Treeings, quasi-invariance, and ergodic combinatorics

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GENERAL SCOPE The proposal lies within the broad scope 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 proposal is to deepen the understanding of these connections and further the theory of CBERs having major open questions as guiding targets. The projects of the proposal can be grouped into three topics. TREEINGS AND SUBTREEINGS In the study of CBERs, a principal role is played by those that are treeable, i.e. there is an acyclic Borel graph (a treeing) whose connected components are exactly the equivalence classes. These not only form a particularly interesting subclass of CBERs, but also serve as a critical tool for studying all CBERs in general. One project concerns a closure property of the class of treeable CBERs, while others require the development of new constructions of subtreeings, subgraphings, and subforests. The latter concerns the endeavor of structurally witnessing the nonamenability of a group within its free action on a probability space. To attack the main problems here, I propose the development of percolation theory on measured graphs. This is interesting on its own and opens new prospects and questions. QUASI-INVARIANCE A CBER $E$ is better understood when it is probability measure preserving (pmp), Borel automorphism of $E$ preserve a probability measure. This is because a number of techniques from various areas is available for pmp CBERs, including the theory of cost, $\ell^2$-(co)homology, operator algebras, and percolation theory. However, when the CBER is merely quasi-pmp, that is, every Borel automorphism of $E$ only preserves the non-nullness of sets, none of these techniques are available. In my recent work, I develop new tools for dealing with quasi-invariance, using which, I propose to generalize several results known in the pmp setting to the quasi-pmp setting. Furthermore, I have an idea for a definition of cost in this setting, which should be investigated. If promising, I propose developing a theory of quasi-pmp cost. ERGODIC THEOREMS The last topic 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 my recent proof the Birkhoff ergodic theorem.