Quotients of Poisson boundaries, entropy, and spectral gap

Poisson boundary is a measurable $\Gamma$-space canonically associated with a group $\Gamma$ and a probability measure $\mu$ on it. The collection of all measurable $\Gamma$-equivariant quotients, known as $\mu$-boundaries, of the Poisson boundary forms a partially ordered set, equipped with a strictly monotonic non-negative function, known as Furstenberg or differential entropy. In this paper we demonstrate the richness and the complexity of this lattice of quotients for the case of free groups and surface groups and rather general measures. In particular, we show that there are continuum many unrelated $\mu$-boundaries at each, sufficiently low, entropy level, and there are continuum many distinct order-theoretic cubes of $\mu$-boundaries. These $\mu$-boundaries are constructed from dense linear representations $\rho:\Gamma\to G$ to semi-simple Lie groups, like $\PSL_2(\bbC)^d$ with absolutely continuous stationary measures on $\hat\bbC^d$.