Flexibility and Complexity in Stationary Boundaries of Negatively Curved Groups

The goal of this thesis is to understand the structure of the lattice of stationary boundaries, or equivalently the quotients of the Poisson boundary, for reasonable random walks on groups Γ which are flexible in a suitable sense. In particular, we will prove two main results of a similar f lavor. 1. Using the presence of actions with spectral gap we show that the (order-theoretic) lattice of stationary boundaries associated to random walks on torsion-free, (group-theoretic) lattices in SO(2,1) with an exponential moment is highly complex. Specifically, If Γ is Fn or π1(Σg) with n or g sufficiently large, and µ has an exponential moment then A The boundary lattice admits continuous deformations. B The boundary lattice contains cubes of arbitrarily high dimension. 2. Using the small cancellation theories associated to acylindrically hyperbolic groups, we show that the boundary lattice of any acylindrically hyperbolic group has an embedded copy of Cantor space, and thus has the cardinality of the continuum.