Nonamenable

Nonamenable describes a class of mathematical objects that fail to satisfy the notion of amenability. In group theory, a discrete group G is amenable if there exists a left-invariant finitely additive probability measure defined on all subsets of G, or equivalently if it satisfies the Følner condition: for every finite subset K ⊂ G and every ε > 0 there exists a finite F ⊂ G with |KF \ F| < ε|F|. A group is nonamenable when no such invariant mean exists; equivalently, a paradoxical decomposition exists for the group (Tarski's theorem).

Examples and implications are central to the concept. The free group F2 on two generators is nonamenable. Any group containing F2 as a subgroup is nonamenable. Lattices in noncompact semisimple Lie groups of rank at least 2, and many hyperbolic and linear groups, are nonamenable. By contrast, amenable groups include finite groups, abelian groups, and more generally solvable groups. Nonamenability is a robust property in many contexts, governing how a group or space can be decomposed or measured.

In graph theory and analysis, nonamenability is reflected in a positive isoperimetric constant and often in a spectral gap. The Cayley graph of a nonamenable group is itself nonamenable and is frequently an expander. Nonamenability implies phenomena such as rapid escape of random walks and exponential decay of return probabilities. It also intersects with operator-algebraic properties, influencing the structure of associated von Neumann algebras and related analytical frameworks.