Let M be a closed Riemannian manifold and denote by X its universal covering space equipped with the pulled back Riemannian metric.

There is an intimate relation between the Laplace operator on X and the fundamental group of M. One example of this is the result of Brooks from 1981: the fundamental group of M is amenable if and only if the Laplace operator on X has a spectral gap around 0.

There are further relations between harmonic functions on X and the fundamental group G of M. The first one we want to discuss is the Liouville property:

- Lyons and Sullivan showed 1984 that if G is non-amenable, then there exist non-constant bounded harmonic functions on X.
- Kaimanovich on the other hand proved in 1986 that if G has sub-exponential growth or is polycyclic, then any bounded harmonic function on X is constant.
- In the remaining case, i.e., that G is an amenable group of exponential growth (and is not polycyclic), it is only known by a 2004 result of Erschler that there exists a closed manifold M with solvable fundamental group G and such that the universal cover of it admits a non-constant bounded harmonic function.

The strong Liouville property asks about the existence of non-constant positive harmonic functions on the universal cover:

- Lyons and Sullivan showed (in the same paper from 1984) that if G has polynomial growth, then any positive harmonic function on M is constant.
- In an arXiv preprint from yesterday Polymerakis showed that if G has exponential growth, then there are non-constant positive harmonic functions on the universal cover X.
- The remaining unsolved case is now the one of G having intermediate growth. But note that though there are some examples of groups with intermediate growth, none of these examples is finitely presentable. Hence it is currently unknown whether the remaining case here actually occurs (since fundamental groups of compact manifolds are finitely presentable).

Note that this week there was actually a school on harmonic maps: link.