Harmonic projections, recurrent extensions and stability of the Liouville property

The Liouville property is a strong form of amenability, but contrary to amenability, it is not well-behaved under extensions. In this paper it is shown that, for every probability measure whose support generates the group as a semigroup, bounded harmonic functions are constant on the cosets of the FC-hypercentre, and more generally of any normal subgroup generated by elements whose centraliser is a recurrent set for the random walk; iterating this construction yields the hyperrecurrent subgroup. Hence the Liouville property passes to hyperrecurrent extensions; this applies for instance to many metabelian groups, to Houghton's groups and to permutational wreath products. For [FC-]hypercentral extensions, measures whose support only generates the group are also considered, using a projection from $\ell^\infty$ onto the space of harmonic functions. Finally, it is shown that adding to a symmetric finitely supported measure a Dirac mass at an element of finite order whose centraliser is a recurrent set does not change the bounded harmonic functions.