⭕Geometric Group Theory Unit 9 – Amenable Groups: Følner Sequences

Key Concepts and Definitions

• Amenable groups are a class of groups that admit a finitely additive, translation-invariant probability measure
• The concept of amenability was introduced by John von Neumann in 1929 to study the Banach-Tarski paradox
• Amenability is a measure-theoretic property that captures the idea of a group having "almost invariant" finite subsets
• A group $G$ is amenable if for every $\varepsilon > 0$ and every finite subset $K \subseteq G$, there exists a finite subset $F \subseteq G$ such that $|KF \triangle F| < \varepsilon |F|$
  • $KF$ denotes the set $\{kf : k \in K, f \in F\}$
  • $\triangle$ represents the symmetric difference between sets
• The class of amenable groups includes all finite groups, abelian groups, and solvable groups
• Non-amenable groups include free groups on two or more generators and groups containing free subgroups (e.g., $SL(2, \mathbb{Z})$)

Historical Context and Development

• The study of amenable groups originated from John von Neumann's work on the Banach-Tarski paradox in 1929
• Von Neumann introduced the concept of amenability to characterize groups for which the Banach-Tarski paradox does not occur
• The term "amenable" was coined by M. M. Day in 1949, derived from the German word "messbar" meaning "measurable"
• In the 1940s and 1950s, the theory of amenable groups was further developed by mathematicians such as M. M. Day, E. Følner, and A. N. Kolmogorov
• Følner's criterion, introduced by Erling Følner in 1955, provided a combinatorial characterization of amenability using finite subsets with small boundary
• The study of amenable groups gained momentum in the 1960s and 1970s with contributions from mathematicians like F. P. Greenleaf, E. Granirer, and A. Hulanicki
• In recent decades, amenable groups have found applications in various areas of mathematics, including ergodic theory, operator algebras, and geometric group theory

Properties of Amenable Groups

• Amenable groups are closed under taking subgroups, quotients, extensions, and direct limits
• The class of amenable groups is closed under group extensions: if $N$ is a normal subgroup of $G$ and both $N$ and $G/N$ are amenable, then $G$ is amenable
• Amenable groups satisfy the Følner condition: for every $\varepsilon > 0$ and every finite subset $K \subseteq G$, there exists a finite subset $F \subseteq G$ such that $|KF \triangle F| < \varepsilon |F|$
• Amenable groups have invariant means: there exists a linear functional $m: \ell^\infty(G) \to \mathbb{R}$ such that $m(\mathbf{1}) = 1$ and $m(f) = m(f_g)$ for all $f \in \ell^\infty(G)$ and $g \in G$, where $f_g(x) = f(gx)$
• The class of amenable groups is strictly larger than the class of groups with subexponential growth
• Amenable groups satisfy the strong Følner condition: for every $\varepsilon > 0$ and every finite subset $K \subseteq G$, there exists a finite subset $F \subseteq G$ such that $|kF \triangle F| < \varepsilon |F|$ for all $k \in K$

Følner Sequences: Introduction and Significance

• A Følner sequence is a sequence of finite subsets of a group that asymptotically minimizes the ratio of the boundary size to the subset size
• Formally, a sequence $(F_n)_{n \in \mathbb{N}}$ of finite subsets of a group $G$ is a Følner sequence if for every $g \in G$, $\lim_{n \to \infty} \frac{|gF_n \triangle F_n|}{|F_n|} = 0$
• The existence of a Følner sequence characterizes amenability: a group is amenable if and only if it admits a Følner sequence
• Følner sequences provide a way to approximate invariant measures on amenable groups
• The concept of Følner sequences is central to the study of asymptotic properties of amenable groups, such as growth and isoperimetric inequalities
• Følner sequences have applications in ergodic theory, particularly in the study of invariant measures and entropy

Constructing Følner Sequences

• Constructing Følner sequences for specific amenable groups can be a challenging task
• For finitely generated abelian groups, Følner sequences can be constructed using rectangular boxes or balls in the Cayley graph
• In nilpotent groups, Følner sequences can be obtained by considering sets of the form $\{g \in G : \|g\| \leq n\}$, where $\|\cdot\|$ is a suitable norm on the group
• For solvable groups, Følner sequences can be constructed inductively using the extensions of Følner sequences for the derived series
• In some cases, Følner sequences can be obtained by considering level sets of suitable functions on the group (e.g., word length or distance functions)
• The construction of Følner sequences often relies on the specific structure and properties of the group under consideration

Examples and Applications

• The integers $\mathbb{Z}$ are amenable, and a Følner sequence is given by the intervals $F_n = \{-n, -n+1, \ldots, n-1, n\}$
• The free group on two generators, $F_2$, is not amenable, as it contains a subgroup isomorphic to the free group on countably many generators
• Amenable groups have found applications in the study of the Banach-Tarski paradox: the paradox does not occur in dimensions three and higher for amenable groups
• In ergodic theory, amenable groups are precisely the groups for which the mean ergodic theorem holds for all unitary representations
• Amenability plays a role in the study of group actions on measure spaces and the existence of invariant measures
• The Følner condition has been used to study the growth and isoperimetric properties of finitely generated groups
• Amenable groups have been studied in relation to the Novikov conjecture and the Baum-Connes conjecture in topology and operator algebras

Connections to Other Areas of Mathematics

• Amenable groups have deep connections to various areas of mathematics, including:
  • Ergodic theory: amenability is closely related to the existence of invariant measures and the study of group actions on measure spaces
  • Operator algebras: amenability plays a role in the study of group C*-algebras and von Neumann algebras
  • Geometric group theory: amenability is related to growth properties, isoperimetric inequalities, and the geometry of Cayley graphs
  • Harmonic analysis: amenability is connected to the existence of invariant means and the study of unitary representations
• The Følner condition has analogues in other settings, such as measured equivalence relations and metric spaces
• Amenability has been generalized to various other structures, including semigroups, groupoids, and measured equivalence relations
• The study of amenable groups has inspired the development of new concepts and techniques in related areas, such as sofic groups and hyperlinear groups

Advanced Topics and Open Problems

• The Følner-Adyan theorem states that a finitely presented group is amenable if and only if it does not contain a non-abelian free subgroup
• The von Neumann-Day problem asks whether every non-amenable group contains a non-abelian free subgroup; it remains open
• The Greenleaf-Ruzsa problem concerns the existence of Følner sequences with additional regularity properties, such as polynomial growth of the boundary
• The study of amenable actions and the associated Zimmer program aim to understand the structure of group actions on measure spaces
• The Baum-Connes conjecture, which relates the K-theory of a group C*-algebra to the equivariant K-homology of the group's classifying space, has been verified for large classes of amenable groups
• The Novikov conjecture, which concerns the homotopy invariance of higher signatures, has been proven for amenable groups
• The relationship between amenability and other group properties, such as Kazhdan's property (T) and the Haagerup property, is an active area of research
• Generalizations of amenability, such as coarse amenability and measured amenability, have been studied in the context of metric spaces and measured equivalence relations