Ergodic Theory of Non-Amenable Group Actions

University Of Texas At Austin, Austin TX

Investigators

Abstract

Classical dynamics studies how systems change in time. Ergodic theory focuses on the statistical behavior of dynamical systems. Applications of ergodic theory are widespread: from traffic modeling to aerospace engineering and population dynamics. It is natural and of practical importance to generalize the role of time in a dynamical system to more complicated groups of symmetries. This generalized notion of dynamics leads to applications in statistical physics, number theory and geometry. However, new tools are needed especially in the particular case when the group of symmetries is non-amenable which means that boundary phenomena are too significant to be safely ignored. Non-amenable groups naturally arise in many parts of mathematics such as geometry and number theory. This project is concerned with developing the tools needed to analyze the statistical behavior of non-amenable group actions by generalizing ergodic theory to this context. The research goals of this project are: 1) Continue developing sofic entropy theory, which is a vast generalization of Kolmogorov-Sinai entropy to actions of sofic groups, a class of groups that contains all amenable groups and residually finite groups. One ambitious goal is to classify the mixing Markov chains over free groups using asymptotic geometric invariants of associated model spaces. 2) Classify sofic approximations of low-dimensional groups (using geometric techniques), with an eye towards using the results to construct the first known non-sofic groups from amalgams of low-dimensional groups. 3) Explore the limits of pointwise ergodic theorems for geometrically defined groups (e.g. Lie groups, CAT(0) cubulated groups, relatively hyperbolic groups) via the measured-equivalence techniques developed by the principal investigator and Nevo and the ideas behind the recent L^1-counterexample of Tao. Also develop new multiplicative ergodic theorems for cocycles taking values in tracial von Neumann algebras. 4) Import tools from geometric group theory into the study of measured equivalence relations (MERs) through the use of metric bundles in place of actions, graphings in place of Cayley graphs and so on. More specific goals include proving analogs of the Tits alternative for MERs and classifying the normal sub-equivalence relations of low-dimensional MERs. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.