libterm.com Amenable groups are an important concept in abstract algebra and harmonic analysis, characterized by the existence of a finitely additive, invariant mean on bounded functions defined over the group. These groups exhibit properties like fixed point theorems, measure equivalence, and have applications in ergodic theory and probability. Explore the article to understand how amenable groups influence modern mathematical theory and their practical implications.
Table of Comparison
| Property | Amenable Group | Kazhdan Group (Property (T)) |
|---|---|---|
| Definition | Group with an invariant mean on bounded functions | Group with a spectral gap in its unitary representations |
| Examples | Abelian groups, finite groups, solvable groups | Higher rank lattices (e.g., SL(n,Z), n >= 3), compact simple Lie groups |
| Fixed Point Property | May lack fixed points in affine actions on Banach spaces | Has fixed point property for affine isometric actions on Hilbert spaces |
| Growth Type | Can have subexponential or polynomial growth | Often exhibits exponential growth |
| Relation to Expander Graphs | Does not produce expanders via Cayley graphs | Source of expander graphs by Cayley constructions |
| Stability Under Extensions | Closed under extensions and directed unions | Stable under taking quotients and finite index subgroups |
| Connection to Random Walks | Random walks tend to be recurrent or null-recurrent | Random walks exhibit rapid mixing and spectral gap |
| Applications | Ergodic theory, harmonic analysis, probability | Rigidity theory, group representations, geometric group theory |
Introduction to Amenable and Kazhdan Groups
Amenable groups are characterized by the existence of invariant means, allowing the averaging of functions over the group, which plays a crucial role in harmonic analysis and ergodic theory. Kazhdan groups, defined by Kazhdan's property (T), exhibit strong rigidity properties and admit no almost invariant vectors except trivial ones, influencing representation theory and geometric group theory. Understanding the contrast between amenability, which implies a form of averaging and flexibility, and property (T), which enforces rigidity, is fundamental in analyzing group actions and spectral gaps.
Fundamental Definitions
Amenable groups are defined by the existence of an invariant mean on bounded functions, which implies the group admits a Folner sequence, reflecting a form of asymptotic invariance and "averaging" behavior. Kazhdan groups, characterized by Kazhdan's Property (T), exhibit strong rigidity through fixed-point properties for unitary representations, ensuring that almost invariant vectors exist only when actual invariant vectors are present. The fundamental distinction lies in amenable groups supporting flexible averaging processes, while Kazhdan groups enforce strict spectral gaps that inhibit such flexibility.
Historical Background and Motivation
Amenable groups, introduced by John von Neumann in 1929, arose from efforts to solve the Banach-Tarski paradox, highlighting groups permitting invariant means on bounded functions. Kazhdan groups emerged from David Kazhdan's 1967 work on property (T), motivated by rigidity phenomena in Lie groups and representation theory, emphasizing groups with strong fixed point properties and spectral gaps. The contrasting motivations reflect amenability's ties to paradoxical decompositions and measure theory, while Kazhdan property (T) connects to rigidity, spectral theory, and ergodic actions.
Key Properties of Amenable Groups
Amenable groups are characterized by the existence of an invariant mean on bounded functions, which implies they satisfy the fixed point property for affine actions on compact convex sets. These groups include abelian, finite, and solvable groups, and they lack Kazhdan's property (T), distinguishing them from Kazhdan groups known for strong rigidity and spectral gap properties. Key properties of amenable groups also involve closure under taking subgroups, quotients, extensions, and directed unions, making them stable under various group operations.
Core Features of Kazhdan (Property (T)) Groups
Kazhdan groups, characterized by Property (T), exhibit strong rigidity reflected in the existence of a spectral gap for unitary representations, ensuring any almost invariant vector is close to an invariant vector. Amenable groups lack this rigidity, often possessing invariant means and exhibiting flexible, averaging properties. Core features of Kazhdan groups include fixed point properties for affine isometric actions on Hilbert spaces and rapid decay of matrix coefficients, distinguishing them significantly from amenable groups.
Notable Examples and Non-Examples
Amenable groups include abelian groups, solvable groups, and compact groups, with the additive group of integers \(\mathbb{Z}\) and finite groups as key examples, while non-amenable groups feature free groups on two or more generators such as \(F_2\). Kazhdan groups, characterized by property (T), prominently include higher-rank lattices like \(\mathrm{SL}_n(\mathbb{Z})\) for \(n \geq 3\), whereas amenable groups such as abelian groups and free groups fail to have property (T) and thus are non-Kazhdan. Hyperbolic groups and most non-abelian free groups serve as fundamental non-examples of Kazhdan groups, illustrating the distinction from amenability where the existence of invariant means or fixed point properties diverge significantly.
Main Differences: Amenability vs Property (T)
Amenable groups are characterized by the existence of an invariant mean, reflecting flexibility and averaging properties, while Kazhdan groups possess Property (T), signifying strong fixed-point rigidity and resistance to almost invariant vectors in unitary representations. Amenability implies the ability to approximate the identity in a weak sense, facilitating harmonic analysis and ergodic theory applications. In contrast, Property (T) enforces spectral gaps that ensure stability and robustness in representation theory, highlighting a fundamental opposition between averaging behavior and rigidity.
Intersections and Mutual Exclusion
Amenable groups and Kazhdan groups (groups with Property (T)) intersect trivially in many cases, as amenability involves the existence of invariant means while Kazhdan groups enforce strong rigidity and spectral gap properties. Most Kazhdan groups are non-amenable due to their fixed point and representation rigidity, leading to mutual exclusion except in the case of finite groups which are both amenable and have Property (T). The structural tension between amenability's averaging properties and Kazhdan's spectral gap manifests in their intersection being limited to virtually trivial or finite groups.
Applications in Mathematics and Physics
Amenable groups, characterized by the existence of invariant means, play a crucial role in ergodic theory and harmonic analysis, facilitating the study of dynamical systems and probability measures on groups. Kazhdan groups, defined by property (T), exhibit strong rigidity and spectral gap properties, which are instrumental in geometric group theory, representation theory, and quantum computation for constructing expander graphs and studying unitary representations. The contrasting features of amenable and Kazhdan groups enable diverse applications in mathematical physics, particularly in statistical mechanics, operator algebras, and the analysis of symmetry and phase transitions.
Open Problems and Current Research Directions
Open problems in the study of amenable groups versus Kazhdan groups center on clarifying their boundaries and interactions, particularly the characterization of groups exhibiting intermediate properties between amenability and property (T). Current research explores the role of expanders and their embeddings in infinite groups to understand rigidity phenomena and fixed-point properties linked to Kazhdan's property (T). Investigations also focus on quantum group analogs and the impact of these group properties on operator algebras, geometric group theory, and ergodic theory.
Amenable group Infographic
