The combination of differential geometric and dynamical methods has been very successful in the study of negatively curved manifolds and metric spaces. For instance, delicate information about geodesics and geometry at different scales in these spaces can be gleaned from the behavior of random walks on discrete models and the study of certain ergodic measures on their geometric boundaries. For random walks on nonamenable groups, a basic question has been to understand the relationship of its Poisson boundary to other natural geometric boundaries. Starting with the work of Furstenberg, and continuing with the work of many others, much progress has been made in understanding when Poisson boundary measures for a random walks on important classes of nonpositively curved groups can be supported on their geodesic boundary. However, much less is known about what measures can arise this way. The first part of the proposed research seeks to show that many of the classes of ergodic measures arising from geometric constructions on the ideal boundary are represented by Poisson boundaries. We are also interested in groups which are not nonpositively curved, yet share some common features such as the mapping class groups or the diffeomorphism group of a circle. This represents a natural outgrowth of the PI's work with R. Muchnik. The second proposed direction of study examines the barycenter method as a tool for understanding manifolds admitting nontrivial maps to nonpositively curved manifolds. This is a differential geometric application of the study of boundary measures. By relating the volume and large scale geometry of a manifold, we wish to use these methods to realize further extensions of Mostow rigidity.
A number of remarkable developments in both mathematics and the physical sciences have revealed how random processes in a given system often reflect certain structural features of that system. For example, an ant randomly stepping one unit north, south, east or west in the Euclidean plane will eventually return to its starting point with probabilistic certainty. However, this no longer holds when one allows an additional degree of freedom of movement, say up and down in the third dimension. Hence, the recurrence property of this "random walk" detects the dimension of the ambient space. We propose to study the flexibility of such connections between the geometry of the underlying space and certain random processes. We especially are interested in understanding when generalized random walks on important families of spaces can produce a prescribed set of measurements. From another point of view, these auxiliary measurements themselves capture other intrinsic aspects of these spaces, and can sometimes indicate "rigidity" of the space. This refers to the phenomenon whereby a weak equivalence between spaces implies a strong equivalence. We hope to discover new ways in which rigidity arises.
