Measure on the Ideal Boundary of a Nonpositively Curved Space: Random Walks and Rigidity

微分几何方法与动力学方法的结合在负弯曲流形和度量空间的研究中取得了很大的成功。例如，关于这些空间中不同尺度的测地线和几何的精细信息可以从离散模型上的随机行走行为和对其几何边界上某些遍历度量的研究中收集到。对于不可服从群上的随机漫步，一个基本问题是如何理解其泊松边界与其他自然几何边界的关系。从Furstenberg的工作开始，并继续进行许多其他人的工作，在理解重要类别的非正弯曲群上随机漫步的泊松边界测度何时可以在其测地线边界上得到支持方面取得了很大进展。然而，人们对以这种方式产生的措施知之甚少。提出的研究的第一部分试图表明，由理想边界上的几何结构产生的许多类遍历测度由泊松边界表示。我们也对非正弯曲但有一些共同特征的群感兴趣，如映射类群或圆的微分同构群。这代表了PI与R. Muchnik合作的自然结果。第二个提出的研究方向是检验质心方法作为理解流形的工具，允许非平凡映射到非正弯曲流形。这是对边界测度的微分几何应用研究。通过联系流形的体积和大尺度几何，我们希望利用这些方法来实现Mostow刚性的进一步扩展。数学和物理科学的许多显著发展揭示了一个给定系统中的随机过程如何经常反映该系统的某些结构特征。例如，一只蚂蚁在欧几里得平面上随机向北、南、东或西移动一个单位，最终将以概率确定性返回其起点。然而，当一个人允许额外的运动自由度时，这就不再成立了，比如在三维空间中上下移动。因此，这种“随机漫步”的递归特性可以检测周围空间的维度。我们建议研究基础空间几何与某些随机过程之间的这种联系的灵活性。我们特别感兴趣的是理解在重要的空间族上的广义随机漫步何时能产生一组规定的测量值。从另一个角度来看，这些辅助测量本身捕获了这些空间的其他内在方面，有时可以表明空间的“刚性”。这是指空间之间的弱等价意味着强等价的现象。我们希望发现刚性产生的新途径。

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.