly in the study of Følner sequences, which help analyze the asymptotic behavior of groups acting on spaces. Invariant measures can provide insights into ergodic theory and the long-term behavior of dynamical systems by ensuring that certain properties remain stable under transformations.
M. gromov: M. Gromov is a mathematician known for his foundational contributions to geometric group theory, particularly in the development of the concept of Følner sequences. These sequences are used to analyze the asymptotic behavior of groups and their actions on spaces, providing a way to study the properties of groups through their geometric structures.
Measure-preserving action: A measure-preserving action is a group action on a measurable space that leaves the measure of sets invariant. This means that if a group acts on a space, the measure of any measurable set remains the same after the action is applied, making it essential in the study of dynamical systems and ergodic theory. It connects to key concepts like Følner sequences because these sequences are used to analyze the asymptotic behavior of the measures associated with group actions, particularly in understanding how they relate to invariant measures.
Non-amenable groups: Non-amenable groups are groups that do not admit a Følner sequence, which means they cannot be approximated by finite subsets in a way that their growth is controlled by the group's structure. These groups exhibit certain properties that distinguish them from amenable groups, such as a lack of invariant means or specific types of growth that are not manageable. Understanding non-amenable groups is crucial in studying the interplay between group actions and their geometric properties.
Probabilistic methods: Probabilistic methods refer to a set of techniques in mathematics and computer science that leverage probability theory to address problems and derive results. These methods often provide insights or solutions where deterministic approaches may fall short, particularly in cases involving randomness or uncertainty. They are particularly useful in combinatorics, computer science, and geometric group theory, where they can help demonstrate the existence of certain structures or properties without explicitly constructing them.
Symmetric generating set: A symmetric generating set for a group is a set of group elements that generates the group and is invariant under taking inverses, meaning if an element is in the set, its inverse is also included. This property ensures that the set is symmetric, making it useful for analyzing the structure of the group and understanding its symmetries. In the context of Følner sequences, symmetric generating sets play a crucial role in defining properties related to amenability and growth rates of groups.