Harmonic projection and hypercentral extensions

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 some measures, the Liouville property is preserved by [FC-]hypercentral extensions. To this end a projection from $\ell^\infty$ onto the space of harmonic functions is introduced.

💡 Research Summary

The paper investigates the stability of the Liouville property—a strong form of amenability stating that all bounded harmonic functions are constant—under group extensions. While amenability is known to be preserved under extensions, the Liouville property is notoriously fragile; classic counter‑examples such as the lamplighter group show that even extensions of Abelian groups can destroy it. The author focuses on a particular class of extensions, namely central and FC‑central (hyper‑central) extensions, and proves that for a wide family of probability measures the Liouville property is indeed preserved.

The first technical contribution is the construction of a harmonic projection. Given a countable group Γ and a probability measure P on Γ, the author defines the averaging operator f ↦ f∗P on ℓ∞(Γ). A function is P‑harmonic if it is a fixed point of this operator. By considering a sequence of measures μₙ that are approximately P‑invariant (i.e., μₙ∗(δₑ−P) → 0 in ℓ¹) and taking a weak‑* limit m of this sequence, the map π(f)=f∗m becomes a norm‑one linear projection from ℓ∞(Γ) onto the space BHP(Γ) of bounded P‑harmonic functions. Lemma 2.1 and Proposition 2.2 establish that π is indeed a projection, that its kernel coincides with the image of I−P, and that when the support of P generates an infinite subgroup, the space c₀(Γ) lies inside ker π. This harmonic projection provides a functional‑analytic tool to “average out” non‑harmonic components while preserving harmonic ones.

The core of the paper deals with extensions. Let N◁Γ be a normal subgroup contained in the FC‑hypercenter of Γ, and let ϕ:Γ→Γ/N be the quotient map. Suppose ν is a finitely supported symmetric probability measure on Γ/N whose support generates the quotient, and let μ be any probability measure on Γ that generates N and whose push‑forward ϕ∗μ is a lazy version of ν (i.e., μ = tδₑ + (1−t)ν for some t∈