9.1 Definition and characterizations of amenable groups

Amenable groups are a fascinating class of groups with various equivalent characterizations. They allow for a finitely additive, left-invariant probability measure on all subsets, satisfying conditions like the Følner and Reiter properties. These groups play a crucial role in many areas of mathematics.

The study of amenable groups connects deeply to the Banach-Tarski paradox and paradoxical decompositions. Understanding these relationships provides insights into group-theoretic properties, geometric paradoxes, and their implications in fields like ergodic theory and measure theory.

Amenable Groups: Definitions and Characterizations

Fundamental Definitions and Conditions

Top images from around the web for Fundamental Definitions and Conditions

• 

  Open Source Synthesis and Verification Tool for Fixed-to-Floating and Floating-to-Fixed Points ... View original

  Is this image relevant?

• 

  Category:Paradoxical decomposition - Wikimedia Commons View original

  Is this image relevant?

• 

  On the question of two-step nucleation in protein crystallization - Faraday Discussions (RSC ... View original

  Is this image relevant?

• 

  Open Source Synthesis and Verification Tool for Fixed-to-Floating and Floating-to-Fixed Points ... View original

  Is this image relevant?

• 

  Category:Paradoxical decomposition - Wikimedia Commons View original

  Is this image relevant?

1 of 3

Top images from around the web for Fundamental Definitions and Conditions

• 

  Open Source Synthesis and Verification Tool for Fixed-to-Floating and Floating-to-Fixed Points ... View original

  Is this image relevant?

• 

  Category:Paradoxical decomposition - Wikimedia Commons View original

  Is this image relevant?

• 

  On the question of two-step nucleation in protein crystallization - Faraday Discussions (RSC ... View original

  Is this image relevant?

• 

  Open Source Synthesis and Verification Tool for Fixed-to-Floating and Floating-to-Fixed Points ... View original

  Is this image relevant?

• 

  Category:Paradoxical decomposition - Wikimedia Commons View original

  Is this image relevant?

1 of 3

• Amenable groups admit a finitely additive, left-invariant probability measure on all subsets
• Følner condition characterizes amenable groups through existence of finite subset sequences with specific properties
• Reiter condition provides characterization of amenability using approximate invariant functions
• Fixed point property for affine actions on compact convex sets equivalently characterizes amenability
• Paradoxical decomposition property absence characterizes amenable groups
• Growth rate determines amenability (subexponential growth implies amenability)
• Invariant states existence on group's C*-algebra offers operator-algebraic characterization

Examples and Applications

• Finite groups are amenable (uniform probability measure satisfies conditions)
• Abelian groups are amenable (Haar measure provides invariant measure)
• Solvable groups are amenable (built from amenable subgroups)
• Free groups on two or more generators are non-amenable (exhibit paradoxical decompositions)
• Amenability preserves under group operations (subgroups, quotients, extensions)
• Thompson's group F amenability remains an open problem in group theory
• Amenable groups find applications in ergodic theory, harmonic analysis, and operator algebras

Amenability and Invariant Means

Invariant Mean Properties

• Invariant mean defined as linear functional on bounded real-valued functions space
• Left-invariant mean existence on group equivalent to group amenability
• Amenability and invariant means relationship extends to discrete and continuous groups
• Von Neumann's amenability definition directly relates to invariant means existence
• Bounded real-valued functions space on group ($l^{\infty}(G)$) crucial in invariant means study
• Invariant means concept extends to amenable actions of groups on spaces
• Amenability and invariant means relationship impacts ergodic theory and dynamical systems study

Constructing and Analyzing Invariant Means

• Banach limits used to construct invariant means on amenable groups
• Følner sequences facilitate approximate construction of invariant means
• Invariant means on amenable groups satisfy additional properties (positivity, normalization)
• Day's fixed point theorem connects invariant means to fixed points of affine actions
• Invariant means on non-amenable groups constructed using ultrafilters (non-constructive)
• Invariant means study leads to important results in functional analysis (Riesz representation theorem)
• Amenable groups' invariant means used to prove ergodic theorems (von Neumann's ergodic theorem)

Equivalence of Amenability Characterizations

Proof Strategies and Techniques

• Equivalence proof involves showing each characterization implies another cyclically
• Følner condition serves as starting point to prove equivalence with other characterizations
• Fixed point property and invariant means existence equivalence proven by constructing appropriate affine actions
• Reiter condition and invariant means existence equivalence established using convolution operators
• Paradoxical decomposition property negation shown equivalent to amenability using measure-theoretic arguments
• Amenability and invariant states existence on C*-algebras equivalence proven using functional analysis techniques
• Growth rate characterization connected to other properties through group elements and subsets analysis

Key Theorems and Results

• Følner's theorem establishes equivalence between Følner condition and amenability
• Hulanicki's theorem connects amenability to existence of invariant mean on $l^{\infty}(G)$
• Kesten's theorem relates amenability to spectral properties of random walks on groups
• Day-von Neumann theorem equates amenability with fixed point property for affine actions
• Tarski's alternative links amenability to absence of paradoxical decompositions
• Grigorchuk-Cohen theorem connects amenability to groups with subexponential growth
• Connes-Tauer theorem establishes equivalence between amenability and hyperfiniteness of von Neumann algebras

Amenability and the Banach-Tarski Paradox

Paradoxical Decompositions and Group Actions

• Banach-Tarski paradox states solid ball in three-dimensional space decomposes and reassembles into two identical copies
• Amenable groups do not allow paradoxical decompositions (directly related to Banach-Tarski paradox)
• Free group on two generators (non-amenable) plays crucial role in Banach-Tarski paradox construction
• Amenability and Banach-Tarski paradox connection involves equidecomposability concept in group actions
• Hausdorff paradox (Banach-Tarski precursor) closely related to certain groups' non-amenability
• Amenability study in Banach-Tarski paradox context led to important descriptive set theory and measure theory results
• Connection understanding provides insights into group-theoretic properties and geometric paradoxes relationship

Implications and Extensions

• Tarski's circle-squaring problem resolved using amenability and Banach-Tarski paradox connection
• Amenable groups actions preserve Lebesgue measure (contrasts with Banach-Tarski paradox)
• Banach-Tarski paradox generalized to higher dimensions and other geometries (hyperbolic spaces)
• Paradoxical decompositions study led to development of paradoxical measure theory
• Amenability and Banach-Tarski paradox relationship extends to more general group actions (amenable actions)
• Connection between amenability and Banach-Tarski paradox impacts logic and set theory (axiom of choice role)
• Understanding this relationship crucial for analyzing measure-preserving group actions in ergodic theory