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.
