By McMullen C.

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An amenable group is ‘small’. • If G is amenable then any quotient H = G/N is amenable. ) • If H ⊂ G and G is amenable, then so is H. ) • If 0 → A → B → C → 0 and A and C are amenable, then so is B. ) • G is amenable iff every finitely generated subgroup of G is amenable. (For a finitely generated subgroup H, we get an H-invariant mean on G with the aid of a transversal. ) 52 • Abelian groups are amenable (since finitely generated ones are). Solvable groups are amenable. 4. The free group G = a, b is not amenable.

Orbit counting in dimension one. Theorem. For any n = 0, N (R) = |{(x, t) ∈ Z2 : q(x, t) = n}| ∼ C log R for some C > 0. Proof. The group Γ = SO(q, R) acts discretely, by orientation-preserving isometries on H1 , so its action is cyclic and X = H1 /Γ is a compact manifold (a circle). The integral solutions to q(x, t) = 1 descend to a discrete, hence finite subset of X. Thus the number of solutions N (R) meeting the Euclidean ball B(R) of radius R grows like vol(B(R) ∩ H1 for the invariant volume form.

For completeness we also record: Theorem. Any A-invariant vector is also SLn R-invariant. Proof. As above, ξ is invariant under each Gi by the A-trick for SL2 R, and these generate. 12. Theorem: any unitary representation of SLn R without invariant vectors is mixing. Proof. Decompose the representation relative to P , obtaining a spectral measure µ on P ∼ = Rn−1 . Since there are no invariant vectors, we have µ(0) = 0. For any measurable set E ⊂ Rn−1 we have as before a(HE ) = HaE , where a ∈ A acts multiplicatively on P .