Topological versus metric dependence of Liouville-type properties on covering spaces

Determine whether the Liouville and strong Liouville properties for a normal Riemannian covering p: M → N of a closed manifold depend only on the topology of the base manifold N (equivalently, on the deck transformation group Γ), or whether the choice of Riemannian metric on the covering space M influences these properties.

Background

The paper revisits classical questions about Liouville-type properties in the setting of normal Riemannian coverings p: M → N of closed manifolds, where Γ is the deck transformation group. It highlights foundational work by Lyons and Sullivan connecting amenability, growth, and harmonic functions on covering spaces.

Within this context, the authors explicitly note that it is not known whether the Liouville and strong Liouville properties are determined purely by topological data (the covering and the deck group) or whether they may depend on the particular Riemannian metric placed on the covering space. Clarifying this would sharpen the relationship between geometric analysis and the large-scale topology of coverings.