Amenable Group - Examples

Examples

• Finite groups are amenable. Use the counting measure with the discrete definition. More generally, compact groups are amenable. The Haar measure is an invariant mean (unique taking total measure 1).
• The group of integers is amenable (a sequence of intervals of length tending to infinity is a Følner sequence).The existence of a shift-invariant, finitely additive probability measure on the group Z also follows easily from the Hahn–Banach theorem this way. Let S be the shift operator on the sequence space ℓ∞(Z), which is defined by (Sx)_i = x_i+1 for all x ∈ ℓ∞(Z), and let u ∈ ℓ∞(Z) be the constant sequence u_i = 1 for all i ∈ Z. Any element y ∈ Y:=Ran(S − I) has a distance larger than or equal to 1 from u (otherwise y_i = x_i+1 - x_i would be positive and bounded away from zero, whence x_i could not be bounded). This implies that there is a well-defined norm-one linear form on the subspace Ru + Y taking tu + y to t. By the Hahn–Banach theorem the latter admits a norm-one linear extension on ℓ∞(Z), which is by construction a shift-invariant finitely additive probability measure on Z.
• By the direct limit property above, a group is amenable if all its finitely generated subgroups are. That is, locally amenable groups are amenable.
  • By the fundamental theorem of finitely generated abelian groups, it follows that abelian groups are amenable.
• It follows from the extension property above that a group is amenable if it has a finite index amenable subgroup. That is, virtually amenable groups are amenable.
• Furthermore, it follows that all solvable groups are amenable.

All examples above are elementary amenable. The next class examples below can be used to exhibit non-elementary amenable examples thanks to the existence of groups of intermediate growth.

• Finitely generated groups of subexponential growth are amenable. A suitable subsequence of balls will provide a Følner sequence.