  Conformal dimension is a fundamental invariant of metric spaces, particularly
suited to the study of self-similar spaces, such as spaces with an expanding
self-covering (e.g. Julia sets of complex rational functions). The dynamics of
these systems are encoded by the associated iterated monodromy groups, which
are examples of contracting self-similar groups. Their amenability is a
well-known open question. We show that if $G$ is an iterated monodromy group,
and if the (Alfhors-regular) conformal dimension of the underlying space is
strictly less than 2, then every symmetric random walk with finite second
moment on $G$ has the Liouville property. As a corollary, every such group is
amenable. This criterion applies to all examples of contracting groups
previously known to be amenable, and to many new ones. In particular, it
implies that for every post-critically finite complex rational function $f$
whose Julia set is not the whole sphere, the iterated monodromy group of $f$ is
amenable.
