Consider a lattice in the isometry group of a pinched negatively curved contractible manifold.
There are two natural averaging procedures on this group: averaging with respect to balls in the group and taking a finitely supported random walk.
These correspond to two natural measures on the boundary the Patterson-Sullivan measure (which for symetric manifolds is in the Lebesgue measure class) and the harmonic measure.
These two measures satisfy conformality properties with respect to two metrics on the lattice: the metric induced by the orbit map $d$ and the so called Green metric $d_G$ associated to the random walk, which is quasi-isometric to the word metric.
We will show that the harmonic and Patterson-Sullivan measures are singular unless the two metrics are roughly similar: $|d-c_1 d_G|<c_2$ for uniform constants $c_1, c_2$. Thus, they are always singular for *nonuniform* lattices (when the quotient manifold is noncompact).
Everything can be generalized to geometrically finite actions on proper Gromov hyperbolic spaces.
If time permits, we will also show how the two measures on the boundary correspond to two measures on the unit tangent bundle (the measure of maximal entropy and the harmonic invariant measure) and that closed geodesics on the quotient manifold satisfy two different equidistribution properties with respect to the two measures.