CAREER: Ergodic Theory of Nonamenable Group Actions

这个项目涉及不可服从群作用的遍历理论中的三个主题：熵、刚性/弹性现象和逐点遍历定理。自从科尔莫戈罗夫(1958)的最初突破以来，动力熵在服从行动的遍历理论中发挥了重要作用。这位校长最近发现了柯尔莫戈洛夫熵对SOFIC群作用的推广。这类群包含许多熟悉的和有趣的群，包括所有的线性群，从而将熵理论第一次扩展到某些不可服从的群。这一新的发展打开了大量问题的大门，这些问题可能会对遍历理论和其他领域(例如，统计物理、概率组合数学、算子代数)产生重大影响。在过去的十年里，人们对不可服从的群体诉讼的刚性理论产生了极大的兴趣(例如，参见A.Furman、D.Gaboriau和S.Popa的著作)。新的结果表明，在特殊条件下，测度共轭等价于作用的先验弱等价形式，如轨道等价和von Neumann等价。期望主要研究者的新的熵理论能够更准确地确定它何时是轨道等价或von Neumann等价不变量，从而对刚性理论做出重要贡献。首席研究人员最近证明了第一个轨道等价灵活性在不可服从的设置中的结果。预计为证明而开发的技术将更广泛地适用，并可能产生适用于某些团体和行动的一般轨道等价灵活性理论。点态遍历定理有很长的历史，始于Birkhoff。在与A·内沃的合作中，首席研究员正在开发一般方法来证明离散的、不可服从的群的作用的逐点遍历定理。到目前为止的成功包括自由群作用的新的逐点遍历平均序列和关于球平均和球面平均的一般字双曲群作用的第一个逐点遍历定理。新技术消除了一个根本障碍，因为它们将问题简化为服从的情况，然后可以通过经典的Wiener-Calderon-FolNer理论来解决。该项目的教育部分包括在德克萨斯农工大学举办的年度研讨会，目的是向研究生介绍最新的研究进展，并将在该项目重要领域工作的不同研究人员群体聚集在一起。这位首席研究员计划通过他所在系的荣誉项目指导本科生，并在数学俱乐部会议和本科生和高中生暑期项目期间发表受欢迎的演讲。首席调查员还计划邀请一位专家讲解员作为杰出讲师，就当前感兴趣的主题进行一系列鼓舞人心的演讲。该项目直接为研究生提供培训。这应该很容易吸引一个人，因为首席研究员最近的结果是可以访问的，并导致了一个开放的问题领域，其解决方案可能对遍历理论和其他领域产生重大影响。

This project is concerned with three subjects within the ergodic theory of nonamenable group actions: entropy, rigidity/flexibility phenomena, and pointwise ergodic theorems. Since Kolmogorov's initial breakthrough (1958), dynamical entropy has played a major role in the ergodic theory of amenable actions. The principal recently discovered a generalization of Kolmogorov's entropy to actions by sofic groups. This class of groups contains many familiar and interesting groups, including all linear groups, thereby providing the first extension of entropy theory to certain nonamenable groups. This new development opens the door to an abundance of problems that could have a great impact in ergodic theory and other fields (e.g., statistical physics, probabilistic combinatorics, operator algebras). In the last decade, there has been an explosion of interest in rigidity theory for nonamenable group actions (see, for example, works of A.Furman, D. Gaboriau, and S. Popa). New results indicate that, under special conditions, measure-conjugacy is equivalent to a priori weaker forms of equivalence of actions, such as orbit equivalence and von Neumann equivalence. It is expected that the principal investigator's new entropy theory can make an important contribution to rigidity theory by determining more precisely when it is an orbit equivalence or von Neumann equivalence invariant. The principal investigator has recently proven the first orbit equivalence flexibility results in the nonamenable setting. It is expected that the techniques developed for the proof will be more broadly applicable and could result in a general orbit equivalence flexibility theory for certain groups and actions. Pointwise ergodic theorems have a long history, beginning with Birkhoff. In collaboration with A. Nevo, the principal investigator is developing general methods for proving pointwise ergodic theorems for actions of discrete, nonamenable groups. Successes so far include new pointwise ergodic averaging sequences for actions of free groups and the first pointwise ergodic theorems for actions of general word-hyperbolic groups with respect to ball and spherical averages. The new techniques remove a fundamental obstacle in that they reduce the problem to the amenable case, which can then be solved by the classical Wiener-Calderon-Folner theory. The educational component of the project includes yearly workshops at Texas A&M for the purposes of introducing graduate students to the latest research developments and bringing together the diverse groups of researchers working in areas important to the project. The principal investigator plans to mentor undergraduates through his department's honors program and to give popular talks at math club meetings and during his department's summer programs for undergraduates and high school students. The principal investigator also plans to invite an expert expositor as a distinguished lecturer to give a series of inspiring talks on topics of current interest. This project directly provides for the training of a graduate student. It should be easy to attract one because the principal investigator's recent results are accessible and lead to an open field of problems whose solutions could have significant impact in ergodic theory and other areas.