Ergodicity of the bicentralizer flow and Kadison's problem

We show that the relative bicentralizer flow of a type $\mathrm{III}_1$ irreducible subfactor with expectation is always ergodic. As a consequence, every irreducible subfactor with expectation in a factor with separable predual contains a maximal abelian subalgebra. This completes the solution to Kadison's problem on maximal abelian subalgebras from 1967.