This talk is a follow-up of the talk of V. Nekrashevych, and is based on joint work with him and with T. Zheng.
Conformal dimension is a fundamental invariant of metric spaces, particularly suited to study self-similar metric spaces, such as spaces with an expanding self-covering (e.g. Julia set of rational functions). Such maps are encoded by their associated iterated monodromy group, which is an example of contracting self-similar group. The amenability of such groups is a well-known open problem. In this talk I will explain that if G is an iterated monodromy group, and if the associated space has conformal dimension strictly less than 2, then every symmetric random walk on G satisfying a suitable moment condition has trivial Poisson boundary (also called the Liouville property); as a consequence G is amenable. This criterion recovers all previous results establishing amenability or the Liouville property for some more restricted classes of iterated monodromy groups, and applies to many new examples whose amenability was open. In particular it shows amenability of the iterated monodromy group of every post-critically finite rational function whose Julia set is not the whole sphere.