Treeability, Quasi-Invariance, and Ergodic Combinatorics

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

This award supports the principal investigator's research projects within the general framework of definable equivalence relations on Polish spaces, which is a modern focus of descriptive set theory. This theory provides a general framework for understanding the nature of classification of mathematical objects up to some notion of equivalence, and, due to its broad scope, it has natural interactions with many areas of mathematics. A central place in this theory is occupied by countable Borel equivalence relations (CBERs), which arise via actions of countable groups as well as via locally countable graphs. These connections between equivalence relations, group actions, and graphs create an extremely fruitful interplay between descriptive set theory, ergodic theory, measured group theory, and descriptive graph combinatorics. The overarching goal of the reseach is to deepen the understanding of these connections and further the theory of CBERs having major open questions as guiding targets. The projects can be grouped into four topics. The first concerns closure properties of the class of treeable CBERs, namely, whether abstract quasi-isometries and finite index extensions preserve treeability in the Borel or measurable contexts. Next is the investigation of the quasi-invariant setting aiming to develop a theory of quasi-invariant cost. The third topic is the existence of desirable subgraphs of graphings of a CBER, e.g., the existence of cost-approximating subgraphings or subtreeings, as well as that of maximal hyperfinite free factors. The last part is devoted to the pointwise ergodic theorems, namely, proving new instances and finding new proofs of known ones using a pointwise-combinatorial tiling argument in the style of the PI's recent proof the Birkhoff ergodic theorem. 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.