Lyons–Sullivan conjecture relating exponential growth of the deck group and positive harmonic functions on the cover

Determine whether, for any normal Riemannian covering p: M → N of a closed manifold with deck transformation group Γ, the group Γ has exponential growth if and only if the covering M admits a non-constant positive harmonic function, without assuming that Γ is linear.

Background

The authors recall a conjecture of Lyons and Sullivan asserting a precise correspondence between exponential growth of the deck transformation group Γ and the existence of non-constant positive harmonic functions on the covering space M. This bridges group growth and potential theory on manifolds.

They note that this conjecture has been proved under the additional hypothesis that Γ is linear (a closed subgroup of GL_n(R)), but the general case is not settled within the text, indicating an outstanding problem beyond the linear setting.